9459:
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9454:{\displaystyle {\begin{alignedat}{6}a\cdot (\mathbf {v} \otimes \mathbf {w} )~&=~(a\cdot \mathbf {v} )\otimes \mathbf {w} ~=~\mathbf {v} \otimes (a\cdot \mathbf {w} ),&&~~{\text{ where }}a{\text{ is a scalar}}\\(\mathbf {v} _{1}+\mathbf {v} _{2})\otimes \mathbf {w} ~&=~\mathbf {v} _{1}\otimes \mathbf {w} +\mathbf {v} _{2}\otimes \mathbf {w} &&\\\mathbf {v} \otimes (\mathbf {w} _{1}+\mathbf {w} _{2})~&=~\mathbf {v} \otimes \mathbf {w} _{1}+\mathbf {v} \otimes \mathbf {w} _{2}.&&\\\end{alignedat}}}
6259:
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as the unit "balls" enclose each other, a sequence converges to zero in one norm if and only if it so does in the other norm. In the infinite-dimensional case, however, there will generally be inequivalent topologies, which makes the study of topological vector spaces richer than that of vector
184:, which, roughly speaking, specifies the number of independent directions in the space. This means that, for two vector spaces over a given field and with the same dimension, the properties that depend only on the vector-space structure are exactly the same (technically the vector spaces are
9103:
14129:
can be interpreted in terms of
Hilbert spaces. For example, a great many fields in physics and engineering lead to such equations, and frequently solutions with particular physical properties are used as basis functions, often orthogonal. As an example from physics, the time-dependent
10556:
2832:. In the special case of two arrows on the same line, their sum is the arrow on this line whose length is the sum or the difference of the lengths, depending on whether the arrows have the same direction. Another operation that can be done with arrows is scaling: given any positive
3200:
3984:
12458:
10363:
7690:
2655:
was the first to give the modern definition of vector spaces and linear maps in 1888, although he called them "linear systems". Peano's axiomatization allowed for vector spaces with infinite dimension, but Peano did not develop that theory further. In 1897,
3989:
5508:
4716:
12344:
13515:
11009:
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15115:
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studies the isomorphism classes of all vector bundles over some topological space. In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real
13141:
8995:
1720:
14050:, and is much applied in engineering. More generally, and more conceptually, the theorem yields a simple description of what "basic functions", or, in abstract Hilbert spaces, what basic vectors suffice to generate a Hilbert space
12984:
12624:
5364:. It is an isomorphism, by its very definition. Therefore, two vector spaces over a given field are isomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space over a given field is
11352:
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13873:
10213:
9805:
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14379:: using that the sum of two polynomials is a polynomial, they form a vector space; they form an algebra since the product of two polynomials is again a polynomial. Rings of polynomials (in several variables) and their
12193:
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12863:
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are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property. Therefore, the set of such functions are vector spaces, whose study belongs to
1220:
11857:
3001:
11863:—focuses on infinite-dimensional vector spaces, since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence. The image at the right shows the equivalence of the
9819:
From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space over a given field is characterized, up to isomorphism, by its dimension. However, vector spaces
8922:
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6159:
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under addition, and requiring that scalar multiplication commute with the tensor product ⊗, much the same way as with the tensor product of two vector spaces introduced in the above section on
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if and only if they have the same coefficients. Also equivalently, they are linearly independent if a linear combination results in the zero vector if and only if all its coefficients are zero.
13332:
8300:
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3526:. The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic. The example of complex numbers is essentially the same as (that is, it is
1404:
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An equivalent definition of a vector space can be given, which is much more concise but less elementary: the first four axioms (related to vector addition) say that a vector space is an
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1974:. A linear subspace is a vector space for the induced addition and scalar multiplication; this means that the closure property implies that the axioms of a vector space are satisfied.
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4100:{\displaystyle {\begin{alignedat}{9}&&a\,&&+\,3b\,&\,+&\,&c&\,=0\\4&&a\,&&+\,2b\,&\,+&\,2&c&\,=0\\\end{alignedat}}}
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is a scalar that tells whether the associated map is an isomorphism or not: to be so it is sufficient and necessary that the determinant is nonzero. The linear transformation of
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are a fundamental tool for the study of vector spaces, especially when the dimension is finite. In the infinite-dimensional case, the existence of infinite bases, often called
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From a conceptual point of view, all notions related to topological vector spaces should match the topology. For example, instead of considering all linear maps (also called
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Der
Barycentrische Calcul : ein neues Hülfsmittel zur analytischen Behandlung der Geometrie (Barycentric calculus: a new utility for an analytic treatment of geometry)
14319:
10442:
8947:
8853:
4404:
19472:
18486:"Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales (On operations in abstract sets and their application to integral equations)"
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of
Lebesgue measure, being unbounded, cannot be integrated with the classical Riemann integral. So spaces of Riemann integrable functions would not be complete in the
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Banach and
Hilbert spaces are complete topological vector spaces whose topologies are given, respectively, by a norm and an inner product. Their study—a key piece of
7179:
5731:{\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n})\mapsto \left(\sum _{j=1}^{n}a_{1j}x_{j},\sum _{j=1}^{n}a_{2j}x_{j},\ldots ,\sum _{j=1}^{n}a_{mj}x_{j}\right),}
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4845:{\displaystyle {\begin{aligned}f(\mathbf {v} +\mathbf {w} )&=f(\mathbf {v} )+f(\mathbf {w} ),\\f(a\cdot \mathbf {v} )&=a\cdot f(\mathbf {v} )\end{aligned}}}
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complete, which may be seen as a justification for
Lebesgue's integration theory.) Concretely this means that for any sequence of Lebesgue-integrable functions
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of the tensor product, an instance of the method—much used in advanced abstract algebra—to indirectly define objects by specifying maps from or to this object.
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In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones.
16272:
A basis of a
Hilbert space is not the same thing as a basis of a linear algebra. For distinction, a linear algebra basis for a Hilbert space is called a
11628:. Roughly, a vector space is complete provided that it contains all necessary limits. For example, the vector space of polynomials on the unit interval
4206:
They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too.
2647:
are present. Grassmann's 1844 work exceeds the framework of vector spaces as well since his considering multiplication led him to what are today called
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2509:
between vectors and their coordinate vectors maps vector addition to vector addition and scalar multiplication to scalar multiplication. It is thus a
15908:
It is also common, especially in higher mathematics, to not use any typographical method for distinguishing vectors from other mathematical objects.
19104:
9098:{\displaystyle \mathbf {v} _{1}\otimes \mathbf {w} _{1}+\mathbf {v} _{2}\otimes \mathbf {w} _{2}+\cdots +\mathbf {v} _{n}\otimes \mathbf {w} _{n},}
7283:
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studied the barycentric calculus initiated by Möbius. He envisaged sets of abstract objects endowed with operations. In his work, the concepts of
2136:
12115:
8162:
18743:
17847:
598:
15494:, yield modules. The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not have
10551:{\displaystyle \mathbf {x} \cdot \mathbf {y} =\cos \left(\angle (\mathbf {x} ,\mathbf {y} )\right)\cdot |\mathbf {x} |\cdot |\mathbf {y} |.}
13730:
The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red).
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if its elements are linearly independent and span the vector space. Every vector space has at least one basis, or many in general (see
8880:
6863:
6923:
1777:
18108:, The Wadsworth & Brooks/Cole Mathematics Series, Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software,
8219:
and the second and third isomorphism theorem can be formulated and proven in a way very similar to the corresponding statements for
19615:
19277:
10875:. Note that in other conventions time is often written as the first, or "zeroeth" component so that the Lorentz product is written
6161:
By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in
14110:. Not only does the theorem exhibit suitable basis functions as sufficient for approximation purposes, but also together with the
10561:
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19948:
19513:
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14079:
of their span (that is, finite linear combinations and limits of those) is the whole space. Such a set of functions is called a
10153:
respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm
10116:
5461:
are a useful notion to encode linear maps. They are written as a rectangular array of scalars as in the image at the right. Any
3246:
20006:
16499:
14992:
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3228:
itself with its addition viewed as vector addition and its multiplication viewed as scalar multiplication. More generally, all
3195:{\displaystyle {\begin{aligned}(x_{1},y_{1})+(x_{2},y_{2})&=(x_{1}+x_{2},y_{1}+y_{2}),\\a(x,y)&=(ax,ay).\end{aligned}}}
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norm, and the orthogonal decomposition would not apply to them. This shows one of the advantages of
Lebesgue integration.",
8332:
19147:
6989:
18673:
13979:
with desirable properties that approximate a given limit function is equally crucial. Early analysis, in the guise of the
16388:
16037:
13952:
By definition, in a
Hilbert space, any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions
3845:
2513:, which allows translating reasonings and computations on vectors into reasonings and computations on their coordinates.
2102:
16579:
12453:{\displaystyle \|\mathbf {x} \|_{p}:=\left(\sum _{i}|x_{i}|^{p}\right)^{\frac {1}{p}}\qquad {\text{ for }}p<\infty .}
18387:
13272:
10358:{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =\mathbf {x} \cdot \mathbf {y} =x_{1}y_{1}+\cdots +x_{n}y_{n}.}
8258:
7395:
3798:
also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions
1147:
under addition, and the four remaining axioms (related to the scalar multiplication) say that this operation defines a
591:
9575:
3710:
3559:
then we see that the rules for addition and scalar multiplication correspond exactly to those in the earlier example.
11700:
6435:
are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set
2818:
spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the
1361:
18527:
Betrachtungen über einige
Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry)
15929:
a scalar that produces a vector, while the scalar product is a multiplication of two vectors that produces a scalar.
9934:
20057:
19996:
7685:{\displaystyle a_{0}f+a_{1}{\frac {df}{dx}}+a_{2}{\frac {d^{2}f}{dx^{2}}}+\cdots +a_{n}{\frac {d^{n}f}{dx^{n}}}=0,}
5127:
For example, the arrows in the plane and the ordered pairs of numbers vector spaces in the introduction above (see
173:, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying
99:
15374:
Properties of certain vector bundles provide information about the underlying topological space. For example, the
14324:
General vector spaces do not possess a multiplication between vectors. A vector space equipped with an additional
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19894:
17947:
15187:
11664:
10771:
9539:
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A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces where any
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3606:
1469:
1410:
18942:
18730:
Calcolo
Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva
18414:
16727:
15777:
13337:
12991:
11163:
7875:
16468:
15328:. Despite their locally trivial character, vector bundles may (depending on the shape of the underlying space
1725:
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of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the
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584:
16531:
4546:
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with the same topology is complete. A norm gives rise to a topology by defining that a sequence of vectors
8062:
5967:
4684:
3968:
221:
17:
14118:. Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensional
12055:
is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.
8484:
19841:
19691:
19581:
19416:
18059:
16759:
15332:) be "twisted" in the large (that is, the bundle need not be (globally isomorphic to) the trivial bundle
14400:, which are neither commutative nor associative, but the failure to be so is limited by the constraints (
13221:
12339:{\displaystyle \|\mathbf {x} \|_{\infty }:=\sup _{i}|x_{i}|\qquad {\text{ for }}p=\infty ,{\text{ and }}}
11361:
3978:
3972:
3530:
to) the vector space of ordered pairs of real numbers mentioned above: if we think of the complex number
2549:
introduced, in 1804, certain operations on points, lines, and planes, which are predecessors of vectors.
2378:
1836:
453:
174:
13878:
13549:
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adopted Peano's axioms and made initial inroads into the theory of infinite-dimensional vector spaces.
2328:
1253:
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18127:
15595:
14752:
13510:{\displaystyle \lim _{k,\ n\to \infty }\int _{\Omega }\left|f_{k}(x)-f_{n}(x)\right|^{p}\,{d\mu (x)}=0}
12004:
11906:
11004:{\displaystyle \langle \mathbf {x} |\mathbf {y} \rangle =-x_{0}y_{0}+x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}.}
10368:
9824:
do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions
9633:
8478:. Addition and scalar multiplication is performed componentwise. A variant of this construction is the
8249:
of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space.
8148:
6189:
5955:
2613:
2219:
208:
and related areas. Infinite-dimensional vector spaces occur in many areas of mathematics. For example,
18683:
11742:
11509:
10763:{\displaystyle \langle \mathbf {x} |\mathbf {y} \rangle =x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}-x_{4}y_{4}.}
10611:
9893:
8720:
8425:
7077:
7048:
2105:). This is a fundamental property of vector spaces, which is detailed in the remainder of the section.
19986:
19635:
13745:
8611:
8152:
8116:
7500:, which is precisely the set of solutions to the system of homogeneous linear equations belonging to
6168:
5279:
2708:
2506:
1979:
19492:
16230:
11193:
10083:
19978:
19861:
18054:
15590:
are vector spaces whose origins are not specified. More precisely, an affine space is a set with a
14299:
14111:
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11238:
also has to carry a topology in this context; a common choice is the reals or the complex numbers.
11019:
10422:
8989:
as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called
8927:
8833:
8156:
4384:
2510:
2080:
248:
181:
19576:
15110:{\displaystyle \mathbf {v} _{1}\otimes \mathbf {v} _{2}=-\mathbf {v} _{2}\otimes \mathbf {v} _{1}}
12032:
11771:
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11059:
10400:
8858:
8811:
7448:
7238:
7216:
6751:
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6328:
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2998:. The sum of two such pairs and the multiplication of a pair with a number is defined as follows:
20024:
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15879:
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14197:
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8647:
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2495:
1119:. These two cases are the most common ones, but vector spaces with scalars in an arbitrary field
1018:
544:
158:
38:
14907:{\displaystyle \mathbf {v} _{1}\otimes \mathbf {v} _{2}\otimes \cdots \otimes \mathbf {v} _{n},}
14131:
14042:
can be approximated as closely as desired by a polynomial. A similar approximation technique by
13695:{\displaystyle \lim _{k\to \infty }\int _{\Omega }\left|f(x)-f_{k}(x)\right|^{p}\,{d\mu (x)}=0.}
12713:
9848:
9510:
5319:
are completely determined by specifying the images of the basis vectors, because any element of
4713:
that reflect the vector space structure, that is, they preserve sums and scalar multiplication:
3916:
and similarly for multiplication. Such function spaces occur in many geometric situations, when
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228:
162:
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16557:
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16398:
16202:
16115:. For technical reasons, in the context of functions one has to identify functions that agree
15665:{\displaystyle V\times V\to W,\;(\mathbf {v} ,\mathbf {a} )\mapsto \mathbf {a} +\mathbf {v} .}
11596:
9931:, which relies on the ability to express a function as a difference of two positive functions
7184:
4344:
4169:
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19803:
19798:
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An important example is the space of solutions of a system of inhomogeneous linear equations
14695:
14537:
14151:
14126:
13155:
13136:{\displaystyle \|f\|_{p}:=\left(\int _{\Omega }|f(x)|^{p}\,{d\mu (x)}\right)^{\frac {1}{p}}.}
11946:
11886:
11489:
8236:
7152:
6253:
6213:
5763:
3205:
1951:
711:
111:
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can be used to condense multiple linear equations as above into one vector equation, namely
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parametrized by the points of a differentiable manifold. The tangent bundle of the circle
14349:
13520:
12742:
8143:, that is, a corpus of mathematical objects and structure-preserving maps between them (a
1715:{\displaystyle a_{1}\mathbf {g} _{1}+a_{2}\mathbf {g} _{2}+\cdots +a_{k}\mathbf {g} _{k},}
8:
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11972:
maps between topological vector spaces are required to be continuous. In particular, the
11860:
11493:
11355:
10871:, as opposed to three space-dimensions—makes it useful for the mathematical treatment of
10067:
10063:
9835:
9825:
8955:
8714:
8220:
7278:
6614:
6549:
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3952:
3939:
2684:
2657:
2636:
1886:
1507:
1443:
620:
549:
539:
390:
290:
282:
273:
256:
252:
240:
135:
19361:
14478:
14086:
14053:
12686:
12492:
11460:
11407:
9721:
9674:
8969:
7722:
5074:; they are then essentially identical as vector spaces, since all identities holding in
4942:
4899:
4110:
3204:
The first example above reduces to this example if an arrow is represented by a pair of
1915:. Equivalently, they are linearly independent if two linear combinations of elements of
19701:
19537:. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press.
19393:
19336:
19312:
19227:
19208:
19123:
19052:
18986:
18423:
17627:
15541:
15525:
15507:
15483:
15405:
14920:
14826:
14675:
14655:
14455:
14435:
14384:
14075:
13989:
12979:{\displaystyle \|\mathbf {x} _{n}\|_{1}=\sum _{i=1}^{2^{n}}2^{-n}=2^{n}\cdot 2^{-n}=1.}
12619:{\displaystyle \mathbf {x} _{n}=\left(2^{-n},2^{-n},\ldots ,2^{-n},0,0,\ldots \right),}
12199:
11866:
11631:
11539:
11221:
11143:
11123:
11103:
11027:
10872:
10073:
10010:
9873:
9808:
9701:
9609:
9464:
8791:
8696:
8676:
8574:
8405:
8382:
8122:
8044:
7855:
7503:
7428:
7372:
7352:
7260:
7126:
7123:. This way, the quotient space "forgets" information that is contained in the subspace
7106:
6773:
6664:
6659:
6642:
6506:
6500:
6482:
6462:
6438:
6418:
6398:
6374:
6350:
6308:
6288:
5769:
5741:
5476:
4922:
4639:
4619:
2605:
2585:
1599:
616:
In this article, vectors are represented in boldface to distinguish them from scalars.
355:
346:
304:
213:
201:
19368:(1971), "Schnelle Multiplikation großer Zahlen (Fast multiplication of big numbers)",
18224:, Carus Mathematical Monographs, Washington, DC: Mathematical Association of America,
15351:
14403:
14013:
11710:
11670:
8571:), where only tuples with finitely many nonzero vectors are allowed. If the index set
6183:
is large enough to contain a zero of this polynomial (which automatically happens for
5157:-component of the arrow, as shown in the image at the right. Conversely, given a pair
19899:
19856:
19783:
19676:
19556:
19538:
19505:
19458:
19436:
19385:
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19318:
19300:
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18871:
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18393:
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18271:
18246:
18225:
18207:
18189:
18171:
18153:
18131:
18109:
18091:
18040:
18033:
Differential equations and their applications: an introduction to applied mathematics
18018:
17993:
17959:
17925:
17905:
17887:
17851:
17827:
17819:
17798:
17773:
17747:
17718:
17698:
17660:
17634:
17612:
16116:
16014:
15462:
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14529:
14325:
14146:. Definite values for physical properties such as energy, or momentum, correspond to
14135:
14047:
13914:
13023:
11045:
8756:
6083:
5268:
5142:
5041:
3947:
2574:
2538:
2450:
2122:. It follows that, in general, no base can be explicitly described. For example, the
1954:
under vector addition and scalar multiplication; that is, the sum of two elements of
1571:
1158:
1148:
627:
146:
95:
11488:
of the series depends on the topology imposed on the function space. In such cases,
11347:{\displaystyle \sum _{i=1}^{\infty }f_{i}~=~\lim _{n\to \infty }f_{1}+\cdots +f_{n}}
6325:
that is closed under addition and scalar multiplication (and therefore contains the
6216:
of the map. The set of all eigenvectors corresponding to a particular eigenvalue of
5889:
is the absolute value of the determinant of the 3-by-3 matrix formed by the vectors
3661:
2530:
1518:
19904:
19808:
19661:
19497:
19432:
19428:
19421:
19397:
19377:
19113:
19035:
18978:
18917:
18892:
18840:
18759:
18712:
18579:
18552:
18502:
18434:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
18347:
18006:
17981:
17765:
17690:
17652:
15561:
15450:
15421:
15118:
14938:
14915:
14164:
14160:
14115:
13262:
8717:
which deals with extending notions such as linear maps to several variables. A map
8140:
7147:
2791:
2570:
2565:
on directed line segments that share the same length and direction which he called
2546:
2534:
944:
864:
635:
375:
142:
15876:
It is also common, especially in physics, to denote vectors with an arrow on top:
13868:{\displaystyle \langle f\ ,\ g\rangle =\int _{\Omega }f(x){\overline {g(x)}}\,dx,}
400:
227:
Many vector spaces that are considered in mathematics are also endowed with other
19963:
19756:
19716:
19706:
19548:
19365:
19350:
19283:
19183:
19161:
19085:
19019:
19005:
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18900:
18881:
18825:
18778:
18650:
18604:
18589:
18521:
18325:
18281:
18071:
18036:
18014:
17989:
17955:
17919:
17883:
17882:, Graduate Texts in Mathematics, vol. 135 (2nd ed.), Berlin, New York:
17865:
17808:
17741:
17712:
15454:
14843:
to obtain an algebra. As a vector space, it is spanned by symbols, called simple
14645:
14388:
14376:
14227:
14119:
11621:
11481:
10605:
10208:{\textstyle |\mathbf {v} |:={\sqrt {\langle \mathbf {v} ,\mathbf {v} \rangle }}.}
9829:
9800:{\displaystyle u(\mathbf {v} \otimes \mathbf {w} )=g(\mathbf {v} ,\mathbf {w} ).}
6499:. Expressed in terms of elements, the span is the subspace consisting of all the
6268:
6249:
3964:
3577:
3563:
3421:, form a vector space over the reals with the usual addition and multiplication:
3378:
2664:
2624:
2581:
2522:
2318:{\displaystyle \mathbf {v} =a_{1}\mathbf {b} _{1}+\cdots +a_{n}\mathbf {b} _{n},}
2127:
2119:
1926:
915:
639:
467:
461:
448:
428:
419:
385:
322:
236:
209:
131:
19102:
Halpern, James D. (Jun 1966), "Bases in Vector Spaces and the Axiom of Choice",
15833:
The set of one-dimensional subspaces of a fixed finite-dimensional vector space
15343:
14937:
varies. The multiplication is given by concatenating such symbols, imposing the
11535:
consist of plane vectors of norm 1. Depicted are the unit spheres in different
6546:-dimensional vector space, any subspace of dimension 1 less, i.e., of dimension
6258:
3566:
provide another class of examples of vector spaces, particularly in algebra and
2541:
by identifying solutions to an equation of two variables with points on a plane
2019:, in the sense that it is the intersection of all linear subspaces that contain
19968:
19889:
19624:
19530:
19404:
19200:
19077:
18997:
18848:
18836:
18811:
18803:
18725:
18600:
18538:(1833), "Sopra alcune applicazioni di un nuovo metodo di geometria analitica",
18465:
18300:
18259:
18238:
18079:
17673:
15918:
15375:
15272:
15139:
14789:
11455:
11053:
10445:
8232:
7042:
6741:{\displaystyle \mathbf {v} +W=\{\mathbf {v} +\mathbf {w} :\mathbf {w} \in W\},}
5886:
4407:
3656:
3418:
2787:
2668:
2652:
2644:
1535:
1112:
1030:
509:
217:
193:
166:
18565:
17769:
17694:
17656:
10080:, which measures angles between vectors. Norms and inner products are denoted
20041:
20001:
19924:
19884:
19851:
19831:
19509:
19454:
19389:
19060:
18774:
18701:
Moore, Gregory H. (1995), "The axiomatization of linear algebra: 1875–1940",
18514:
18481:
18449:
18397:
18382:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
18371:
18352:
18186:
Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets
17608:
17600:
17551:
16286:
15850:
15537:
15503:
15379:
15149:
15135:
14782:
14380:
13739:
13726:
13721:
13709:
12076:
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9843:
5932:
5881:
3556:
2815:
2696:
2676:
2672:
2620:
1144:
837:
802:
642:
that satisfy the eight axioms listed below. In this context, the elements of
395:
360:
317:
260:
19560:
18967:
Eisenberg, Murray; Guy, Robert (1979), "A proof of the hairy ball theorem",
9832:, since the addition operation allows only finitely many terms to be added.
7342:{\displaystyle \operatorname {im} (f)=\{f(\mathbf {v} ):\mathbf {v} \in V\}}
5394:. However, there is no "canonical" or preferred isomorphism; an isomorphism
5296:; the map is an isomorphism if and only if the space is finite-dimensional.
4312:{\displaystyle A={\begin{bmatrix}1&3&1\\4&2&2\end{bmatrix}}}
2966:
A second key example of a vector space is provided by pairs of real numbers
2529:
in the plane or three-dimensional space. Around 1636, French mathematicians
2197:{\displaystyle (\mathbf {b} _{1},\mathbf {b} _{2},\ldots ,\mathbf {b} _{n})}
19823:
19773:
18717:
18584:
18506:
17875:
17121:
16931:
16835:
16325:
15952:
15598:. In particular, a vector space is an affine space over itself, by the map
15571:
15557:
15397:
14167:
acting on functions in terms of these eigenfunctions and their eigenvalues.
14143:
12188:{\displaystyle \mathbf {x} =\left(x_{1},x_{2},\ldots ,x_{n},\ldots \right)}
12071:
12065:
11250:
11052:. Compatible here means that addition and scalar multiplication have to be
8785:
8591:
is finite, the two constructions agree, but in general they are different.
7841:{\displaystyle f\mapsto D(f)=\sum _{i=0}^{n}a_{i}{\frac {d^{i}f}{dx^{i}}},}
5972:
3665:
Addition of functions: the sum of the sine and the exponential function is
3397:
2983:
2061:
569:
334:
264:
127:
31:
19134:
Hughes-Hallett, Deborah; McCallum, William G.; Gleason, Andrew M. (2013),
18557:Éléments d'histoire des mathématiques (Elements of history of mathematics)
18525:
15921:, which is an additional operation on some specific vector spaces, called
13704:
Imposing boundedness conditions not only on the function, but also on its
12858:{\displaystyle \|\mathbf {x} _{n}\|_{\infty }=\sup(2^{-n},0)=2^{-n}\to 0,}
9828:
to another function. Likewise, linear algebra is not adapted to deal with
37:"Linear space" redirects here. For a structure in incidence geometry, see
19914:
19879:
19836:
19681:
16273:
15956:
15517:
15240:
14396:
14181:
12707:
10255:
9924:
6456:
6212:, a basis consisting of eigenvectors. This phenomenon is governed by the
5921:
5445:
4985:
4966:
2833:
2123:
2115:
2094:
1986:
1104:
882:
559:
554:
443:
433:
407:
244:
79:
18206:(2nd ed.), Harlow, Essex, England: Prentice-Hall (published 2002),
17539:
15951:
This is typically the case when a vector space is also considered as an
15516:
shows; those modules that do (including all vector spaces) are known as
14186:
8951:
2759:
2663:
An important development of vector spaces is due to the construction of
19943:
19686:
19381:
19304:
19127:
18990:
18317:
18292:
17786:
16943:
16907:
15433:
14746:
14344:
14155:
14147:
13705:
10601:
7849:
6574:
6415:, when the ambient space is unambiguously a vector space. Subspaces of
6209:
5055:
4696:
2628:
2593:
2526:
2091:
Basis (linear algebra) § Proof that every vector space has a basis
1215:{\displaystyle \mathbf {v} -\mathbf {w} =\mathbf {v} +(-\mathbf {w} ).}
309:
185:
19501:
17902:
Abstract Algebra with Applications: Volume 1: Vector spaces and groups
17287:
17239:
16420:
15540:. The algebro-geometric interpretation of commutative rings via their
11852:{\displaystyle \lim _{n\to \infty }|\mathbf {v} _{n}-\mathbf {v} |=0.}
8115:
The existence of kernels and images is part of the statement that the
7918:, for example). Since differentiation is a linear procedure (that is,
3591:-vector space, by the given multiplication and addition operations of
1067:
Distributivity of scalar multiplication with respect to field addition
1033:
of scalar multiplication with respect to vector addition
19741:
17797:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
16895:
15853:
generalize this by parametrizing linear subspaces of fixed dimension
15823:
15532:, with the elements being called vectors. Some authors use the term
15486:
what vector spaces are to fields: the same axioms, applied to a ring
15144:
14191:
11503:
8525:
8400:
5759:
5337:
5051:
3921:
2640:
2632:
564:
370:
327:
295:
19118:
18982:
17139:
5101:
4450:
is the zero vector. In a similar vein, the solutions of homogeneous
2893:
is defined as the arrow pointing in the opposite direction instead.
104:, can be added together and multiplied ("scaled") by numbers called
19909:
16120:
15440:
15416:
15409:
13215:
12083:
11699:
can be uniformly approximated by a sequence of polynomials, by the
11218:
To make sense of specifying the amount a scalar changes, the field
10868:
5372:
isomorphism) by its dimension, a single number. In particular, any
3943:
2799:
2794:, starting at one fixed point. This is used in physics to describe
2723:
2688:
2609:
365:
205:
154:
44:
19409:
A Comprehensive Introduction to Differential Geometry (Volume Two)
19004:, Graduate Texts in Mathematics, vol. 150, Berlin, New York:
17359:
17335:
16955:
16799:
15566:
14138:
describes the change of physical properties in time by means of a
4319:
is the matrix containing the coefficients of the given equations,
3981:
are closely tied to vector spaces. For example, the solutions of
19593:
17841:
17407:
17323:
17263:
17087:
15280:
11536:
10397:
this reflects the common notion of the angle between two vectors
6102:
is finite-dimensional, this can be rephrased using determinants:
2680:
2648:
83:
17479:
17151:
15804:
in this equation. The space of solutions is the affine subspace
11499:
10057:
8917:{\displaystyle \mathbf {v} \mapsto g(\mathbf {v} ,\mathbf {w} )}
6913:{\displaystyle \left(\mathbf {v} _{1}+\mathbf {v} _{2}\right)+W}
6202:) any linear map has at least one eigenvector. The vector space
1911:
can be written as a linear combination of the other elements of
973:
Compatibility of scalar multiplication with field multiplication
126:
are kinds of vector spaces based on different kinds of scalars:
19919:
17764:, Undergraduate Texts in Mathematics (3rd ed.), Springer,
14844:
12196:
11024:
Convergence questions are treated by considering vector spaces
8990:
6979:{\displaystyle a\cdot (\mathbf {v} +W)=(a\cdot \mathbf {v} )+W}
6283:
3929:
1826:{\displaystyle \mathbf {g} _{1},\ldots ,\mathbf {g} _{k}\in G.}
1123:
are also commonly considered. Such a vector space is called an
299:
17826:(3rd ed.), American Mathematical Soc., pp. 193–222,
17383:
17371:
17187:
16979:
16883:
16859:
10604:. An important variant of the standard dot product is used in
19133:
18620:
Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik
17203:
17051:
17027:
17015:
17003:
16991:
16823:
16588:, ch. "Algèbre linéaire et algèbre multilinéaire", pp. 78–91.
16324:, Corollary 8.3. The sections of the tangent bundle are just
15705:
15520:. Nevertheless, a vector space can be compactly defined as a
9534:
7520:. This concept also extends to linear differential equations
5369:
3372:
3229:
2795:
2542:
2446:
751:
150:
17063:
16847:
16408:
16364:
15169:
is a family of vector spaces parametrized continuously by a
8065:. In particular, the solutions to the differential equation
6154:{\displaystyle \det(f-\lambda \cdot \operatorname {Id} )=0.}
4534:{\displaystyle f^{\prime \prime }(x)+2f^{\prime }(x)+f(x)=0}
3342:
is the above-mentioned simplest example, in which the field
2545:. To achieve geometric solutions without using coordinates,
1966:. This implies that every linear combination of elements of
18188:, Texts in Applied Mathematics, New York: Springer-Verlag,
17419:
17347:
16711:
16013:
This requirement implies that the topology gives rise to a
15959:, while an affine subspace does not necessarily contain it.
15822:
is the space of solutions of the homogeneous equation (the
13022:
are endowed with a norm that replaces the above sum by the
11048:, a structure that allows one to talk about elements being
10593:{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =0}
3597:. For example, the complex numbers are a vector space over
1513:
17395:
17251:
15925:. Scalar multiplication is the multiplication of a vector
15676:
is a vector space, then an affine subspace is a subset of
10145:{\displaystyle \langle \mathbf {v} ,\mathbf {w} \rangle ,}
9814:
8962:
The tensor product is a particular vector space that is a
8212:{\displaystyle V/\ker(f)\;\equiv \;\operatorname {im} (f)}
6526:
Linear subspace of dimension 1 and 2 are referred to as a
5352:
gives rise to a linear map that maps any basis element of
5325:
is expressed uniquely as a linear combination of them. If
2027:
is also the set of all linear combinations of elements of
1224:
Direct consequences of the axioms include that, for every
18566:"A general outline of the genesis of vector space theory"
17491:
17455:
17443:
17099:
17039:
16811:
15029:{\displaystyle \mathbf {v} _{2}\otimes \mathbf {v} _{1}.}
12988:
More generally than sequences of real numbers, functions
11358:
of the corresponding finite partial sums of the sequence
10805:{\displaystyle \langle \mathbf {x} |\mathbf {x} \rangle }
5878:
is uniquely represented by a matrix via this assignment.
5450:
3348:
is also regarded as a vector space over itself. The case
18866:, Contemporary Mathematics volume 31, Providence, R.I.:
18631:, translated by Kannenberg, Lloyd C., Providence, R.I.:
17651:, vol. 242, Springer Science & Business Media,
17515:
17467:
17311:
17275:
17175:
16967:
16871:
16687:
16473:
16471:
14982:{\displaystyle \mathbf {v} _{1}\otimes \mathbf {v} _{2}}
5989:, are particularly important since in this case vectors
3700:{\displaystyle \sin +\exp :\mathbb {R} \to \mathbb {R} }
2847:, but is dilated or shrunk by multiplying its length by
1534:(blue) expressed in terms of different bases: using the
17689:, Applied and Numerical Harmonic Analysis, Birkhäuser,
17431:
16919:
16787:
16775:
16663:
16444:
15578:. It is a two-dimensional subspace shifted by a vector
14823:
is a formal way of adding products to any vector space
11616:
The bigger diamond depicts points of 1-norm equal to 2.
11454:
could be (real or complex) functions belonging to some
9919:
can be ordered by comparing its vectors componentwise.
9870:
under which some vectors can be compared. For example,
8034:{\displaystyle (c\cdot f)^{\prime }=c\cdot f^{\prime }}
7976:{\displaystyle (f+g)^{\prime }=f^{\prime }+g^{\prime }}
5145:
can be expressed as an ordered pair by considering the
2498:
addition and scalar multiplication, whose dimension is
2093:). Moreover, all bases of a vector space have the same
134:. Scalars can also be, more generally, elements of any
17299:
17163:
16615:
16487:
16303:
which restricts to linear isomorphisms between fibers.
10160:
10076:, a datum which measures lengths of vectors, or by an
8958:
depicting the universal property of the tensor product
8534:
8487:
8372:{\displaystyle \left(\mathbf {v} _{i}\right)_{i\in I}}
8262:
7887:
5744:
4701:
The relation of two vector spaces can be expressed by
4471:
4268:
204:. Finite-dimensional vector spaces occur naturally in
18897:
Elements of Mathematics : Algebra I Chapters 1-3
16651:
16233:
16205:
16165:
16138:
16040:
15982:
15917:
Scalar multiplication is not to be confused with the
15882:
15780:
15743:
15604:
15190:
15046:
14995:
14951:
14923:
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14797:
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8794:
8764:
8723:
8699:
8679:
8650:
8614:
8577:
8457:
8428:
8408:
8385:
8335:
8308:
8261:
8165:
8125:
8071:
8047:
7989:
7924:
7878:
7858:
7748:
7725:
7698:
7526:
7506:
7473:
7451:
7431:
7398:
7375:
7355:
7286:
7263:
7241:
7219:
7187:
7155:
7129:
7109:
7080:
7051:
7035:{\displaystyle \mathbf {v} _{1}+W=\mathbf {v} _{2}+W}
6992:
6926:
6866:
6831:
6796:
6776:
6754:
6687:
6667:
6645:
6617:
6591:
6552:
6509:
6485:
6465:
6441:
6421:
6401:
6377:
6353:
6331:
6311:
6291:
6120:
5816:
5792:
5772:
5511:
5479:
4945:
4925:
4902:
4880:
4858:
4719:
4662:
4642:
4622:
4549:
4462:
4416:
4387:
4347:
4325:
4256:
4218:
4172:
4135:
4113:
3987:
3848:
3816:
3713:
3671:
3609:
3249:
3004:
2687:
began to interact, notably with key concepts such as
2627:
which allows for harmonization and simplification of
2465:
2381:
2375:, and that this decomposition is unique. The scalars
2331:
2250:
2222:
2139:
1839:
1780:
1728:
1630:
1472:
1446:
1413:
1364:
1327:
1290:
1256:
1230:
1170:
18462:
Topological vector spaces, distributions and kernels
17563:
17527:
17127:
17075:
16639:
8924:
is linear in the sense above and likewise for fixed
7445:. The kernel of this map is the subspace of vectors
6274:
is a linear subspace. It is the intersection of two
3216:
The simplest example of a vector space over a field
48:
Vector addition and scalar multiplication: a vector
19451:
Lie groups, Lie algebras, and their representations
19299:
18627:Grassmann, Hermann (2000), Kannenberg, L.C. (ed.),
17503:
16591:
16456:
16432:
16104:{\displaystyle \|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p}}
15774:on linear equations, which can be found by setting
15770:generalizing the homogeneous case discussed in the
15498:. For example, modules need not have bases, as the
15036:Forcing two such elements to be equal leads to the
11667:is not complete because any continuous function on
10770:In contrast to the standard dot product, it is not
7392:An important example is the kernel of a linear map
6681:") is defined as follows: as a set, it consists of
2580:Vectors were reconsidered with the presentation of
2126:form an infinite-dimensional vector space over the
19420:
19226:
18782:
18264:Introductory functional analysis with applications
17626:
16699:
16627:
16603:
16376:
16352:
16255:
16220:
16178:
16151:
16103:
15996:
15900:
15796:
15762:
15664:
15208:
15109:
15028:
14981:
14929:
14906:
14835:
14815:
14773:
14737:
14684:
14664:
14636:
14520:
14464:
14444:
14424:
14367:
14313:
14288:
14218:
14098:
14065:
14034:
13998:
13971:
13941:
13905:
13867:
13773:
13694:
13576:
13538:
13509:
13375:
13326:
13251:
13206:
13164:
13135:
13014:
12978:
12857:
12760:
12731:
12698:
12675:
12645:
12618:
12504:
12481:
12452:
12338:
12247:
12208:
12187:
12103:
12043:
12021:
11993:
11964:
11927:
11895:
11875:
11851:
11782:
11760:
11731:
11691:
11655:
11608:
11585:
11548:
11527:
11472:
11446:
11419:
11396:
11346:
11230:
11210:
11182:
11152:
11132:
11112:
11092:
11070:
11036:
11003:
10859:
10804:
10762:
10629:
10592:
10550:
10436:
10411:
10389:
10357:
10246:
10216:Vector spaces endowed with such data are known as
10207:
10144:
10104:
10046:
10019:
9999:
9972:
9911:
9882:
9862:
9799:
9733:
9710:
9686:
9663:
9618:
9594:
9564:
9525:
9499:
9473:
9453:
9097:
8981:
8941:
8916:
8869:
8847:
8822:
8800:
8776:
8747:
8705:
8685:
8665:
8636:
8583:
8563:
8516:
8470:
8443:
8414:
8391:
8371:
8321:
8294:
8211:
8131:
8092:
8053:
8033:
7975:
7910:
7864:
7840:
7734:
7711:
7684:
7512:
7492:
7459:
7437:
7417:
7381:
7361:
7341:
7269:
7249:
7227:
7205:
7173:
7135:
7115:
7095:
7066:
7034:
6978:
6912:
6852:
6817:
6782:
6762:
6740:
6673:
6651:
6631:
6603:
6564:
6515:
6491:
6471:
6447:
6427:
6407:
6383:
6359:
6339:
6317:
6297:
6153:
5839:{\displaystyle \mathbf {x} \mapsto A\mathbf {x} .}
5838:
5800:
5778:
5750:
5730:
5485:
4954:
4931:
4911:
4888:
4866:
4844:
4675:
4648:
4628:
4608:
4533:
4442:
4398:
4374:
4333:
4311:
4240:
4198:
4158:
4122:
4099:
3908:
3834:
3776:
3699:
3633:
3300:
3194:
2952:has the opposite direction and the same length as
2478:
2413:
2363:
2317:
2236:
2196:
1871:
1825:
1766:
1714:
1491:
1458:
1432:
1398:
1349:
1312:
1273:
1242:
1214:
611:
165:. The concept of vector spaces is fundamental for
19360:
18377:
18245:(6th ed.), New York: John Wiley & Sons,
18202:Ifeachor, Emmanuel C.; Jervis, Barrie W. (2001),
16519:
14389:rings of functions of algebraic geometric objects
14328:defining the multiplication of two vectors is an
13559:
13327:{\displaystyle f_{1},f_{2},\ldots ,f_{n},\ldots }
13231:
12462:The topologies on the infinite-dimensional space
12112:consisting of infinite vectors with real entries
8295:{\displaystyle \textstyle {\prod _{i\in I}V_{i}}}
7418:{\displaystyle \mathbf {x} \mapsto A\mathbf {x} }
3938:. Many notions in topology and analysis, such as
2982:is significant, so such a pair is also called an
2958:(blue vector pointing down in the second image).
20039:
19105:Proceedings of the American Mathematical Society
18857:"Existence of bases implies the axiom of choice"
18540:Il poligrafo giornale di scienze, lettre ed arti
18085:
17329:
13592:
13391:
12802:
12284:
11798:
11300:
9694:shown in the diagram with a dotted arrow, whose
9602:is bilinear. The universality states that given
9595:{\displaystyle \mathbf {v} \otimes \mathbf {w} }
8226:
6243:
6121:
3777:{\displaystyle (\sin +\exp )(x)=\sin(x)+\exp(x)}
2786:The first example of a vector space consists of
64:is stretched by a factor of 2, yielding the sum
18835:
18204:Digital Signal Processing: A Practical Approach
18183:
17946:
17593:Elementary Linear Algebra: Applications Version
16675:
16414:
16340:
15955:. In this case, a linear subspace contains the
15551:
15548:, the algebraic counterpart to vector bundles.
8151:. Because of this, many statements such as the
5961:
2216:. The definition of a basis implies that every
1399:{\displaystyle (-1)\mathbf {v} =-\mathbf {v} ,}
19032:Classic Set Theory: A guided independent study
18773:
18378:Narici, Lawrence; Beckenstein, Edward (2011).
18201:
17954:(in German) (9th ed.), Berlin, New York:
15818:is a particular solution of the equation, and
5212:between two vector spaces form a vector space
19609:
19473:"The JPEG still picture compression standard"
18422:
18344:An introduction to abstract harmonic analysis
17988:, Elements of mathematics, Berlin, New York:
17209:
17145:
16312:A line bundle, such as the tangent bundle of
15352:identifying open intervals with the real line
15346:can be seen as a line bundle over the circle
14945:. In general, there are no relations between
10812:also takes negative values, for example, for
10058:Normed vector spaces and inner product spaces
1164:Subtraction of two vectors can be defined as
592:
216:infinite-dimensional vector spaces, and many
18966:
18609:(in French), Chez Firmin Didot, père et fils
18086:Dennery, Philippe; Krzywicki, Andre (1996),
17497:
16092:
16085:
16073:
16066:
16054:
16041:
15408:, there is no (tangent) vector field on the
14154:and the associated wavefunctions are called
13806:
13788:
13355:
13341:
13261:These spaces are complete. (If one uses the
13186:
13179:
13039:
13032:
12888:
12872:
12790:
12774:
12361:
12352:
12271:
12262:
11013:
10900:
10882:
10799:
10781:
10662:
10644:
10581:
10565:
10281:
10265:
10197:
10181:
10136:
10120:
10072:"Measuring" vectors is done by specifying a
9833:
7336:
7305:
6732:
6702:
5958:if and only if its determinant is positive.
5292:, any vector space can be embedded into its
5001:, which is a map such that the two possible
3325:form a vector space that is usually denoted
750:To have a vector space, the eight following
19448:
18184:Gasquet, Claude; Witomski, Patrick (1999),
17590:
17437:
16509:
15544:allows the development of concepts such as
14230:of functions on this hyperbola is given by
13734:Complete inner product spaces are known as
11624:has a limit; such a vector space is called
9565:{\displaystyle (\mathbf {v} ,\mathbf {w} )}
6986:. The key point in this definition is that
4690:
4241:{\displaystyle A\mathbf {x} =\mathbf {0} ,}
2961:
2918:, but is stretched to the double length of
2841:, the arrow that has the same direction as
2431:on the basis. They are also said to be the
1350:{\displaystyle s\mathbf {0} =\mathbf {0} ,}
1313:{\displaystyle 0\mathbf {v} =\mathbf {0} ,}
19616:
19602:
19250:
19180:Riemannian Geometry and Geometric Analysis
19155:
18742:: CS1 maint: location missing publisher (
18534:
17843:Matrix Analysis and Applied Linear Algebra
16621:
16321:
15841:; it may be used to formalize the idea of
15763:{\displaystyle A\mathbf {v} =\mathbf {b} }
15716:) and consists of all vectors of the form
15680:obtained by translating a linear subspace
15623:
14296:an infinite-dimensional vector space over
14289:{\displaystyle \mathbf {R} /(x\cdot y-1),}
13558:
13230:
9838:require considering additional structures.
8193:
8189:
7493:{\displaystyle A\mathbf {x} =\mathbf {0} }
5408:is equivalent to the choice of a basis of
5058:). If there exists an isomorphism between
3634:{\displaystyle \mathbf {Q} (i{\sqrt {5}})}
3386:, numbers that can be written in the form
3373:Complex numbers and other field extensions
3301:{\displaystyle (a_{1},a_{2},\dots ,a_{n})}
2558:
1492:{\displaystyle \mathbf {v} =\mathbf {0} .}
1433:{\displaystyle s\mathbf {v} =\mathbf {0} }
599:
585:
114:must satisfy certain requirements, called
19491:
19480:IEEE Transactions on Consumer Electronics
19117:
19051:
18810:, Advanced Book Classics (2nd ed.),
18716:
18626:
18613:
18583:
18351:
17937:
17863:
17680:, vol. 7, Princeton University Press
17389:
17069:
17021:
16853:
16657:
16477:
15797:{\displaystyle \mathbf {b} =\mathbf {0} }
14159:
13855:
13668:
13483:
13376:{\displaystyle \|f_{n}\|_{p}<\infty ,}
13096:
13015:{\displaystyle f:\Omega \to \mathbb {R} }
13008:
11183:{\displaystyle \mathbf {x} +\mathbf {y} }
9859:
9852:
7911:{\displaystyle f^{\prime \prime }(x)^{2}}
4086:
4075:
4069:
4064:
4057:
4050:
4030:
4022:
4016:
4011:
4004:
3997:
3693:
3685:
2130:, for which no specific basis is known.
1982:of linear subspaces is a linear subspace.
1502:Even more concisely, a vector space is a
1000:Identity element of scalar multiplication
180:Vector spaces are characterized by their
19279:Categories for the Working Mathematician
19272:
19224:
19199:
19136:Calculus : Single and Multivariable
18996:
18916:
18891:
18668:
18551:
18473:
18258:
18237:
18005:
17980:
17917:
17899:
17818:
17739:
17485:
17425:
17377:
17233:
17221:
17197:
17157:
17081:
16749:
16733:
16645:
16585:
16426:
16394:
16018:
15565:
15143:
14185:
13725:
11498:
10558:Because of this, two vectors satisfying
8950:
6920:, and scalar multiplication is given by
6257:
5880:
5449:
5100:
3660:
1767:{\displaystyle a_{1},\ldots ,a_{k}\in F}
1517:
1514:Bases, vector coordinates, and subspaces
110:. The operations of vector addition and
43:
19470:
19415:
19101:
19029:
18941:
18599:
18520:
18165:
18070:
17687:A Basis Theory Primer: Expanded Edition
17646:
17569:
17533:
17365:
17245:
16597:
15176:. More precisely, a vector bundle over
14171:
13214:and equipped with this norm are called
10860:{\displaystyle \mathbf {x} =(0,0,0,1).}
9815:Vector spaces with additional structure
5995:can be compared with their image under
5123:yields an isomorphism of vector spaces.
2896:The following shows a few examples: if
1976:The closure property also implies that
14:
20040:
20007:Comparison of linear algebra libraries
19535:An introduction to homological algebra
19529:
19403:
19076:
18802:
18682:
18563:
18480:
18456:
18341:
18219:
18121:
18103:
17672:
17595:(10th ed.), John Wiley & Sons
17509:
17473:
17317:
17293:
17281:
17257:
17193:
16889:
16717:
16705:
16633:
16609:
16569:
16187:
15354:). It is, however, different from the
15314:) is isomorphic to the trivial bundle
8061:) this assignment is linear, called a
6235:
5358:to the corresponding basis element of
4609:{\displaystyle f(x)=ae^{-x}+bxe^{-x},}
2713:
2550:
1616:, a linear combination of elements of
19597:
19330:
18854:
18766:
18724:
18700:
18404:
18168:Fourier Analysis and Its Applications
18143:
18052:
18030:
17874:
17839:
17730:
17714:Vector Spaces and Matrices in Physics
17624:
17599:
17591:Anton, Howard; Rorres, Chris (2010),
17557:
17545:
17521:
17305:
17181:
17169:
17133:
17117:
17093:
17057:
17033:
17009:
16997:
16985:
16961:
16913:
16829:
16721:
16693:
16669:
16537:
16525:
16493:
16462:
16450:
16438:
16402:
16382:
16370:
16358:
16316:is trivial if and only if there is a
15969:
15124:
14816:{\displaystyle \operatorname {T} (V)}
14652:Examples include the vector space of
13207:{\displaystyle \|f\|_{p}<\infty ,}
13172:(for example an interval) satisfying
12512:For example, the sequence of vectors
12248:{\displaystyle (1\leq p\leq \infty )}
12079:, are complete normed vector spaces.
11160:vary by a bounded amount, then so do
8517:{\textstyle \bigoplus _{i\in I}V_{i}}
6459:, and it is the smallest subspace of
6226:corresponding to the eigenvalue (and
2765:Scalar multiplication: the multiples
2441:on the basis. One also says that the
27:Algebraic structure in linear algebra
19253:Optimization by vector space methods
19177:
18316:
18291:
18266:, Wiley Classics Library, New York:
17785:
17759:
17710:
17684:
17461:
17449:
17413:
17401:
17353:
17341:
17269:
17105:
17045:
16973:
16949:
16937:
16925:
16901:
16877:
16865:
16841:
16817:
16805:
16793:
16781:
16765:
16553:
16505:
16346:
15310:such that the restriction of π to π(
4107:are given by triples with arbitrary
2596:by the latter. They are elements in
1570:(black), and using a different, non-
19333:The geometry of Minkowski spacetime
18845:Introduction to Commutative Algebra
18648:
17733:Foundations of Discrete Mathematics
16681:
14942:
13252:{\displaystyle L^{\;\!p}(\Omega ).}
12001:consists of continuous functionals
11397:{\displaystyle f_{1},f_{2},\ldots }
10867:Singling out the fourth coordinate—
8564:{\textstyle \coprod _{i\in I}V_{i}}
5414:, by mapping the standard basis of
3958:
3909:{\displaystyle (f+g)(w)=f(w)+g(w),}
3369:) reduces to the previous example.
3211:
2414:{\displaystyle a_{1},\ldots ,a_{n}}
2103:Dimension theorem for vector spaces
2011:is the smallest linear subspace of
1872:{\displaystyle a_{1},\ldots ,a_{k}}
1111:, and when the scalar field is the
24:
19623:
19282:(2nd ed.), Berlin, New York:
19182:(4th ed.), Berlin, New York:
19160:(3rd ed.), Berlin, New York:
18949:(2nd ed.), Berlin, New York:
17938:Stoll, R. R.; Wong, E. T. (1968),
16247:
15396:, since there is a global nonzero
15232:) is a vector space. The case dim
14798:
13983:, established an approximation of
13906:{\displaystyle {\overline {g(x)}}}
13817:
13762:
13612:
13602:
13577:{\displaystyle L^{\;\!p}(\Omega )}
13568:
13420:
13410:
13367:
13240:
13198:
13159:
13062:
13001:
12794:
12723:
12444:
12325:
12275:
12239:
11890:
11808:
11600:
11310:
11275:
10482:
10254:can be equipped with the standard
8329:consists of the set of all tuples
8026:
8007:
7968:
7955:
7942:
7884:
6853:{\displaystyle \mathbf {v} _{2}+W}
6818:{\displaystyle \mathbf {v} _{1}+W}
6222:forms a vector space known as the
5848:Moreover, after choosing bases of
5193:is negative) turns back the arrow
4496:
4468:
4443:{\displaystyle \mathbf {0} =(0,0)}
3650:
2924:(the second image). Equivalently,
2822:of the two arrows, and is denoted
2364:{\displaystyle a_{1},\dots ,a_{n}}
1274:{\displaystyle \mathbf {v} \in V,}
60:(red, upper illustration). Below,
54:(blue) is added to another vector
25:
20079:
20063:Vectors (mathematics and physics)
19569:
19057:Introduction to Quantum Mechanics
18970:The American Mathematical Monthly
15976:and derive the concrete shape of
15129:
14774:{\displaystyle \mathbf {R} ^{3},}
13715:
12022:{\displaystyle V\to \mathbf {R} }
11928:{\displaystyle \mathbf {R} ^{2}:}
11701:Weierstrass approximation theorem
10637:endowed with the Lorentz product
10390:{\displaystyle \mathbf {R} ^{2},}
9664:{\displaystyle g:V\times W\to X,}
8713:is one of the central notions of
8594:
6581:The counterpart to subspaces are
5175:to the right (or to the left, if
5131:) are isomorphic: a planar arrow
5086:, transported to similar ones in
4454:form vector spaces. For example,
3543:as representing the ordered pair
2237:{\displaystyle \mathbf {v} \in V}
1958:and the product of an element of
251:, which include function spaces,
20020:
20019:
19997:Basic Linear Algebra Subprograms
19755:
19411:, Houston, TX: Publish or Perish
19229:Advanced Engineering Mathematics
19082:Finite-dimensional vector spaces
18606:Théorie analytique de la chaleur
18243:Advanced Engineering Mathematics
17948:van der Waerden, Bartel Leendert
17867:Linear Algebra with Applications
17743:Advanced Engineering Mathematics
17678:Finite Dimensional Vector Spaces
17647:Grillet, Pierre Antoine (2007),
16306:
16279:
15845:lines intersecting at infinity.
15790:
15782:
15756:
15748:
15655:
15647:
15636:
15628:
15184:equipped with a continuous map
15097:
15082:
15064:
15049:
15013:
14998:
14969:
14954:
14891:
14870:
14855:
14758:
14304:
14238:
14114:, it enables one to construct a
14106:its cardinality is known as the
12877:
12779:
12521:
12356:
12266:
12120:
12059:
12037:
12015:
11936:spaces without additional data.
11912:
11834:
11820:
11776:
11761:{\displaystyle \mathbf {v} _{n}}
11748:
11528:{\displaystyle \mathbf {R} ^{2}}
11515:
11201:
11176:
11168:
11086:
11064:
10896:
10886:
10820:
10795:
10785:
10658:
10648:
10630:{\displaystyle \mathbf {R} ^{4}}
10617:
10577:
10569:
10536:
10518:
10497:
10489:
10464:
10456:
10427:
10405:
10374:
10296:
10288:
10277:
10269:
10193:
10185:
10167:
10132:
10124:
10093:
9912:{\displaystyle \mathbf {R} ^{n}}
9899:
9787:
9779:
9762:
9754:
9588:
9580:
9555:
9547:
9461:These rules ensure that the map
9432:
9423:
9409:
9400:
9373:
9358:
9346:
9335:
9321:
9312:
9298:
9279:
9262:
9247:
9206:
9189:
9175:
9164:
9134:
9126:
9082:
9067:
9046:
9031:
9016:
9001:
8932:
8907:
8899:
8885:
8863:
8838:
8816:
8748:{\displaystyle g:V\times W\to X}
8444:{\displaystyle \mathbf {v} _{i}}
8431:
8343:
7486:
7478:
7453:
7411:
7400:
7326:
7315:
7243:
7221:
7096:{\displaystyle \mathbf {v} _{2}}
7083:
7067:{\displaystyle \mathbf {v} _{1}}
7054:
7016:
6995:
6963:
6937:
6889:
6874:
6834:
6799:
6756:
6722:
6714:
6706:
6689:
6333:
5829:
5818:
5794:
5513:
5493:gives rise to a linear map from
5254:. The space of linear maps from
5070:, the two spaces are said to be
4882:
4860:
4831:
4804:
4777:
4760:
4739:
4731:
4418:
4392:
4327:
4231:
4223:
3611:
2758:
2722:
2302:
2271:
2252:
2224:
2181:
2160:
2145:
2085:A subset of a vector space is a
1804:
1783:
1699:
1668:
1643:
1482:
1474:
1426:
1418:
1389:
1378:
1340:
1332:
1303:
1295:
1258:
1202:
1188:
1180:
1172:
670:assigns to any two vectors
19895:Seven-dimensional cross product
18943:Coxeter, Harold Scott MacDonald
18222:A Panorama of Harmonic Analysis
17227:
17215:
17111:
16266:
16193:
16126:
16024:
16007:
15962:
15945:
14010:, every continuous function on
13774:{\displaystyle L^{2}(\Omega ),}
12489:are inequivalent for different
12432:
12313:
11665:topology of uniform convergence
8637:{\displaystyle V\otimes _{F}W,}
8601:Tensor product of vector spaces
8379:, which specify for each index
7872:appear linearly (as opposed to
6790:. The sum of two such elements
2671:. This was later formalized by
1115:, the vector space is called a
1107:, the vector space is called a
612:Definition and basic properties
169:, together with the concept of
18922:General Topology. Chapters 1-4
18146:Partial differential equations
18090:, Courier Dover Publications,
17560:, Exercise 5.13.15–17, p. 442.
16940:, ch. V.3., Corollary, p. 106.
16844:, ch. IV.4, Corollary, p. 106.
16256:{\displaystyle L^{p}(\Omega )}
16250:
16244:
16119:to get a norm, and not only a
15932:
15911:
15889:
15870:
15771:
15643:
15640:
15624:
15614:
15378:consists of the collection of
15287:and some (fixed) vector space
15200:
14810:
14804:
14711:
14699:
14625:
14622:
14610:
14601:
14595:
14592:
14580:
14571:
14565:
14562:
14550:
14541:
14515:
14503:
14494:
14482:
14419:
14407:
14375:forms an algebra known as the
14362:
14356:
14280:
14262:
14254:
14242:
14029:
14017:
13933:
13927:
13894:
13888:
13846:
13840:
13831:
13825:
13765:
13759:
13682:
13676:
13654:
13648:
13632:
13626:
13599:
13571:
13565:
13546:belonging to the vector space
13533:
13527:
13497:
13491:
13469:
13463:
13447:
13441:
13407:
13243:
13237:
13110:
13104:
13086:
13081:
13075:
13068:
13004:
12846:
12827:
12805:
12406:
12390:
12309:
12294:
12242:
12224:
12011:
11953:
11839:
11814:
11805:
11726:
11714:
11686:
11674:
11647:
11635:
11480:in which case the series is a
11307:
11211:{\displaystyle a\mathbf {x} .}
10891:
10851:
10827:
10790:
10653:
10541:
10531:
10523:
10513:
10501:
10485:
10172:
10162:
10105:{\displaystyle |\mathbf {v} |}
10098:
10088:
9973:{\displaystyle f=f^{+}-f^{-}.}
9842:A vector space may be given a
9791:
9775:
9766:
9750:
9652:
9559:
9543:
9383:
9353:
9272:
9242:
9210:
9196:
9168:
9154:
9138:
9122:
8911:
8895:
8889:
8739:
8206:
8200:
8186:
8180:
8081:
8075:
8003:
7990:
7938:
7925:
7899:
7892:
7764:
7758:
7752:
7742:too. In the corresponding map
7404:
7319:
7311:
7299:
7293:
7197:
7168:
7162:
6967:
6953:
6947:
6933:
6142:
6124:
5822:
5568:
5565:
5520:
4835:
4827:
4808:
4794:
4781:
4773:
4764:
4756:
4743:
4727:
4559:
4553:
4522:
4516:
4507:
4501:
4482:
4476:
4437:
4425:
4366:
4348:
3900:
3894:
3885:
3879:
3870:
3864:
3861:
3849:
3829:
3817:
3771:
3765:
3753:
3747:
3735:
3729:
3726:
3714:
3689:
3628:
3615:
3295:
3250:
3182:
3164:
3154:
3142:
3129:
3077:
3067:
3041:
3035:
3009:
2974:. The order of the components
2191:
2140:
1374:
1365:
1206:
1195:
13:
1:
18868:American Mathematical Society
18789:, Toronto: Thomson Learning,
18633:American Mathematical Society
18601:Fourier, Jean Baptiste Joseph
18150:American Mathematical Society
18106:Real analysis and probability
17795:Graduate Texts in Mathematics
17578:
15415:which is everywhere nonzero.
14314:{\displaystyle \mathbf {R} .}
14142:, whose solutions are called
14140:partial differential equation
11703:. In contrast, the space of
10437:{\displaystyle \mathbf {y} ,}
10007:denotes the positive part of
8942:{\displaystyle \mathbf {v} .}
8848:{\displaystyle \mathbf {w} .}
8302:of a family of vector spaces
8227:Direct product and direct sum
8147:) that behaves much like the
6244:Subspaces and quotient spaces
5810:
5128:
4656:are arbitrary constants, and
4456:
4452:linear differential equations
4399:{\displaystyle A\mathbf {x} }
4212:
3788:Functions from any fixed set
2802:. Given any two such arrows,
2679:, around 1920. At that time,
1103:When the scalar field is the
715:, assigns to any scalar
690:which is commonly written as
662:The binary operation, called
98:whose elements, often called
19737:Eigenvalues and eigenvectors
19431:Mathematics Series, London:
18426:; Wolff, Manfred P. (1999).
18322:Real and functional analysis
17900:Spindler, Karlheinz (1993),
17864:Nicholson, W. Keith (2018),
17330:Dennery & Krzywicki 1996
16334:
15972:, choose to start with this
15861:of subspaces, respectively.
15552:Affine and projective spaces
15461:of that bundle are known as
15148:A Möbius strip. Locally, it
14394:Another crucial example are
14343:For example, the set of all
13898:
13850:
13781:with inner product given by
12044:{\displaystyle \mathbf {C} }
11783:{\displaystyle \mathbf {v} }
11496:are two prominent examples.
11093:{\displaystyle \mathbf {y} }
11071:{\displaystyle \mathbf {x} }
10412:{\displaystyle \mathbf {x} }
8870:{\displaystyle \mathbf {w} }
8823:{\displaystyle \mathbf {v} }
8808:is linear in both variables
8063:linear differential operator
7460:{\displaystyle \mathbf {x} }
7250:{\displaystyle \mathbf {0} }
7228:{\displaystyle \mathbf {v} }
6763:{\displaystyle \mathbf {v} }
6604:{\displaystyle W\subseteq V}
6340:{\displaystyle \mathbf {0} }
5968:Eigenvalues and eigenvectors
5962:Eigenvalues and eigenvectors
5801:{\displaystyle \mathbf {x} }
4889:{\displaystyle \mathbf {w} }
4867:{\displaystyle \mathbf {v} }
4685:natural exponential function
4334:{\displaystyle \mathbf {x} }
3979:homogeneous linear equations
3969:Linear differential equation
2986:. Such a pair is written as
2459:on the basis, since the set
754:must be satisfied for every
709:The binary function, called
222:cardinality of the continuum
7:
19582:Encyclopedia of Mathematics
19449:Varadarajan, V. S. (1974),
18692:(in German), archived from
18559:(in French), Paris: Hermann
18409:(2 ed.), McGraw-Hill,
18166:Folland, Gerald B. (1992),
18144:Evans, Lawrence C. (1998),
18104:Dudley, Richard M. (1989),
18060:Encyclopedia of Mathematics
17973:
16415:Atiyah & Macdonald 1969
16320:that vanishes nowhere, see
15901:{\displaystyle {\vec {v}}.}
15694:; this space is denoted by
15371:whereas the former is not.
15209:{\displaystyle \pi :E\to X}
14219:{\displaystyle x\cdot y=1.}
14116:basis of orthogonal vectors
12683:and the following ones are
11973:
8966:recipient of bilinear maps
8666:{\displaystyle V\otimes W,}
6262:A line passing through the
5786:with the coordinate vector
5435:
5105:Describing an arrow vector
3973:Systems of linear equations
2702:
2614:systems of linear equations
2109:
1919:define the same element of
790:
650:, and the elements of
247:. This is also the case of
175:systems of linear equations
10:
20084:
19471:Wallace, G.K. (Feb 1992),
19331:Naber, Gregory L. (2003),
19251:Luenberger, David (1997),
19233:(8th ed.), New York:
19059:, Upper Saddle River, NJ:
18847:, Advanced Book Classics,
18649:Guo, Hongyu (2021-06-16),
18220:Krantz, Steven G. (1999),
18128:Princeton University Press
18088:Mathematics for Physicists
17685:Heil, Christopher (2011),
17629:Matrices and vector spaces
17625:Brown, William A. (1991),
17583:
16952:, Theorem VII.9.8, p. 198.
15555:
15472:
15468:
15386:is globally isomorphic to
15133:
14175:
13719:
12732:{\displaystyle p=\infty ,}
12063:
11017:
10061:
9863:{\displaystyle \,\leq ,\,}
9671:there exists a unique map
9526:{\displaystyle V\otimes W}
8855:That is to say, for fixed
8598:
8230:
8149:category of abelian groups
8100:form a vector space (over
6770:is an arbitrary vector in
6247:
6208:may or may not possess an
6042:is a scalar, is called an
5965:
5439:
4984:such that there exists an
4694:
3962:
3654:
3603:, and the field extension
2912:has the same direction as
2706:
2612:in 1867, who also defined
2525:, via the introduction of
2516:
2449:of the coordinates is the
1883:of the linear combination.
931:, there exists an element
200:, and its dimension is an
145:, which allow modeling of
36:
29:
20015:
19977:
19933:
19870:
19822:
19764:
19753:
19649:
19631:
19156:Husemoller, Dale (1994),
18652:What Are Tensors Exactly?
18564:Dorier, Jean-Luc (1995),
18428:Topological Vector Spaces
18380:Topological Vector Spaces
18342:Loomis, Lynn H. (2011) ,
17986:Topological vector spaces
17770:10.1007/978-1-4757-1949-9
17746:, John Wiley & Sons,
17695:10.1007/978-0-8176-4687-5
17657:10.1007/978-0-387-71568-1
17548:, Example 5.13.5, p. 436.
17210:Schaefer & Wolff 1999
17146:Schaefer & Wolff 1999
16916:, Th. 2.5 and 2.6, p. 49.
14125:The solutions to various
14008:Stone–Weierstrass theorem
13383:satisfying the condition
12482:{\displaystyle \ell ^{p}}
12104:{\displaystyle \ell ^{p}}
11974:(topological) dual space
11243:topological vector spaces
11014:Topological vector spaces
9500:{\displaystyle V\times W}
8777:{\displaystyle V\times W}
8245:of vector spaces and the
8159:in matrix-related terms)
8153:first isomorphism theorem
8117:category of vector spaces
6455:of vectors is called its
6169:characteristic polynomial
2729:Vector addition: the sum
2709:Examples of vector spaces
2553:introduced the notion of
2507:one-to-one correspondence
2101:of the vector space (see
1946:is a non-empty subset of
1891:The elements of a subset
249:topological vector spaces
141:Vector spaces generalize
19225:Kreyszig, Erwin (1999),
19034:(1st ed.), London:
18684:Möbius, August Ferdinand
18122:Dunham, William (2005),
17740:Kreyszig, Erwin (2020),
17498:Eisenberg & Guy 1979
16904:, Theorem IV.2.1, p. 95.
16221:{\displaystyle p\neq 2,}
15864:
15367:, because the latter is
14194:, given by the equation
13985:differentiable functions
13517:there exists a function
11707:continuous functions on
11609:{\displaystyle \infty .}
11020:Topological vector space
9890:-dimensional real space
9834:Therefore, the needs of
7206:{\displaystyle f:V\to W}
5946:corresponding to a real
4691:Linear maps and matrices
4375:{\displaystyle (a,b,c),}
4199:{\displaystyle c=-5a/2.}
2962:Ordered pairs of numbers
2651:. Italian mathematician
2631:. Around the same time,
2521:Vector spaces stem from
2511:vector space isomorphism
2435:of the decomposition of
870:There exists an element
737:, which is denoted
30:Not to be confused with
20058:Mathematical structures
19309:Wheeler, John Archibald
19030:Goldrei, Derek (1996),
18855:Blass, Andreas (1984),
18837:Atiyah, Michael Francis
18804:Atiyah, Michael Francis
18675:Lectures on Quaternions
18670:Hamilton, William Rowan
18494:Fundamenta Mathematicae
17921:Linear Algebraic Groups
17918:Springer, T.A. (2000),
17880:Advanced Linear Algebra
17840:Meyer, Carl D. (2000),
17735:, John Wiley & Sons
17633:, New York: M. Dekker,
17296:, Theorem 11.2, p. 102.
16510:Anton & Rorres 2010
16263:is not a Hilbert space.
15536:to mean modules over a
15496:multiplicative inverses
15273:"trivial" vector bundle
15243:. For any vector space
15180:is a topological space
14738:{\displaystyle =xy-yx,}
14637:{\displaystyle ]+]+]=0}
14432:denotes the product of
14108:Hilbert space dimension
14044:trigonometric functions
14006:by polynomials. By the
13165:{\displaystyle \Omega }
11965:{\displaystyle V\to W,}
11896:{\displaystyle \infty }
7692:where the coefficients
7174:{\displaystyle \ker(f)}
5340:between fixed bases of
5305:is chosen, linear maps
3641:is a vector space over
3568:algebraic number theory
2903:, the resulting vector
2739:(black) of the vectors
2555:barycentric coordinates
2490:-tuples of elements of
1019:multiplicative identity
157:, that have not only a
39:Linear space (geometry)
19722:Row and column vectors
18752:Formulario mathematico
18718:10.1006/hmat.1995.1025
18623:(in German), O. Wigand
18585:10.1006/hmat.1995.1024
18507:10.4064/fm-3-1-133-181
18405:Rudin, Walter (1991),
18031:Braun, Martin (1993),
17248:, Proposition III.7.2.
16257:
16222:
16180:
16153:
16105:
15998:
15968:Some authors, such as
15902:
15798:
15764:
15666:
15583:
15463:differential one-forms
15404:. In contrast, by the
15275:. Vector bundles over
15210:
15162:
15111:
15030:
14983:
14931:
14908:
14837:
14817:
14775:
14739:
14686:
14666:
14638:
14522:
14466:
14446:
14426:
14369:
14336:-algebra if the field
14321:
14315:
14290:
14220:
14150:of a certain (linear)
14127:differential equations
14100:
14073:in the sense that the
14067:
14036:
14000:
13973:
13943:
13907:
13869:
13775:
13731:
13696:
13578:
13540:
13511:
13377:
13328:
13265:instead, the space is
13253:
13208:
13166:
13137:
13016:
12980:
12927:
12859:
12762:
12733:
12700:
12677:
12676:{\displaystyle 2^{-n}}
12647:
12620:
12506:
12483:
12454:
12340:
12249:
12210:
12189:
12105:
12045:
12023:
11995:
11966:
11929:
11897:
11877:
11853:
11784:
11762:
11733:
11693:
11657:
11617:
11610:
11587:
11586:{\displaystyle p=1,2,}
11550:
11529:
11474:
11448:
11421:
11398:
11348:
11279:
11232:
11212:
11184:
11154:
11134:
11114:
11094:
11072:
11044:carrying a compatible
11038:
11005:
10861:
10806:
10764:
10631:
10594:
10552:
10438:
10413:
10391:
10359:
10248:
10209:
10146:
10106:
10048:
10021:
10001:
9974:
9913:
9884:
9864:
9801:
9735:
9712:
9688:
9665:
9620:
9596:
9566:
9527:
9501:
9475:
9455:
9099:
8983:
8959:
8943:
8918:
8871:
8849:
8824:
8802:
8778:
8749:
8707:
8687:
8667:
8638:
8585:
8565:
8518:
8472:
8445:
8416:
8393:
8373:
8323:
8296:
8213:
8133:
8094:
8093:{\displaystyle D(f)=0}
8055:
8035:
7977:
7912:
7866:
7842:
7790:
7736:
7713:
7686:
7514:
7494:
7461:
7439:
7425:for some fixed matrix
7419:
7383:
7363:
7343:
7271:
7251:
7229:
7207:
7175:
7137:
7117:
7097:
7068:
7036:
6980:
6914:
6854:
6819:
6784:
6764:
6742:
6675:
6653:
6633:
6605:
6583:quotient vector spaces
6566:
6517:
6493:
6473:
6449:
6429:
6409:
6385:
6361:
6341:
6319:
6299:
6279:
6155:
5956:orientation preserving
5917:
5840:
5802:
5780:
5752:
5732:
5696:
5643:
5596:
5487:
5455:
5124:
4956:
4933:
4913:
4890:
4868:
4846:
4677:
4650:
4630:
4610:
4535:
4444:
4400:
4376:
4335:
4313:
4242:
4200:
4160:
4159:{\displaystyle b=a/2,}
4124:
4101:
3910:
3836:
3785:
3778:
3701:
3635:
3302:
3196:
2604:; treating them using
2494:is a vector space for
2480:
2415:
2365:
2319:
2238:
2198:
2097:, which is called the
1962:by a scalar belong to
1873:
1827:
1768:
1716:
1595:
1493:
1460:
1434:
1400:
1351:
1314:
1275:
1244:
1243:{\displaystyle s\in F}
1216:
619:A vector space over a
231:. This is the case of
192:if its dimension is a
75:
19727:Row and column spaces
19672:Scalar multiplication
19257:John Wiley & Sons
19235:John Wiley & Sons
19205:Differential geometry
19178:Jost, Jürgen (2005),
19140:John Wiley & Sons
18678:, Royal Irish Academy
18474:Historical references
18268:John Wiley & Sons
17731:Joshi, K. D. (1989),
17344:, Th. XIII.6, p. 349.
17120:, Th. 14.3. See also
16808:, ch. XII.3., p. 335.
16478:Stoll & Wong 1968
16429:, §1.1, Definition 2.
16258:
16223:
16181:
16179:{\displaystyle L^{2}}
16154:
16152:{\displaystyle L^{2}}
16106:
15999:
15938:This axiom is not an
15903:
15799:
15765:
15667:
15569:
15211:
15147:
15112:
15031:
14984:
14932:
14909:
14838:
14818:
14776:
14749:of two matrices, and
14740:
14687:
14667:
14639:
14523:
14467:
14447:
14427:
14370:
14316:
14291:
14221:
14189:
14152:differential operator
14101:
14068:
14037:
14001:
13974:
13972:{\displaystyle f_{n}}
13944:
13942:{\displaystyle g(x),}
13908:
13870:
13776:
13729:
13697:
13579:
13541:
13512:
13378:
13329:
13254:
13209:
13167:
13138:
13017:
12981:
12900:
12860:
12763:
12734:
12701:
12678:
12648:
12646:{\displaystyle 2^{n}}
12621:
12507:
12484:
12455:
12341:
12250:
12211:
12190:
12106:
12046:
12024:
11996:
11994:{\displaystyle V^{*}}
11967:
11930:
11898:
11878:
11854:
11785:
11763:
11734:
11694:
11658:
11611:
11588:
11551:
11530:
11502:
11490:pointwise convergence
11475:
11449:
11447:{\displaystyle f_{i}}
11422:
11399:
11349:
11259:
11233:
11213:
11185:
11155:
11135:
11115:
11095:
11073:
11039:
11006:
10869:corresponding to time
10862:
10807:
10765:
10632:
10595:
10553:
10439:
10414:
10392:
10360:
10249:
10247:{\displaystyle F^{n}}
10210:
10147:
10107:
10049:
10047:{\displaystyle f^{-}}
10022:
10002:
10000:{\displaystyle f^{+}}
9975:
9927:, are fundamental to
9921:Ordered vector spaces
9914:
9885:
9865:
9802:
9736:
9713:
9689:
9666:
9621:
9597:
9567:
9528:
9502:
9476:
9456:
9105:subject to the rules
9100:
8984:
8954:
8944:
8919:
8872:
8850:
8825:
8803:
8779:
8750:
8708:
8688:
8673:of two vector spaces
8668:
8639:
8586:
8566:
8519:
8473:
8471:{\displaystyle V_{i}}
8446:
8417:
8394:
8374:
8324:
8322:{\displaystyle V_{i}}
8297:
8237:Direct sum of modules
8214:
8134:
8095:
8056:
8036:
7978:
7913:
7867:
7843:
7770:
7737:
7714:
7712:{\displaystyle a_{i}}
7687:
7515:
7495:
7462:
7440:
7420:
7384:
7364:
7344:
7277:. The kernel and the
7272:
7252:
7230:
7208:
7176:
7138:
7118:
7098:
7069:
7037:
6981:
6915:
6855:
6820:
6785:
6765:
6743:
6676:
6654:
6634:
6611:, the quotient space
6606:
6585:. Given any subspace
6567:
6518:
6494:
6474:
6450:
6430:
6410:
6386:
6362:
6342:
6320:
6300:
6261:
6254:Quotient vector space
6214:Jordan canonical form
6156:
6067:is an element of the
6012:. Any nonzero vector
5884:
5841:
5803:
5781:
5764:matrix multiplication
5753:
5733:
5676:
5623:
5576:
5488:
5453:
5366:completely classified
5338:1-to-1 correspondence
5169:, the arrow going by
5104:
5092:, and vice versa via
4957:
4934:
4914:
4891:
4869:
4847:
4707:linear transformation
4678:
4676:{\displaystyle e^{x}}
4651:
4631:
4611:
4536:
4445:
4401:
4377:
4336:
4314:
4243:
4201:
4161:
4125:
4102:
3911:
3837:
3835:{\displaystyle (f+g)}
3779:
3702:
3664:
3636:
3303:
3237:(sequences of length
3206:Cartesian coordinates
3197:
2693:-integrable functions
2683:and the new field of
2592:and the inception of
2481:
2479:{\displaystyle F^{n}}
2416:
2366:
2320:
2239:
2199:
1874:
1828:
1769:
1717:
1521:
1494:
1461:
1435:
1401:
1352:
1315:
1276:
1245:
1217:
712:scalar multiplication
704:of these two vectors.
188:). A vector space is
124:complex vector spaces
112:scalar multiplication
47:
19862:Gram–Schmidt process
19814:Gaussian elimination
19211:, pp. xiv+352,
19084:, Berlin, New York:
18924:, Berlin, New York:
18899:, Berlin, New York:
18864:Axiomatic set theory
18841:Macdonald, Ian Grant
18704:Historia Mathematica
18655:, World Scientific,
18571:Historia Mathematica
18324:, Berlin, New York:
18148:, Providence, R.I.:
18124:The Calculus Gallery
18035:, Berlin, New York:
18013:, Berlin, New York:
17760:Lang, Serge (1987),
17711:Jain, M. C. (2001),
17416:, ch. III.1, p. 121.
17272:, Cor. 4.1.2, p. 69.
16964:, ch. 8, p. 135–156.
16285:That is, there is a
16231:
16203:
16163:
16136:
16113:Minkowski inequality
16038:
15980:
15974:equivalence relation
15940:associative property
15923:inner product spaces
15880:
15778:
15741:
15602:
15546:locally free modules
15342:). For example, the
15216:such that for every
15188:
15044:
14993:
14949:
14921:
14850:
14827:
14795:
14753:
14696:
14676:
14656:
14538:
14479:
14456:
14436:
14404:
14368:{\displaystyle p(t)}
14350:
14330:algebra over a field
14300:
14234:
14198:
14178:Algebra over a field
14172:Algebras over fields
14163:decomposes a linear
14132:Schrödinger equation
14112:Gram–Schmidt process
14087:
14054:
14014:
13990:
13981:Taylor approximation
13956:
13921:
13879:
13785:
13746:
13742:. The Hilbert space
13588:
13550:
13539:{\displaystyle f(x)}
13521:
13387:
13338:
13273:
13222:
13176:
13156:
13147:integrable functions
13029:
12992:
12869:
12771:
12761:{\displaystyle p=1:}
12743:
12714:
12687:
12657:
12630:
12516:
12493:
12466:
12349:
12259:
12221:
12200:
12116:
12088:
12033:
12005:
11978:
11947:
11907:
11887:
11867:
11794:
11772:
11743:
11711:
11671:
11632:
11597:
11562:
11540:
11510:
11461:
11431:
11408:
11362:
11256:
11222:
11194:
11164:
11144:
11124:
11104:
11082:
11060:
11028:
10879:
10816:
10778:
10641:
10612:
10562:
10452:
10423:
10401:
10369:
10262:
10231:
10222:inner product spaces
10218:normed vector spaces
10158:
10117:
10084:
10031:
10011:
9984:
9935:
9929:Lebesgue integration
9894:
9874:
9849:
9744:
9722:
9702:
9675:
9634:
9610:
9576:
9540:
9511:
9485:
9465:
9109:
8996:
8970:
8928:
8881:
8859:
8834:
8812:
8792:
8762:
8721:
8697:
8677:
8648:
8612:
8575:
8532:
8485:
8455:
8426:
8406:
8383:
8333:
8306:
8259:
8163:
8157:rank–nullity theorem
8123:
8119:(over a fixed field
8069:
8045:
7987:
7922:
7876:
7856:
7746:
7723:
7696:
7524:
7504:
7471:
7449:
7429:
7396:
7373:
7353:
7284:
7261:
7239:
7217:
7213:consists of vectors
7185:
7153:
7127:
7107:
7078:
7049:
6990:
6924:
6864:
6829:
6794:
6774:
6752:
6685:
6665:
6643:
6615:
6589:
6550:
6507:
6483:
6463:
6439:
6419:
6399:
6375:
6351:
6329:
6309:
6289:
6190:algebraically closed
6118:
5814:
5790:
5770:
5742:
5509:
5477:
5278:. Via the injective
5050:is both one-to-one (
4943:
4923:
4900:
4878:
4856:
4717:
4660:
4640:
4620:
4547:
4460:
4414:
4385:
4345:
4323:
4254:
4216:
4170:
4133:
4111:
3985:
3846:
3814:
3711:
3669:
3607:
3247:
3002:
2563:equivalence relation
2463:
2379:
2329:
2248:
2220:
2137:
1905:linearly independent
1837:
1778:
1726:
1628:
1470:
1444:
1411:
1362:
1325:
1288:
1254:
1228:
1168:
1117:complex vector space
723:and any vector
646:are commonly called
496:Group with operators
439:Complemented lattice
274:Algebraic structures
253:inner product spaces
241:associative algebras
239:, polynomial rings,
198:infinite-dimensional
20048:Concepts in physics
19992:Numerical stability
19872:Multilinear algebra
19847:Inner product space
19697:Linear independence
19053:Griffiths, David J.
19002:Commutative algebra
18947:Projective Geometry
18785:Solid State Physics
18732:(in Italian), Turin
18424:Schaefer, Helmut H.
18407:Functional analysis
17096:, ch. 1, pp. 31–32.
16696:, pp. 268–271.
16132:"Many functions in
16111:is provided by the
16032:triangle inequality
15997:{\displaystyle V/W}
15490:instead of a field
15279:are required to be
14387:, because they are
14046:is commonly called
12626:in which the first
12082:A first example is
12053:Hahn–Banach theorem
12051:). The fundamental
11861:functional analysis
11494:uniform convergence
11486:mode of convergence
11050:close to each other
10068:Inner product space
10064:Normed vector space
10054:the negative part.
9836:functional analysis
9807:This is called the
8956:Commutative diagram
8715:multilinear algebra
7235:that are mapped to
6632:{\displaystyle V/W}
6565:{\displaystyle n-1}
6501:linear combinations
6479:containing the set
6278:(green and yellow).
6236:Basic constructions
5885:The volume of this
5505:, by the following
5111:by its coordinates
3953:functional analysis
2714:Arrows in the plane
2685:functional analysis
2658:Salvatore Pincherle
2637:linear independence
2606:linear combinations
2577:of that relation.
1887:Linear independence
1459:{\displaystyle s=0}
550:Composition algebra
310:Quasigroup and loop
196:. Otherwise, it is
147:physical quantities
19702:Linear combination
19531:Weibel, Charles A.
19486:(1): xviii–xxxiv,
19382:10.1007/bf02242355
19337:Dover Publications
19301:Misner, Charles W.
19274:Mac Lane, Saunders
19209:Dover Publications
18870:, pp. 31–33,
18767:Further references
18615:Grassmann, Hermann
17820:Mac Lane, Saunders
17368:, Lemma III.16.11.
16453:, pp. 99–101.
16253:
16218:
16176:
16149:
16101:
15994:
15898:
15794:
15760:
15684:by a fixed vector
15662:
15584:
15502:-module (that is,
15406:hairy ball theorem
15261:makes the product
15206:
15163:
15125:Related structures
15107:
15040:, whereas forcing
15026:
14979:
14927:
14904:
14833:
14813:
14771:
14735:
14682:
14662:
14634:
14521:{\displaystyle =-}
14518:
14462:
14442:
14422:
14385:algebraic geometry
14383:form the basis of
14365:
14322:
14311:
14286:
14216:
14099:{\displaystyle H,}
14096:
14066:{\displaystyle H,}
14063:
14032:
13996:
13969:
13939:
13903:
13865:
13771:
13732:
13692:
13606:
13574:
13536:
13507:
13414:
13373:
13324:
13249:
13204:
13162:
13133:
13012:
12976:
12855:
12758:
12729:
12699:{\displaystyle 0,}
12696:
12673:
12643:
12616:
12505:{\displaystyle p.}
12502:
12479:
12450:
12388:
12336:
12292:
12245:
12206:
12185:
12101:
12041:
12019:
11991:
11962:
11925:
11893:
11873:
11849:
11812:
11780:
11758:
11729:
11689:
11663:equipped with the
11653:
11618:
11606:
11583:
11546:
11525:
11473:{\displaystyle V,}
11470:
11444:
11420:{\displaystyle V.}
11417:
11394:
11344:
11314:
11228:
11208:
11180:
11150:
11130:
11110:
11090:
11068:
11034:
11001:
10873:special relativity
10857:
10802:
10760:
10627:
10590:
10548:
10434:
10409:
10387:
10355:
10244:
10205:
10142:
10102:
10044:
10017:
9997:
9970:
9909:
9880:
9860:
9809:universal property
9797:
9734:{\displaystyle g:}
9731:
9708:
9687:{\displaystyle u,}
9684:
9661:
9616:
9592:
9562:
9523:
9497:
9471:
9451:
9449:
9095:
8982:{\displaystyle g,}
8979:
8960:
8939:
8914:
8867:
8845:
8820:
8798:
8774:
8745:
8703:
8683:
8663:
8634:
8581:
8561:
8550:
8514:
8503:
8468:
8441:
8412:
8389:
8369:
8319:
8292:
8291:
8279:
8209:
8129:
8090:
8051:
8031:
7973:
7908:
7862:
7838:
7735:{\displaystyle x,}
7732:
7709:
7682:
7510:
7490:
7457:
7435:
7415:
7379:
7359:
7339:
7267:
7247:
7225:
7203:
7171:
7133:
7113:
7093:
7064:
7045:the difference of
7032:
6976:
6910:
6850:
6815:
6780:
6760:
6738:
6671:
6649:
6629:
6601:
6562:
6513:
6489:
6469:
6445:
6425:
6405:
6381:
6357:
6337:
6315:
6305:of a vector space
6295:
6280:
6151:
6108:having eigenvalue
6071:of the difference
5918:
5836:
5798:
5776:
5762:, or by using the
5751:{\textstyle \sum }
5748:
5728:
5483:
5456:
5181:is negative), and
5125:
4955:{\displaystyle F.}
4952:
4929:
4912:{\displaystyle V,}
4909:
4886:
4864:
4842:
4840:
4673:
4646:
4626:
4606:
4531:
4440:
4396:
4372:
4331:
4309:
4303:
4238:
4196:
4156:
4123:{\displaystyle a,}
4120:
4097:
4095:
3906:
3832:
3786:
3774:
3697:
3631:
3298:
3192:
3190:
2476:
2411:
2361:
2315:
2234:
2204:of a vector space
2194:
1995:of a vector space
1942:of a vector space
1869:
1823:
1764:
1712:
1600:Linear combination
1596:
1489:
1456:
1430:
1396:
1347:
1310:
1271:
1240:
1212:
1135:vector space over
918:of vector addition
867:of vector addition
840:of vector addition
805:of vector addition
733:another vector in
686:a third vector in
190:finite-dimensional
120:Real vector spaces
76:
20033:
20032:
19900:Geometric algebra
19857:Kronecker product
19692:Linear projection
19677:Vector projection
19544:978-0-521-55987-4
19502:10.1109/30.125072
19464:978-0-13-535732-3
19442:978-0-412-10800-6
19346:978-0-486-43235-9
19324:978-0-7167-0344-0
19317:, W. H. Freeman,
19293:978-0-387-98403-2
19266:978-0-471-18117-0
19244:978-0-471-15496-9
19218:978-0-486-66721-8
19193:978-3-540-25907-7
19171:978-0-387-94087-8
19095:978-0-387-90093-3
19070:978-0-13-124405-4
19045:978-0-412-60610-6
19015:978-0-387-94269-8
18960:978-0-387-96532-1
18935:978-3-540-64241-1
18918:Bourbaki, Nicolas
18910:978-3-540-64243-5
18893:Bourbaki, Nicolas
18877:978-0-8218-5026-8
18821:978-0-201-09394-0
18796:978-0-03-083993-1
18750:Peano, G. (1901)
18662:978-981-12-4103-1
18642:978-0-8218-2031-5
18553:Bourbaki, Nicolas
18536:Bellavitis, Giuso
18441:978-1-4612-7155-0
18363:978-0-486-48123-4
18353:2027/uc1.b4250788
18335:978-0-387-94001-4
18310:978-0-201-14179-5
18277:978-0-471-50459-7
18252:978-0-471-85824-9
18231:978-0-88385-031-2
18213:978-0-201-59619-9
18195:978-0-387-98485-8
18177:978-0-534-17094-3
18159:978-0-8218-0772-9
18137:978-0-691-09565-3
18115:978-0-534-10050-6
18097:978-0-486-69193-0
18046:978-0-387-97894-9
18024:978-3-540-41129-1
18007:Bourbaki, Nicolas
17999:978-3-540-13627-9
17982:Bourbaki, Nicolas
17965:978-3-540-56799-8
17931:978-0-8176-4840-4
17911:978-0-8247-9144-5
17893:978-0-387-24766-3
17857:978-0-89871-454-8
17833:978-0-8218-1646-2
17804:978-0-387-95385-4
17779:978-1-4757-1949-9
17753:978-1-119-45592-9
17724:978-0-8493-0978-6
17704:978-0-8176-4687-5
17666:978-0-387-71568-1
17640:978-0-8247-8419-5
17618:978-0-89871-510-1
16868:, Example IV.2.6.
16508:, p. 10–11;
16496:, pp. 41–42.
16322:Husemoller (1994)
16117:almost everywhere
16015:uniform structure
15892:
15422:division algebras
15247:, the projection
15171:topological space
15038:symmetric algebra
14930:{\displaystyle n}
14836:{\displaystyle V}
14781:endowed with the
14685:{\displaystyle n}
14665:{\displaystyle n}
14530:anticommutativity
14465:{\displaystyle y}
14445:{\displaystyle x}
14326:bilinear operator
14136:quantum mechanics
14048:Fourier expansion
13999:{\displaystyle f}
13915:complex conjugate
13901:
13853:
13802:
13796:
13591:
13403:
13390:
13127:
13024:Lebesgue integral
12739:but does not for
12706:converges to the
12436:
12429:
12379:
12334:
12317:
12283:
12209:{\displaystyle p}
12084:the vector space
11876:{\displaystyle 1}
11797:
11656:{\displaystyle ,}
11549:{\displaystyle p}
11427:For example, the
11299:
11298:
11292:
11245:one can consider
11231:{\displaystyle F}
11153:{\displaystyle F}
11133:{\displaystyle a}
11113:{\displaystyle V}
11037:{\displaystyle V}
10772:positive definite
10227:Coordinate space
10200:
10020:{\displaystyle f}
9883:{\displaystyle n}
9711:{\displaystyle f}
9619:{\displaystyle X}
9474:{\displaystyle f}
9398:
9388:
9295:
9285:
9236:
9235: is a scalar
9228:
9227: where
9224:
9221:
9187:
9181:
9153:
9143:
8801:{\displaystyle g}
8757:Cartesian product
8706:{\displaystyle W}
8686:{\displaystyle V}
8584:{\displaystyle I}
8535:
8488:
8415:{\displaystyle I}
8392:{\displaystyle i}
8264:
8132:{\displaystyle F}
8054:{\displaystyle c}
7865:{\displaystyle f}
7833:
7719:are functions in
7671:
7618:
7571:
7513:{\displaystyle A}
7438:{\displaystyle A}
7389:, respectively.
7382:{\displaystyle W}
7362:{\displaystyle V}
7349:are subspaces of
7270:{\displaystyle W}
7136:{\displaystyle W}
7116:{\displaystyle W}
6783:{\displaystyle V}
6674:{\displaystyle W}
6652:{\displaystyle V}
6538:respectively. If
6516:{\displaystyle S}
6492:{\displaystyle S}
6472:{\displaystyle V}
6448:{\displaystyle S}
6428:{\displaystyle V}
6408:{\displaystyle V}
6384:{\displaystyle V}
6360:{\displaystyle V}
6318:{\displaystyle V}
6298:{\displaystyle W}
6266:(blue, thick) in
6114:is equivalent to
6082:(where Id is the
5779:{\displaystyle A}
5486:{\displaystyle A}
5388:is isomorphic to
5269:dual vector space
5143:coordinate system
5137:departing at the
4932:{\displaystyle a}
4649:{\displaystyle b}
4629:{\displaystyle a}
3948:differentiability
3626:
3496:for real numbers
3208:of its endpoint.
2575:equivalence class
2559:Bellavitis (1833)
2539:analytic geometry
2451:coordinate vector
2133:Consider a basis
2118:, depends on the
1907:if no element of
1620:is an element of
1608:of elements of a
1159:endomorphism ring
1149:ring homomorphism
1109:real vector space
1101:
1100:
700:, and called the
609:
608:
202:infinite cardinal
143:Euclidean vectors
16:(Redirected from
20075:
20023:
20022:
19905:Exterior algebra
19842:Hadamard product
19759:
19747:Linear equations
19618:
19611:
19604:
19595:
19594:
19590:
19564:
19526:
19525:
19524:
19518:
19512:, archived from
19495:
19477:
19467:
19445:
19433:Chapman and Hall
19429:Chapman and Hall
19426:
19412:
19400:
19376:(3–4): 281–292,
19366:Strassen, Volker
19357:
19327:
19296:
19269:
19247:
19232:
19221:
19196:
19174:
19152:
19149:978-0470-88861-2
19130:
19121:
19098:
19073:
19048:
19036:Chapman and Hall
19026:
18993:
18963:
18938:
18913:
18888:
18861:
18851:
18832:
18799:
18788:
18779:Mermin, N. David
18760:Internet Archive
18747:
18741:
18733:
18721:
18720:
18697:
18679:
18665:
18645:
18629:Extension Theory
18624:
18610:
18596:
18587:
18560:
18547:
18531:
18522:Bolzano, Bernard
18517:
18490:
18468:
18458:Treves, François
18453:
18419:
18401:
18374:
18355:
18338:
18313:
18288:
18255:
18234:
18216:
18198:
18180:
18162:
18140:
18118:
18100:
18082:
18072:Choquet, Gustave
18067:
18049:
18027:
18002:
17968:
17943:
17942:, Academic Press
17934:
17914:
17896:
17871:
17860:
17836:
17815:
17782:
17756:
17736:
17727:
17707:
17681:
17669:
17649:Abstract algebra
17643:
17632:
17621:
17596:
17573:
17567:
17561:
17555:
17549:
17543:
17537:
17531:
17525:
17519:
17513:
17507:
17501:
17495:
17489:
17483:
17477:
17471:
17465:
17459:
17453:
17447:
17441:
17438:Varadarajan 1974
17435:
17429:
17423:
17417:
17411:
17405:
17399:
17393:
17387:
17381:
17375:
17369:
17363:
17357:
17351:
17345:
17339:
17333:
17327:
17321:
17315:
17309:
17303:
17297:
17291:
17285:
17279:
17273:
17267:
17261:
17255:
17249:
17243:
17237:
17231:
17225:
17219:
17213:
17207:
17201:
17191:
17185:
17179:
17173:
17167:
17161:
17155:
17149:
17143:
17137:
17131:
17125:
17115:
17109:
17103:
17097:
17091:
17085:
17079:
17073:
17067:
17061:
17055:
17049:
17043:
17037:
17031:
17025:
17019:
17013:
17007:
17001:
16995:
16989:
16988:, ch. 8, p. 140.
16983:
16977:
16971:
16965:
16959:
16953:
16947:
16941:
16935:
16929:
16923:
16917:
16911:
16905:
16899:
16893:
16887:
16881:
16875:
16869:
16863:
16857:
16851:
16845:
16839:
16833:
16827:
16821:
16815:
16809:
16803:
16797:
16791:
16785:
16779:
16773:
16763:
16757:
16747:
16741:
16731:
16725:
16715:
16709:
16703:
16697:
16691:
16685:
16679:
16673:
16667:
16661:
16655:
16649:
16643:
16637:
16631:
16625:
16619:
16613:
16607:
16601:
16595:
16589:
16583:
16577:
16567:
16561:
16551:
16545:
16535:
16529:
16523:
16517:
16503:
16497:
16491:
16485:
16475:
16466:
16460:
16454:
16448:
16442:
16436:
16430:
16424:
16418:
16412:
16406:
16392:
16386:
16380:
16374:
16368:
16362:
16356:
16350:
16344:
16329:
16310:
16304:
16302:
16283:
16277:
16270:
16264:
16262:
16260:
16259:
16254:
16243:
16242:
16227:
16225:
16224:
16219:
16197:
16191:
16185:
16183:
16182:
16177:
16175:
16174:
16158:
16156:
16155:
16150:
16148:
16147:
16130:
16124:
16110:
16108:
16107:
16102:
16100:
16099:
16081:
16080:
16062:
16061:
16028:
16022:
16011:
16005:
16003:
16001:
16000:
15995:
15990:
15966:
15960:
15949:
15943:
15936:
15930:
15915:
15909:
15907:
15905:
15904:
15899:
15894:
15893:
15885:
15874:
15839:projective space
15813:
15803:
15801:
15800:
15795:
15793:
15785:
15769:
15767:
15766:
15761:
15759:
15751:
15736:
15725:
15703:
15693:
15671:
15669:
15668:
15663:
15658:
15650:
15639:
15631:
15574:(light blue) in
15562:Projective space
15451:cotangent bundle
15395:
15366:
15341:
15327:
15270:
15260:
15238:
15215:
15213:
15212:
15207:
15160:
15119:exterior algebra
15116:
15114:
15113:
15108:
15106:
15105:
15100:
15091:
15090:
15085:
15073:
15072:
15067:
15058:
15057:
15052:
15035:
15033:
15032:
15027:
15022:
15021:
15016:
15007:
15006:
15001:
14988:
14986:
14985:
14980:
14978:
14977:
14972:
14963:
14962:
14957:
14939:distributive law
14936:
14934:
14933:
14928:
14913:
14911:
14910:
14905:
14900:
14899:
14894:
14879:
14878:
14873:
14864:
14863:
14858:
14842:
14840:
14839:
14834:
14822:
14820:
14819:
14814:
14780:
14778:
14777:
14772:
14767:
14766:
14761:
14744:
14742:
14741:
14736:
14691:
14689:
14688:
14683:
14671:
14669:
14668:
14663:
14643:
14641:
14640:
14635:
14527:
14525:
14524:
14519:
14471:
14469:
14468:
14463:
14451:
14449:
14448:
14443:
14431:
14429:
14428:
14425:{\displaystyle }
14423:
14374:
14372:
14371:
14366:
14320:
14318:
14317:
14312:
14307:
14295:
14293:
14292:
14287:
14261:
14241:
14225:
14223:
14222:
14217:
14168:
14165:compact operator
14161:spectral theorem
14105:
14103:
14102:
14097:
14072:
14070:
14069:
14064:
14041:
14039:
14038:
14035:{\displaystyle }
14033:
14005:
14003:
14002:
13997:
13978:
13976:
13975:
13970:
13968:
13967:
13948:
13946:
13945:
13940:
13912:
13910:
13909:
13904:
13902:
13897:
13883:
13874:
13872:
13871:
13866:
13854:
13849:
13835:
13821:
13820:
13800:
13794:
13780:
13778:
13777:
13772:
13758:
13757:
13701:
13699:
13698:
13693:
13685:
13667:
13666:
13661:
13657:
13647:
13646:
13616:
13615:
13605:
13583:
13581:
13580:
13575:
13564:
13563:
13545:
13543:
13542:
13537:
13516:
13514:
13513:
13508:
13500:
13482:
13481:
13476:
13472:
13462:
13461:
13440:
13439:
13424:
13423:
13413:
13401:
13382:
13380:
13379:
13374:
13363:
13362:
13353:
13352:
13333:
13331:
13330:
13325:
13317:
13316:
13298:
13297:
13285:
13284:
13263:Riemann integral
13258:
13256:
13255:
13250:
13236:
13235:
13213:
13211:
13210:
13205:
13194:
13193:
13171:
13169:
13168:
13163:
13142:
13140:
13139:
13134:
13129:
13128:
13120:
13118:
13114:
13113:
13095:
13094:
13089:
13071:
13066:
13065:
13047:
13046:
13021:
13019:
13018:
13013:
13011:
12985:
12983:
12982:
12977:
12969:
12968:
12953:
12952:
12940:
12939:
12926:
12925:
12924:
12914:
12896:
12895:
12886:
12885:
12880:
12864:
12862:
12861:
12856:
12845:
12844:
12820:
12819:
12798:
12797:
12788:
12787:
12782:
12767:
12765:
12764:
12759:
12738:
12736:
12735:
12730:
12705:
12703:
12702:
12697:
12682:
12680:
12679:
12674:
12672:
12671:
12652:
12650:
12649:
12644:
12642:
12641:
12625:
12623:
12622:
12617:
12612:
12608:
12589:
12588:
12567:
12566:
12551:
12550:
12530:
12529:
12524:
12511:
12509:
12508:
12503:
12488:
12486:
12485:
12480:
12478:
12477:
12459:
12457:
12456:
12451:
12437:
12434:
12431:
12430:
12422:
12420:
12416:
12415:
12414:
12409:
12403:
12402:
12393:
12387:
12369:
12368:
12359:
12345:
12343:
12342:
12337:
12335:
12332:
12318:
12315:
12312:
12307:
12306:
12297:
12291:
12279:
12278:
12269:
12254:
12252:
12251:
12246:
12215:
12213:
12212:
12207:
12194:
12192:
12191:
12186:
12184:
12180:
12173:
12172:
12154:
12153:
12141:
12140:
12123:
12110:
12108:
12107:
12102:
12100:
12099:
12075:, introduced by
12056:
12050:
12048:
12047:
12042:
12040:
12028:
12026:
12025:
12020:
12018:
12000:
11998:
11997:
11992:
11990:
11989:
11971:
11969:
11968:
11963:
11934:
11932:
11931:
11926:
11921:
11920:
11915:
11902:
11900:
11899:
11894:
11882:
11880:
11879:
11874:
11858:
11856:
11855:
11850:
11842:
11837:
11829:
11828:
11823:
11817:
11811:
11789:
11787:
11786:
11781:
11779:
11767:
11765:
11764:
11759:
11757:
11756:
11751:
11738:
11736:
11735:
11732:{\displaystyle }
11730:
11698:
11696:
11695:
11692:{\displaystyle }
11690:
11662:
11660:
11659:
11654:
11615:
11613:
11612:
11607:
11592:
11590:
11589:
11584:
11555:
11553:
11552:
11547:
11534:
11532:
11531:
11526:
11524:
11523:
11518:
11479:
11477:
11476:
11471:
11453:
11451:
11450:
11445:
11443:
11442:
11426:
11424:
11423:
11418:
11403:
11401:
11400:
11395:
11387:
11386:
11374:
11373:
11353:
11351:
11350:
11345:
11343:
11342:
11324:
11323:
11313:
11296:
11290:
11289:
11288:
11278:
11273:
11249:of vectors. The
11237:
11235:
11234:
11229:
11217:
11215:
11214:
11209:
11204:
11189:
11187:
11186:
11181:
11179:
11171:
11159:
11157:
11156:
11151:
11139:
11137:
11136:
11131:
11119:
11117:
11116:
11111:
11099:
11097:
11096:
11091:
11089:
11077:
11075:
11074:
11069:
11067:
11043:
11041:
11040:
11035:
11010:
11008:
11007:
11002:
10997:
10996:
10987:
10986:
10974:
10973:
10964:
10963:
10951:
10950:
10941:
10940:
10928:
10927:
10918:
10917:
10899:
10894:
10889:
10866:
10864:
10863:
10858:
10823:
10811:
10809:
10808:
10803:
10798:
10793:
10788:
10769:
10767:
10766:
10761:
10756:
10755:
10746:
10745:
10733:
10732:
10723:
10722:
10710:
10709:
10700:
10699:
10687:
10686:
10677:
10676:
10661:
10656:
10651:
10636:
10634:
10633:
10628:
10626:
10625:
10620:
10599:
10597:
10596:
10591:
10580:
10572:
10557:
10555:
10554:
10549:
10544:
10539:
10534:
10526:
10521:
10516:
10508:
10504:
10500:
10492:
10467:
10459:
10443:
10441:
10440:
10435:
10430:
10418:
10416:
10415:
10410:
10408:
10396:
10394:
10393:
10388:
10383:
10382:
10377:
10364:
10362:
10361:
10356:
10351:
10350:
10341:
10340:
10322:
10321:
10312:
10311:
10299:
10291:
10280:
10272:
10253:
10251:
10250:
10245:
10243:
10242:
10224:, respectively.
10215:
10214:
10212:
10211:
10206:
10201:
10196:
10188:
10180:
10175:
10170:
10165:
10152:
10151:
10149:
10148:
10143:
10135:
10127:
10111:
10109:
10108:
10103:
10101:
10096:
10091:
10053:
10051:
10050:
10045:
10043:
10042:
10026:
10024:
10023:
10018:
10006:
10004:
10003:
9998:
9996:
9995:
9979:
9977:
9976:
9971:
9966:
9965:
9953:
9952:
9918:
9916:
9915:
9910:
9908:
9907:
9902:
9889:
9887:
9886:
9881:
9869:
9867:
9866:
9861:
9839:
9806:
9804:
9803:
9798:
9790:
9782:
9765:
9757:
9740:
9738:
9737:
9732:
9717:
9715:
9714:
9709:
9693:
9691:
9690:
9685:
9670:
9668:
9667:
9662:
9625:
9623:
9622:
9617:
9601:
9599:
9598:
9593:
9591:
9583:
9571:
9569:
9568:
9563:
9558:
9550:
9532:
9530:
9529:
9524:
9506:
9504:
9503:
9498:
9480:
9478:
9477:
9472:
9460:
9458:
9457:
9452:
9450:
9447:
9446:
9441:
9440:
9435:
9426:
9418:
9417:
9412:
9403:
9396:
9386:
9382:
9381:
9376:
9367:
9366:
9361:
9349:
9341:
9340:
9338:
9330:
9329:
9324:
9315:
9307:
9306:
9301:
9293:
9283:
9282:
9271:
9270:
9265:
9256:
9255:
9250:
9237:
9234:
9229:
9226:
9222:
9219:
9217:
9209:
9192:
9185:
9179:
9178:
9167:
9151:
9141:
9137:
9129:
9104:
9102:
9101:
9096:
9091:
9090:
9085:
9076:
9075:
9070:
9055:
9054:
9049:
9040:
9039:
9034:
9025:
9024:
9019:
9010:
9009:
9004:
8988:
8986:
8985:
8980:
8948:
8946:
8945:
8940:
8935:
8923:
8921:
8920:
8915:
8910:
8902:
8888:
8876:
8874:
8873:
8868:
8866:
8854:
8852:
8851:
8846:
8841:
8829:
8827:
8826:
8821:
8819:
8807:
8805:
8804:
8799:
8783:
8781:
8780:
8775:
8754:
8752:
8751:
8746:
8712:
8710:
8709:
8704:
8692:
8690:
8689:
8684:
8672:
8670:
8669:
8664:
8643:
8641:
8640:
8635:
8627:
8626:
8590:
8588:
8587:
8582:
8570:
8568:
8567:
8562:
8560:
8559:
8549:
8523:
8521:
8520:
8515:
8513:
8512:
8502:
8477:
8475:
8474:
8469:
8467:
8466:
8450:
8448:
8447:
8442:
8440:
8439:
8434:
8421:
8419:
8418:
8413:
8398:
8396:
8395:
8390:
8378:
8376:
8375:
8370:
8368:
8367:
8356:
8352:
8351:
8346:
8328:
8326:
8325:
8320:
8318:
8317:
8301:
8299:
8298:
8293:
8290:
8289:
8288:
8278:
8218:
8216:
8215:
8210:
8173:
8141:abelian category
8138:
8136:
8135:
8130:
8111:
8105:
8099:
8097:
8096:
8091:
8060:
8058:
8057:
8052:
8040:
8038:
8037:
8032:
8030:
8029:
8011:
8010:
7982:
7980:
7979:
7974:
7972:
7971:
7959:
7958:
7946:
7945:
7917:
7915:
7914:
7909:
7907:
7906:
7891:
7890:
7871:
7869:
7868:
7863:
7852:of the function
7847:
7845:
7844:
7839:
7834:
7832:
7831:
7830:
7817:
7813:
7812:
7802:
7800:
7799:
7789:
7784:
7741:
7739:
7738:
7733:
7718:
7716:
7715:
7710:
7708:
7707:
7691:
7689:
7688:
7683:
7672:
7670:
7669:
7668:
7655:
7651:
7650:
7640:
7638:
7637:
7619:
7617:
7616:
7615:
7602:
7598:
7597:
7587:
7585:
7584:
7572:
7570:
7562:
7554:
7552:
7551:
7536:
7535:
7519:
7517:
7516:
7511:
7499:
7497:
7496:
7491:
7489:
7481:
7466:
7464:
7463:
7458:
7456:
7444:
7442:
7441:
7436:
7424:
7422:
7421:
7416:
7414:
7403:
7388:
7386:
7385:
7380:
7368:
7366:
7365:
7360:
7348:
7346:
7345:
7340:
7329:
7318:
7276:
7274:
7273:
7268:
7256:
7254:
7253:
7248:
7246:
7234:
7232:
7231:
7226:
7224:
7212:
7210:
7209:
7204:
7181:of a linear map
7180:
7178:
7177:
7172:
7142:
7140:
7139:
7134:
7122:
7120:
7119:
7114:
7102:
7100:
7099:
7094:
7092:
7091:
7086:
7073:
7071:
7070:
7065:
7063:
7062:
7057:
7041:
7039:
7038:
7033:
7025:
7024:
7019:
7004:
7003:
6998:
6985:
6983:
6982:
6977:
6966:
6940:
6919:
6917:
6916:
6911:
6903:
6899:
6898:
6897:
6892:
6883:
6882:
6877:
6859:
6857:
6856:
6851:
6843:
6842:
6837:
6824:
6822:
6821:
6816:
6808:
6807:
6802:
6789:
6787:
6786:
6781:
6769:
6767:
6766:
6761:
6759:
6747:
6745:
6744:
6739:
6725:
6717:
6709:
6692:
6680:
6678:
6677:
6672:
6658:
6656:
6655:
6650:
6638:
6636:
6635:
6630:
6625:
6610:
6608:
6607:
6602:
6571:
6569:
6568:
6563:
6522:
6520:
6519:
6514:
6498:
6496:
6495:
6490:
6478:
6476:
6475:
6470:
6454:
6452:
6451:
6446:
6434:
6432:
6431:
6426:
6414:
6412:
6411:
6406:
6390:
6388:
6387:
6382:
6366:
6364:
6363:
6358:
6346:
6344:
6343:
6338:
6336:
6324:
6322:
6321:
6316:
6304:
6302:
6301:
6296:
6273:
6231:
6221:
6207:
6201:
6188:
6182:
6176:
6166:
6160:
6158:
6157:
6152:
6113:
6107:
6101:
6095:
6081:
6066:
6061:. Equivalently,
6060:
6051:
6041:
6035:
6017:
6011:
6000:
5994:
5988:
5945:
5939:
5930:
5915:
5906:
5897:
5877:
5859:
5853:
5845:
5843:
5842:
5837:
5832:
5821:
5807:
5805:
5804:
5799:
5797:
5785:
5783:
5782:
5777:
5757:
5755:
5754:
5749:
5737:
5735:
5734:
5729:
5724:
5720:
5719:
5718:
5709:
5708:
5695:
5690:
5666:
5665:
5656:
5655:
5642:
5637:
5619:
5618:
5609:
5608:
5595:
5590:
5564:
5563:
5545:
5544:
5532:
5531:
5516:
5504:
5498:
5492:
5490:
5489:
5484:
5472:
5466:
5454:A typical matrix
5431:
5425:
5419:
5413:
5407:
5393:
5387:
5381:
5363:
5357:
5351:
5345:
5335:
5324:
5318:
5304:
5299:Once a basis of
5291:
5277:
5265:
5259:
5253:
5241:
5229:
5211:
5198:
5192:
5186:
5180:
5174:
5168:
5156:
5150:
5141:of some (fixed)
5136:
5122:
5116:
5110:
5097:
5091:
5085:
5079:
5069:
5063:
5049:
5044:. Equivalently,
5039:
5021:
5000:
4983:
4970:is a linear map
4961:
4959:
4958:
4953:
4938:
4936:
4935:
4930:
4918:
4916:
4915:
4910:
4895:
4893:
4892:
4887:
4885:
4873:
4871:
4870:
4865:
4863:
4851:
4849:
4848:
4843:
4841:
4834:
4807:
4780:
4763:
4742:
4734:
4682:
4680:
4679:
4674:
4672:
4671:
4655:
4653:
4652:
4647:
4635:
4633:
4632:
4627:
4615:
4613:
4612:
4607:
4602:
4601:
4580:
4579:
4540:
4538:
4537:
4532:
4500:
4499:
4475:
4474:
4449:
4447:
4446:
4441:
4421:
4405:
4403:
4402:
4397:
4395:
4381:
4379:
4378:
4373:
4340:
4338:
4337:
4332:
4330:
4318:
4316:
4315:
4310:
4308:
4307:
4247:
4245:
4244:
4239:
4234:
4226:
4205:
4203:
4202:
4197:
4192:
4165:
4163:
4162:
4157:
4149:
4129:
4127:
4126:
4121:
4106:
4104:
4103:
4098:
4096:
4052:
4045:
3999:
3992:
3991:
3959:Linear equations
3937:
3919:
3915:
3913:
3912:
3907:
3841:
3839:
3838:
3833:
3810:is the function
3809:
3803:
3797:
3791:
3783:
3781:
3780:
3775:
3706:
3704:
3703:
3698:
3696:
3688:
3646:
3640:
3638:
3637:
3632:
3627:
3622:
3614:
3602:
3596:
3590:
3584:
3575:
3564:field extensions
3562:More generally,
3554:
3542:
3525:
3519:
3513:
3507:
3501:
3495:
3460:
3416:
3410:
3404:
3395:
3385:
3364:
3357:
3347:
3341:
3333:coordinate space
3330:
3324:
3318:
3307:
3305:
3304:
3299:
3294:
3293:
3275:
3274:
3262:
3261:
3242:
3234:
3227:
3221:
3212:Coordinate space
3201:
3199:
3198:
3193:
3191:
3128:
3127:
3115:
3114:
3102:
3101:
3089:
3088:
3066:
3065:
3053:
3052:
3034:
3033:
3021:
3020:
2997:
2981:
2977:
2973:
2969:
2957:
2951:
2940:
2930:
2923:
2917:
2911:
2902:
2892:
2883:
2877:
2869:. It is denoted
2868:
2862:
2852:
2846:
2840:
2831:
2813:
2807:
2778:
2771:
2762:
2750:
2744:
2738:
2726:
2571:Euclidean vector
2535:Pierre de Fermat
2501:
2493:
2489:
2485:
2483:
2482:
2477:
2475:
2474:
2458:
2444:
2440:
2430:
2420:
2418:
2417:
2412:
2410:
2409:
2391:
2390:
2374:
2370:
2368:
2367:
2362:
2360:
2359:
2341:
2340:
2324:
2322:
2321:
2316:
2311:
2310:
2305:
2299:
2298:
2280:
2279:
2274:
2268:
2267:
2255:
2244:may be written
2243:
2241:
2240:
2235:
2227:
2215:
2211:
2207:
2203:
2201:
2200:
2195:
2190:
2189:
2184:
2169:
2168:
2163:
2154:
2153:
2148:
2128:rational numbers
2113:
2072:
2058:
2054:
2044:
2041:, one says that
2040:
2036:
2030:
2026:
2022:
2018:
2014:
2010:
1998:
1994:
1973:
1969:
1965:
1961:
1957:
1949:
1945:
1941:
1922:
1918:
1914:
1910:
1902:
1898:
1894:
1878:
1876:
1875:
1870:
1868:
1867:
1849:
1848:
1832:
1830:
1829:
1824:
1813:
1812:
1807:
1792:
1791:
1786:
1773:
1771:
1770:
1765:
1757:
1756:
1738:
1737:
1721:
1719:
1718:
1713:
1708:
1707:
1702:
1696:
1695:
1677:
1676:
1671:
1665:
1664:
1652:
1651:
1646:
1640:
1639:
1623:
1619:
1615:
1611:
1607:
1593:
1569:
1543:
1533:
1527:
1498:
1496:
1495:
1490:
1485:
1477:
1465:
1463:
1462:
1457:
1439:
1437:
1436:
1431:
1429:
1421:
1405:
1403:
1402:
1397:
1392:
1381:
1356:
1354:
1353:
1348:
1343:
1335:
1319:
1317:
1316:
1311:
1306:
1298:
1280:
1278:
1277:
1272:
1261:
1249:
1247:
1246:
1241:
1221:
1219:
1218:
1213:
1205:
1191:
1183:
1175:
1156:
1138:
1129:
1127:
1122:
1097:
1062:
1024:
1016:
1012:
995:
967:
953:
945:additive inverse
941:
930:
916:Inverse elements
909:
899:
879:
865:Identity element
859:
832:
791:
787:
783:
779:
775:
771:
765:
759:
745:
736:
732:
728:
722:
718:
699:
689:
685:
681:
675:
653:
645:
636:binary operation
634:together with a
633:
625:
601:
594:
587:
376:Commutative ring
305:Rack and quandle
270:
269:
237:field extensions
235:, which include
224:as a dimension.
210:polynomial rings
73:
59:
53:
21:
20083:
20082:
20078:
20077:
20076:
20074:
20073:
20072:
20038:
20037:
20034:
20029:
20011:
19973:
19929:
19866:
19818:
19760:
19751:
19717:Change of basis
19707:Multilinear map
19645:
19627:
19622:
19575:
19572:
19567:
19545:
19522:
19520:
19516:
19493:10.1.1.318.4292
19475:
19465:
19443:
19405:Spivak, Michael
19347:
19325:
19294:
19284:Springer-Verlag
19267:
19245:
19219:
19201:Kreyszig, Erwin
19194:
19184:Springer-Verlag
19172:
19162:Springer-Verlag
19150:
19119:10.2307/2035388
19096:
19086:Springer-Verlag
19078:Halmos, Paul R.
19071:
19046:
19016:
19006:Springer-Verlag
18998:Eisenbud, David
18983:10.2307/2320587
18961:
18951:Springer-Verlag
18936:
18926:Springer-Verlag
18911:
18901:Springer-Verlag
18878:
18859:
18822:
18797:
18769:
18764:
18735:
18734:
18726:Peano, Giuseppe
18663:
18643:
18546:, Verona: 53–61
18488:
18476:
18471:
18442:
18417:
18390:
18364:
18336:
18326:Springer-Verlag
18311:
18278:
18260:Kreyszig, Erwin
18253:
18239:Kreyszig, Erwin
18232:
18214:
18196:
18178:
18170:, Brooks-Cole,
18160:
18138:
18116:
18098:
18055:"Tangent plane"
18053:BSE-3 (2001) ,
18047:
18037:Springer-Verlag
18025:
18015:Springer-Verlag
18000:
17990:Springer-Verlag
17976:
17971:
17966:
17956:Springer-Verlag
17932:
17912:
17894:
17884:Springer-Verlag
17858:
17834:
17805:
17780:
17754:
17725:
17705:
17674:Halmos, Paul R.
17667:
17641:
17619:
17586:
17581:
17576:
17568:
17564:
17556:
17552:
17544:
17540:
17532:
17528:
17520:
17516:
17508:
17504:
17496:
17492:
17484:
17480:
17472:
17468:
17460:
17456:
17448:
17444:
17436:
17432:
17424:
17420:
17412:
17408:
17400:
17396:
17388:
17384:
17376:
17372:
17364:
17360:
17352:
17348:
17340:
17336:
17328:
17324:
17316:
17312:
17304:
17300:
17292:
17288:
17280:
17276:
17268:
17264:
17256:
17252:
17244:
17240:
17232:
17228:
17220:
17216:
17208:
17204:
17192:
17188:
17180:
17176:
17168:
17164:
17160:, ch. 2, p. 48.
17156:
17152:
17144:
17140:
17132:
17128:
17116:
17112:
17104:
17100:
17092:
17088:
17080:
17076:
17068:
17064:
17060:, ch. 2, p. 48.
17056:
17052:
17044:
17040:
17036:, ch. 3, p. 64.
17032:
17028:
17020:
17016:
17012:, ch. 1, p. 35.
17008:
17004:
17000:, ch. 1, p. 29.
16996:
16992:
16984:
16980:
16974:& Lang 1987
16972:
16968:
16960:
16956:
16948:
16944:
16936:
16932:
16924:
16920:
16912:
16908:
16900:
16896:
16892:, p. 28, Ex. 9.
16888:
16884:
16876:
16872:
16864:
16860:
16852:
16848:
16840:
16836:
16832:, ch. 2, p. 45.
16828:
16824:
16816:
16812:
16804:
16800:
16792:
16788:
16780:
16776:
16764:
16760:
16748:
16744:
16732:
16728:
16716:
16712:
16704:
16700:
16692:
16688:
16680:
16676:
16668:
16664:
16656:
16652:
16644:
16640:
16632:
16628:
16622:Bellavitis 1833
16620:
16616:
16608:
16604:
16596:
16592:
16584:
16580:
16568:
16564:
16552:
16548:
16536:
16532:
16524:
16520:
16504:
16500:
16492:
16488:
16476:
16469:
16461:
16457:
16449:
16445:
16437:
16433:
16425:
16421:
16413:
16409:
16393:
16389:
16381:
16377:
16373:, ch. 1, p. 27.
16369:
16365:
16357:
16353:
16345:
16341:
16337:
16332:
16311:
16307:
16294:
16284:
16280:
16271:
16267:
16238:
16234:
16232:
16229:
16228:
16204:
16201:
16200:
16198:
16194:
16190:, §5.3, p. 125.
16170:
16166:
16164:
16161:
16160:
16143:
16139:
16137:
16134:
16133:
16131:
16127:
16095:
16091:
16076:
16072:
16057:
16053:
16039:
16036:
16035:
16029:
16025:
16021:, loc = ch. II.
16019:Bourbaki (1989)
16012:
16008:
15986:
15981:
15978:
15977:
15967:
15963:
15950:
15946:
15937:
15933:
15916:
15912:
15884:
15883:
15881:
15878:
15877:
15875:
15871:
15867:
15805:
15789:
15781:
15779:
15776:
15775:
15755:
15747:
15742:
15739:
15738:
15727:
15717:
15695:
15685:
15654:
15646:
15635:
15627:
15603:
15600:
15599:
15592:free transitive
15564:
15556:Main articles:
15554:
15477:
15471:
15455:cotangent space
15387:
15358:
15333:
15315:
15262:
15248:
15233:
15189:
15186:
15185:
15152:
15142:
15134:Main articles:
15132:
15127:
15101:
15096:
15095:
15086:
15081:
15080:
15068:
15063:
15062:
15053:
15048:
15047:
15045:
15042:
15041:
15017:
15012:
15011:
15002:
14997:
14996:
14994:
14991:
14990:
14973:
14968:
14967:
14958:
14953:
14952:
14950:
14947:
14946:
14943:tensor products
14922:
14919:
14918:
14895:
14890:
14889:
14874:
14869:
14868:
14859:
14854:
14853:
14851:
14848:
14847:
14828:
14825:
14824:
14796:
14793:
14792:
14762:
14757:
14756:
14754:
14751:
14750:
14697:
14694:
14693:
14692:matrices, with
14677:
14674:
14673:
14657:
14654:
14653:
14646:Jacobi identity
14539:
14536:
14535:
14480:
14477:
14476:
14457:
14454:
14453:
14437:
14434:
14433:
14405:
14402:
14401:
14377:polynomial ring
14351:
14348:
14347:
14340:is specified).
14303:
14301:
14298:
14297:
14257:
14237:
14235:
14232:
14231:
14228:coordinate ring
14199:
14196:
14195:
14184:
14176:Main articles:
14174:
14120:Euclidean space
14088:
14085:
14084:
14055:
14052:
14051:
14015:
14012:
14011:
13991:
13988:
13987:
13963:
13959:
13957:
13954:
13953:
13949:is a key case.
13922:
13919:
13918:
13884:
13882:
13880:
13877:
13876:
13836:
13834:
13816:
13812:
13786:
13783:
13782:
13753:
13749:
13747:
13744:
13743:
13724:
13718:
13669:
13662:
13642:
13638:
13622:
13618:
13617:
13611:
13607:
13595:
13589:
13586:
13585:
13557:
13553:
13551:
13548:
13547:
13522:
13519:
13518:
13484:
13477:
13457:
13453:
13435:
13431:
13430:
13426:
13425:
13419:
13415:
13394:
13388:
13385:
13384:
13358:
13354:
13348:
13344:
13339:
13336:
13335:
13312:
13308:
13293:
13289:
13280:
13276:
13274:
13271:
13270:
13229:
13225:
13223:
13220:
13219:
13216:Lebesgue spaces
13189:
13185:
13177:
13174:
13173:
13157:
13154:
13153:
13119:
13097:
13090:
13085:
13084:
13067:
13061:
13057:
13056:
13052:
13051:
13042:
13038:
13030:
13027:
13026:
13007:
12993:
12990:
12989:
12961:
12957:
12948:
12944:
12932:
12928:
12920:
12916:
12915:
12904:
12891:
12887:
12881:
12876:
12875:
12870:
12867:
12866:
12837:
12833:
12812:
12808:
12793:
12789:
12783:
12778:
12777:
12772:
12769:
12768:
12744:
12741:
12740:
12715:
12712:
12711:
12688:
12685:
12684:
12664:
12660:
12658:
12655:
12654:
12653:components are
12637:
12633:
12631:
12628:
12627:
12581:
12577:
12559:
12555:
12543:
12539:
12538:
12534:
12525:
12520:
12519:
12517:
12514:
12513:
12494:
12491:
12490:
12473:
12469:
12467:
12464:
12463:
12435: for
12433:
12421:
12410:
12405:
12404:
12398:
12394:
12389:
12383:
12378:
12374:
12373:
12364:
12360:
12355:
12350:
12347:
12346:
12333: and
12331:
12316: for
12314:
12308:
12302:
12298:
12293:
12287:
12274:
12270:
12265:
12260:
12257:
12256:
12222:
12219:
12218:
12201:
12198:
12197:
12168:
12164:
12149:
12145:
12136:
12132:
12131:
12127:
12119:
12117:
12114:
12113:
12095:
12091:
12089:
12086:
12085:
12068:
12062:
12036:
12034:
12031:
12030:
12014:
12006:
12003:
12002:
11985:
11981:
11979:
11976:
11975:
11948:
11945:
11944:
11916:
11911:
11910:
11908:
11905:
11904:
11888:
11885:
11884:
11868:
11865:
11864:
11838:
11833:
11824:
11819:
11818:
11813:
11801:
11795:
11792:
11791:
11790:if and only if
11775:
11773:
11770:
11769:
11752:
11747:
11746:
11744:
11741:
11740:
11712:
11709:
11708:
11672:
11669:
11668:
11633:
11630:
11629:
11622:Cauchy sequence
11598:
11595:
11594:
11563:
11560:
11559:
11541:
11538:
11537:
11519:
11514:
11513:
11511:
11508:
11507:
11482:function series
11462:
11459:
11458:
11438:
11434:
11432:
11429:
11428:
11409:
11406:
11405:
11404:of elements of
11382:
11378:
11369:
11365:
11363:
11360:
11359:
11338:
11334:
11319:
11315:
11303:
11284:
11280:
11274:
11263:
11257:
11254:
11253:
11223:
11220:
11219:
11200:
11195:
11192:
11191:
11175:
11167:
11165:
11162:
11161:
11145:
11142:
11141:
11125:
11122:
11121:
11105:
11102:
11101:
11085:
11083:
11080:
11079:
11063:
11061:
11058:
11057:
11054:continuous maps
11029:
11026:
11025:
11022:
11016:
10992:
10988:
10982:
10978:
10969:
10965:
10959:
10955:
10946:
10942:
10936:
10932:
10923:
10919:
10913:
10909:
10895:
10890:
10885:
10880:
10877:
10876:
10819:
10817:
10814:
10813:
10794:
10789:
10784:
10779:
10776:
10775:
10751:
10747:
10741:
10737:
10728:
10724:
10718:
10714:
10705:
10701:
10695:
10691:
10682:
10678:
10672:
10668:
10657:
10652:
10647:
10642:
10639:
10638:
10621:
10616:
10615:
10613:
10610:
10609:
10606:Minkowski space
10576:
10568:
10563:
10560:
10559:
10540:
10535:
10530:
10522:
10517:
10512:
10496:
10488:
10481:
10477:
10463:
10455:
10453:
10450:
10449:
10426:
10424:
10421:
10420:
10404:
10402:
10399:
10398:
10378:
10373:
10372:
10370:
10367:
10366:
10346:
10342:
10336:
10332:
10317:
10313:
10307:
10303:
10295:
10287:
10276:
10268:
10263:
10260:
10259:
10238:
10234:
10232:
10229:
10228:
10192:
10184:
10179:
10171:
10166:
10161:
10159:
10156:
10155:
10154:
10131:
10123:
10118:
10115:
10114:
10113:
10097:
10092:
10087:
10085:
10082:
10081:
10070:
10062:Main articles:
10060:
10038:
10034:
10032:
10029:
10028:
10012:
10009:
10008:
9991:
9987:
9985:
9982:
9981:
9961:
9957:
9948:
9944:
9936:
9933:
9932:
9903:
9898:
9897:
9895:
9892:
9891:
9875:
9872:
9871:
9850:
9847:
9846:
9830:infinite series
9817:
9786:
9778:
9761:
9753:
9745:
9742:
9741:
9723:
9720:
9719:
9703:
9700:
9699:
9676:
9673:
9672:
9635:
9632:
9631:
9611:
9608:
9607:
9587:
9579:
9577:
9574:
9573:
9554:
9546:
9541:
9538:
9537:
9512:
9509:
9508:
9486:
9483:
9482:
9466:
9463:
9462:
9448:
9445:
9436:
9431:
9430:
9422:
9413:
9408:
9407:
9399:
9389:
9377:
9372:
9371:
9362:
9357:
9356:
9345:
9342:
9339:
9334:
9325:
9320:
9319:
9311:
9302:
9297:
9296:
9286:
9278:
9266:
9261:
9260:
9251:
9246:
9245:
9239:
9238:
9233:
9225:
9216:
9205:
9188:
9174:
9163:
9144:
9133:
9125:
9112:
9110:
9107:
9106:
9086:
9081:
9080:
9071:
9066:
9065:
9050:
9045:
9044:
9035:
9030:
9029:
9020:
9015:
9014:
9005:
9000:
8999:
8997:
8994:
8993:
8971:
8968:
8967:
8931:
8929:
8926:
8925:
8906:
8898:
8884:
8882:
8879:
8878:
8862:
8860:
8857:
8856:
8837:
8835:
8832:
8831:
8815:
8813:
8810:
8809:
8793:
8790:
8789:
8763:
8760:
8759:
8722:
8719:
8718:
8698:
8695:
8694:
8678:
8675:
8674:
8649:
8646:
8645:
8622:
8618:
8613:
8610:
8609:
8603:
8597:
8576:
8573:
8572:
8555:
8551:
8539:
8533:
8530:
8529:
8508:
8504:
8492:
8486:
8483:
8482:
8462:
8458:
8456:
8453:
8452:
8435:
8430:
8429:
8427:
8424:
8423:
8407:
8404:
8403:
8384:
8381:
8380:
8357:
8347:
8342:
8341:
8337:
8336:
8334:
8331:
8330:
8313:
8309:
8307:
8304:
8303:
8284:
8280:
8268:
8263:
8260:
8257:
8256:
8239:
8231:Main articles:
8229:
8169:
8164:
8161:
8160:
8124:
8121:
8120:
8107:
8101:
8070:
8067:
8066:
8046:
8043:
8042:
8041:for a constant
8025:
8021:
8006:
8002:
7988:
7985:
7984:
7967:
7963:
7954:
7950:
7941:
7937:
7923:
7920:
7919:
7902:
7898:
7883:
7879:
7877:
7874:
7873:
7857:
7854:
7853:
7826:
7822:
7818:
7808:
7804:
7803:
7801:
7795:
7791:
7785:
7774:
7747:
7744:
7743:
7724:
7721:
7720:
7703:
7699:
7697:
7694:
7693:
7664:
7660:
7656:
7646:
7642:
7641:
7639:
7633:
7629:
7611:
7607:
7603:
7593:
7589:
7588:
7586:
7580:
7576:
7563:
7555:
7553:
7547:
7543:
7531:
7527:
7525:
7522:
7521:
7505:
7502:
7501:
7485:
7477:
7472:
7469:
7468:
7452:
7450:
7447:
7446:
7430:
7427:
7426:
7410:
7399:
7397:
7394:
7393:
7374:
7371:
7370:
7354:
7351:
7350:
7325:
7314:
7285:
7282:
7281:
7262:
7259:
7258:
7242:
7240:
7237:
7236:
7220:
7218:
7215:
7214:
7186:
7183:
7182:
7154:
7151:
7150:
7128:
7125:
7124:
7108:
7105:
7104:
7087:
7082:
7081:
7079:
7076:
7075:
7058:
7053:
7052:
7050:
7047:
7046:
7020:
7015:
7014:
6999:
6994:
6993:
6991:
6988:
6987:
6962:
6936:
6925:
6922:
6921:
6893:
6888:
6887:
6878:
6873:
6872:
6871:
6867:
6865:
6862:
6861:
6838:
6833:
6832:
6830:
6827:
6826:
6803:
6798:
6797:
6795:
6792:
6791:
6775:
6772:
6771:
6755:
6753:
6750:
6749:
6721:
6713:
6705:
6688:
6686:
6683:
6682:
6666:
6663:
6662:
6644:
6641:
6640:
6621:
6616:
6613:
6612:
6590:
6587:
6586:
6551:
6548:
6547:
6508:
6505:
6504:
6503:of elements of
6484:
6481:
6480:
6464:
6461:
6460:
6440:
6437:
6436:
6420:
6417:
6416:
6400:
6397:
6396:
6376:
6373:
6372:
6369:linear subspace
6352:
6349:
6348:
6332:
6330:
6327:
6326:
6310:
6307:
6306:
6290:
6287:
6286:
6267:
6256:
6250:Linear subspace
6248:Main articles:
6246:
6238:
6232:) in question.
6227:
6217:
6203:
6193:
6184:
6178:
6177:. If the field
6172:
6162:
6119:
6116:
6115:
6109:
6103:
6097:
6086:
6072:
6062:
6056:
6047:
6037:
6019:
6013:
6002:
5996:
5990:
5976:
5970:
5964:
5941:
5935:
5924:
5914:
5908:
5905:
5899:
5896:
5890:
5865:
5855:
5849:
5846:
5828:
5817:
5815:
5812:
5811:
5793:
5791:
5788:
5787:
5771:
5768:
5767:
5743:
5740:
5739:
5714:
5710:
5701:
5697:
5691:
5680:
5661:
5657:
5648:
5644:
5638:
5627:
5614:
5610:
5601:
5597:
5591:
5580:
5575:
5571:
5559:
5555:
5540:
5536:
5527:
5523:
5512:
5510:
5507:
5506:
5500:
5494:
5478:
5475:
5474:
5468:
5462:
5448:
5440:Main articles:
5438:
5427:
5421:
5415:
5409:
5395:
5389:
5383:
5377:
5359:
5353:
5347:
5341:
5326:
5320:
5306:
5300:
5283:
5273:
5261:
5255:
5243:
5231:
5230:, also denoted
5219:
5213:
5203:
5194:
5188:
5182:
5176:
5170:
5158:
5152:
5146:
5132:
5129:§ Examples
5118:
5112:
5106:
5093:
5087:
5081:
5075:
5065:
5059:
5045:
5023:
5005:
4988:
4971:
4944:
4941:
4940:
4924:
4921:
4920:
4901:
4898:
4897:
4881:
4879:
4876:
4875:
4859:
4857:
4854:
4853:
4839:
4838:
4830:
4811:
4803:
4788:
4787:
4776:
4759:
4746:
4738:
4730:
4720:
4718:
4715:
4714:
4699:
4693:
4667:
4663:
4661:
4658:
4657:
4641:
4638:
4637:
4621:
4618:
4617:
4594:
4590:
4572:
4568:
4548:
4545:
4544:
4541:
4495:
4491:
4467:
4463:
4461:
4458:
4457:
4417:
4415:
4412:
4411:
4391:
4386:
4383:
4382:
4346:
4343:
4342:
4326:
4324:
4321:
4320:
4302:
4301:
4296:
4291:
4285:
4284:
4279:
4274:
4264:
4263:
4255:
4252:
4251:
4248:
4230:
4222:
4217:
4214:
4213:
4188:
4171:
4168:
4167:
4145:
4134:
4131:
4130:
4112:
4109:
4108:
4094:
4093:
4084:
4079:
4073:
4065:
4051:
4044:
4038:
4037:
4028:
4023:
4020:
4012:
3998:
3988:
3986:
3983:
3982:
3975:
3965:Linear equation
3963:Main articles:
3961:
3933:
3917:
3847:
3844:
3843:
3815:
3812:
3811:
3805:
3799:
3793:
3789:
3712:
3709:
3708:
3692:
3684:
3670:
3667:
3666:
3659:
3653:
3651:Function spaces
3642:
3621:
3610:
3608:
3605:
3604:
3598:
3592:
3586:
3580:
3571:
3544:
3531:
3521:
3515:
3509:
3503:
3497:
3462:
3422:
3412:
3406:
3400:
3387:
3381:
3379:complex numbers
3375:
3359:
3349:
3343:
3336:
3326:
3320:
3317:
3309:
3289:
3285:
3270:
3266:
3257:
3253:
3248:
3245:
3244:
3238:
3230:
3223:
3217:
3214:
3189:
3188:
3157:
3136:
3135:
3123:
3119:
3110:
3106:
3097:
3093:
3084:
3080:
3070:
3061:
3057:
3048:
3044:
3029:
3025:
3016:
3012:
3005:
3003:
3000:
2999:
2987:
2979:
2975:
2971:
2967:
2964:
2953:
2942:
2932:
2925:
2919:
2913:
2904:
2897:
2885:
2879:
2870:
2864:
2858:
2848:
2842:
2836:
2823:
2809:
2803:
2784:
2783:
2782:
2781:
2780:
2773:
2766:
2763:
2754:
2753:
2752:
2751:(red) is shown.
2746:
2740:
2730:
2727:
2716:
2711:
2705:
2665:function spaces
2645:scalar products
2625:matrix notation
2623:introduced the
2582:complex numbers
2523:affine geometry
2519:
2499:
2491:
2487:
2470:
2466:
2464:
2461:
2460:
2454:
2442:
2436:
2426:
2421:are called the
2405:
2401:
2386:
2382:
2380:
2377:
2376:
2372:
2355:
2351:
2336:
2332:
2330:
2327:
2326:
2306:
2301:
2300:
2294:
2290:
2275:
2270:
2269:
2263:
2259:
2251:
2249:
2246:
2245:
2223:
2221:
2218:
2217:
2213:
2209:
2205:
2185:
2180:
2179:
2164:
2159:
2158:
2149:
2144:
2143:
2138:
2135:
2134:
2120:axiom of choice
2070:
2056:
2052:
2042:
2038:
2037:is the span of
2034:
2032:
2028:
2024:
2020:
2016:
2012:
2008:
1996:
1992:
1991:Given a subset
1975:
1971:
1967:
1963:
1959:
1955:
1947:
1943:
1939:
1937:vector subspace
1933:linear subspace
1927:Linear subspace
1920:
1916:
1912:
1908:
1903:are said to be
1900:
1896:
1892:
1879:are called the
1863:
1859:
1844:
1840:
1838:
1835:
1834:
1808:
1803:
1802:
1787:
1782:
1781:
1779:
1776:
1775:
1752:
1748:
1733:
1729:
1727:
1724:
1723:
1703:
1698:
1697:
1691:
1687:
1672:
1667:
1666:
1660:
1656:
1647:
1642:
1641:
1635:
1631:
1629:
1626:
1625:
1621:
1617:
1613:
1609:
1605:
1592:
1585:
1575:
1568:
1558:
1545:
1539:
1529:
1523:
1516:
1481:
1473:
1471:
1468:
1467:
1445:
1442:
1441:
1425:
1417:
1412:
1409:
1408:
1388:
1377:
1363:
1360:
1359:
1339:
1331:
1326:
1323:
1322:
1302:
1294:
1289:
1286:
1285:
1257:
1255:
1252:
1251:
1229:
1226:
1225:
1201:
1187:
1179:
1171:
1169:
1166:
1165:
1161:of this group.
1152:
1151:from the field
1136:
1125:
1124:
1120:
1113:complex numbers
1070:
1036:
1022:
1014:
1003:
976:
955:
949:
932:
922:
901:
887:
871:
843:
808:
785:
781:
777:
773:
767:
761:
755:
738:
734:
730:
724:
720:
716:
691:
687:
683:
677:
671:
664:vector addition
651:
643:
640:binary function
631:
626:is a non-empty
623:
614:
605:
576:
575:
574:
545:Non-associative
527:
516:
515:
505:
485:
474:
473:
462:Map of lattices
458:
454:Boolean algebra
449:Heyting algebra
423:
412:
411:
405:
386:Integral domain
350:
339:
338:
332:
286:
218:function spaces
132:complex numbers
90:(also called a
65:
55:
49:
42:
35:
28:
23:
22:
15:
12:
11:
5:
20081:
20071:
20070:
20065:
20060:
20055:
20050:
20031:
20030:
20028:
20027:
20016:
20013:
20012:
20010:
20009:
20004:
19999:
19994:
19989:
19987:Floating-point
19983:
19981:
19975:
19974:
19972:
19971:
19969:Tensor product
19966:
19961:
19956:
19954:Function space
19951:
19946:
19940:
19938:
19931:
19930:
19928:
19927:
19922:
19917:
19912:
19907:
19902:
19897:
19892:
19890:Triple product
19887:
19882:
19876:
19874:
19868:
19867:
19865:
19864:
19859:
19854:
19849:
19844:
19839:
19834:
19828:
19826:
19820:
19819:
19817:
19816:
19811:
19806:
19804:Transformation
19801:
19796:
19794:Multiplication
19791:
19786:
19781:
19776:
19770:
19768:
19762:
19761:
19754:
19752:
19750:
19749:
19744:
19739:
19734:
19729:
19724:
19719:
19714:
19709:
19704:
19699:
19694:
19689:
19684:
19679:
19674:
19669:
19664:
19659:
19653:
19651:
19650:Basic concepts
19647:
19646:
19644:
19643:
19638:
19632:
19629:
19628:
19625:Linear algebra
19621:
19620:
19613:
19606:
19598:
19592:
19591:
19577:"Vector space"
19571:
19570:External links
19568:
19566:
19565:
19543:
19527:
19468:
19463:
19446:
19441:
19413:
19401:
19358:
19345:
19328:
19323:
19297:
19292:
19270:
19265:
19248:
19243:
19222:
19217:
19197:
19192:
19175:
19170:
19153:
19148:
19138:(6 ed.),
19131:
19112:(3): 670–673,
19099:
19094:
19074:
19069:
19049:
19044:
19027:
19014:
18994:
18977:(7): 572–574,
18964:
18959:
18939:
18934:
18914:
18909:
18889:
18876:
18852:
18849:Addison-Wesley
18833:
18820:
18812:Addison-Wesley
18800:
18795:
18775:Ashcroft, Neil
18770:
18768:
18765:
18763:
18762:
18748:
18722:
18711:(3): 262–303,
18698:
18680:
18666:
18661:
18646:
18641:
18611:
18597:
18578:(3): 227–261,
18561:
18549:
18532:
18518:
18482:Banach, Stefan
18477:
18475:
18472:
18470:
18469:
18466:Academic Press
18464:, Boston, MA:
18454:
18440:
18420:
18415:
18402:
18389:978-1584888666
18388:
18375:
18362:
18339:
18334:
18314:
18309:
18301:Addison-Wesley
18289:
18276:
18256:
18251:
18235:
18230:
18217:
18212:
18199:
18194:
18181:
18176:
18163:
18158:
18141:
18136:
18119:
18114:
18101:
18096:
18083:
18080:Academic Press
18078:, Boston, MA:
18068:
18050:
18045:
18028:
18023:
18003:
17998:
17977:
17975:
17972:
17970:
17969:
17964:
17944:
17940:Linear Algebra
17935:
17930:
17915:
17910:
17897:
17892:
17872:
17861:
17856:
17837:
17832:
17816:
17803:
17783:
17778:
17762:Linear algebra
17757:
17752:
17737:
17728:
17723:
17708:
17703:
17682:
17670:
17665:
17644:
17639:
17622:
17617:
17601:Artin, Michael
17597:
17587:
17585:
17582:
17580:
17577:
17575:
17574:
17562:
17550:
17538:
17526:
17514:
17502:
17490:
17488:, §34, p. 108.
17478:
17466:
17454:
17442:
17430:
17418:
17406:
17394:
17390:Griffiths 1995
17382:
17370:
17358:
17356:, Th. III.1.1.
17346:
17334:
17322:
17310:
17298:
17286:
17274:
17262:
17250:
17238:
17226:
17214:
17202:
17186:
17174:
17162:
17150:
17148:, pp. 204–205.
17138:
17126:
17110:
17098:
17086:
17074:
17070:Nicholson 2018
17062:
17050:
17038:
17026:
17022:Nicholson 2018
17014:
17002:
16990:
16978:
16966:
16954:
16942:
16930:
16918:
16906:
16894:
16882:
16870:
16858:
16854:Nicholson 2018
16846:
16834:
16822:
16810:
16798:
16786:
16774:
16758:
16742:
16726:
16710:
16698:
16686:
16674:
16662:
16658:Grassmann 2000
16650:
16638:
16626:
16614:
16602:
16590:
16578:
16562:
16546:
16530:
16518:
16498:
16486:
16467:
16455:
16443:
16431:
16419:
16407:
16387:
16375:
16363:
16351:
16338:
16336:
16333:
16331:
16330:
16305:
16278:
16265:
16252:
16249:
16246:
16241:
16237:
16217:
16214:
16211:
16208:
16192:
16173:
16169:
16146:
16142:
16125:
16098:
16094:
16090:
16087:
16084:
16079:
16075:
16071:
16068:
16065:
16060:
16056:
16052:
16049:
16046:
16043:
16023:
16006:
15993:
15989:
15985:
15961:
15944:
15931:
15919:scalar product
15910:
15897:
15891:
15888:
15868:
15866:
15863:
15851:flag manifolds
15792:
15788:
15784:
15758:
15754:
15750:
15746:
15661:
15657:
15653:
15649:
15645:
15642:
15638:
15634:
15630:
15626:
15622:
15619:
15616:
15613:
15610:
15607:
15553:
15550:
15473:Main article:
15470:
15467:
15380:tangent spaces
15376:tangent bundle
15205:
15202:
15199:
15196:
15193:
15140:Tangent bundle
15131:
15130:Vector bundles
15128:
15126:
15123:
15104:
15099:
15094:
15089:
15084:
15079:
15076:
15071:
15066:
15061:
15056:
15051:
15025:
15020:
15015:
15010:
15005:
15000:
14976:
14971:
14966:
14961:
14956:
14926:
14903:
14898:
14893:
14888:
14885:
14882:
14877:
14872:
14867:
14862:
14857:
14832:
14812:
14809:
14806:
14803:
14800:
14790:tensor algebra
14770:
14765:
14760:
14734:
14731:
14728:
14725:
14722:
14719:
14716:
14713:
14710:
14707:
14704:
14701:
14681:
14661:
14650:
14649:
14633:
14630:
14627:
14624:
14621:
14618:
14615:
14612:
14609:
14606:
14603:
14600:
14597:
14594:
14591:
14588:
14585:
14582:
14579:
14576:
14573:
14570:
14567:
14564:
14561:
14558:
14555:
14552:
14549:
14546:
14543:
14533:
14517:
14514:
14511:
14508:
14505:
14502:
14499:
14496:
14493:
14490:
14487:
14484:
14461:
14441:
14421:
14418:
14415:
14412:
14409:
14364:
14361:
14358:
14355:
14310:
14306:
14285:
14282:
14279:
14276:
14273:
14270:
14267:
14264:
14260:
14256:
14253:
14250:
14247:
14244:
14240:
14215:
14212:
14209:
14206:
14203:
14173:
14170:
14095:
14092:
14062:
14059:
14031:
14028:
14025:
14022:
14019:
13995:
13966:
13962:
13938:
13935:
13932:
13929:
13926:
13900:
13896:
13893:
13890:
13887:
13864:
13861:
13858:
13852:
13848:
13845:
13842:
13839:
13833:
13830:
13827:
13824:
13819:
13815:
13811:
13808:
13805:
13799:
13793:
13790:
13770:
13767:
13764:
13761:
13756:
13752:
13738:, in honor of
13736:Hilbert spaces
13720:Main article:
13717:
13716:Hilbert spaces
13714:
13710:Sobolev spaces
13691:
13688:
13684:
13681:
13678:
13675:
13672:
13665:
13660:
13656:
13653:
13650:
13645:
13641:
13637:
13634:
13631:
13628:
13625:
13621:
13614:
13610:
13604:
13601:
13598:
13594:
13573:
13570:
13567:
13562:
13556:
13535:
13532:
13529:
13526:
13506:
13503:
13499:
13496:
13493:
13490:
13487:
13480:
13475:
13471:
13468:
13465:
13460:
13456:
13452:
13449:
13446:
13443:
13438:
13434:
13429:
13422:
13418:
13412:
13409:
13406:
13400:
13397:
13393:
13372:
13369:
13366:
13361:
13357:
13351:
13347:
13343:
13323:
13320:
13315:
13311:
13307:
13304:
13301:
13296:
13292:
13288:
13283:
13279:
13268:
13248:
13245:
13242:
13239:
13234:
13228:
13203:
13200:
13197:
13192:
13188:
13184:
13181:
13161:
13132:
13126:
13123:
13117:
13112:
13109:
13106:
13103:
13100:
13093:
13088:
13083:
13080:
13077:
13074:
13070:
13064:
13060:
13055:
13050:
13045:
13041:
13037:
13034:
13010:
13006:
13003:
13000:
12997:
12975:
12972:
12967:
12964:
12960:
12956:
12951:
12947:
12943:
12938:
12935:
12931:
12923:
12919:
12913:
12910:
12907:
12903:
12899:
12894:
12890:
12884:
12879:
12874:
12854:
12851:
12848:
12843:
12840:
12836:
12832:
12829:
12826:
12823:
12818:
12815:
12811:
12807:
12804:
12801:
12796:
12792:
12786:
12781:
12776:
12757:
12754:
12751:
12748:
12728:
12725:
12722:
12719:
12695:
12692:
12670:
12667:
12663:
12640:
12636:
12615:
12611:
12607:
12604:
12601:
12598:
12595:
12592:
12587:
12584:
12580:
12576:
12573:
12570:
12565:
12562:
12558:
12554:
12549:
12546:
12542:
12537:
12533:
12528:
12523:
12501:
12498:
12476:
12472:
12449:
12446:
12443:
12440:
12428:
12425:
12419:
12413:
12408:
12401:
12397:
12392:
12386:
12382:
12377:
12372:
12367:
12363:
12358:
12354:
12330:
12327:
12324:
12321:
12311:
12305:
12301:
12296:
12290:
12286:
12282:
12277:
12273:
12268:
12264:
12244:
12241:
12238:
12235:
12232:
12229:
12226:
12205:
12183:
12179:
12176:
12171:
12167:
12163:
12160:
12157:
12152:
12148:
12144:
12139:
12135:
12130:
12126:
12122:
12098:
12094:
12064:Main article:
12061:
12058:
12039:
12017:
12013:
12010:
11988:
11984:
11961:
11958:
11955:
11952:
11924:
11919:
11914:
11892:
11872:
11848:
11845:
11841:
11836:
11832:
11827:
11822:
11816:
11810:
11807:
11804:
11800:
11778:
11755:
11750:
11728:
11725:
11722:
11719:
11716:
11688:
11685:
11682:
11679:
11676:
11652:
11649:
11646:
11643:
11640:
11637:
11605:
11602:
11582:
11579:
11576:
11573:
11570:
11567:
11545:
11522:
11517:
11504:Unit "spheres"
11469:
11466:
11456:function space
11441:
11437:
11416:
11413:
11393:
11390:
11385:
11381:
11377:
11372:
11368:
11341:
11337:
11333:
11330:
11327:
11322:
11318:
11312:
11309:
11306:
11302:
11295:
11287:
11283:
11277:
11272:
11269:
11266:
11262:
11227:
11207:
11203:
11199:
11178:
11174:
11170:
11149:
11129:
11109:
11088:
11066:
11056:. Roughly, if
11033:
11018:Main article:
11015:
11012:
11000:
10995:
10991:
10985:
10981:
10977:
10972:
10968:
10962:
10958:
10954:
10949:
10945:
10939:
10935:
10931:
10926:
10922:
10916:
10912:
10908:
10905:
10902:
10898:
10893:
10888:
10884:
10856:
10853:
10850:
10847:
10844:
10841:
10838:
10835:
10832:
10829:
10826:
10822:
10801:
10797:
10792:
10787:
10783:
10759:
10754:
10750:
10744:
10740:
10736:
10731:
10727:
10721:
10717:
10713:
10708:
10704:
10698:
10694:
10690:
10685:
10681:
10675:
10671:
10667:
10664:
10660:
10655:
10650:
10646:
10624:
10619:
10589:
10586:
10583:
10579:
10575:
10571:
10567:
10547:
10543:
10538:
10533:
10529:
10525:
10520:
10515:
10511:
10507:
10503:
10499:
10495:
10491:
10487:
10484:
10480:
10476:
10473:
10470:
10466:
10462:
10458:
10446:law of cosines
10433:
10429:
10407:
10386:
10381:
10376:
10354:
10349:
10345:
10339:
10335:
10331:
10328:
10325:
10320:
10316:
10310:
10306:
10302:
10298:
10294:
10290:
10286:
10283:
10279:
10275:
10271:
10267:
10241:
10237:
10204:
10199:
10195:
10191:
10187:
10183:
10178:
10174:
10169:
10164:
10141:
10138:
10134:
10130:
10126:
10122:
10100:
10095:
10090:
10059:
10056:
10041:
10037:
10016:
9994:
9990:
9969:
9964:
9960:
9956:
9951:
9947:
9943:
9940:
9923:, for example
9906:
9901:
9879:
9858:
9855:
9816:
9813:
9796:
9793:
9789:
9785:
9781:
9777:
9774:
9771:
9768:
9764:
9760:
9756:
9752:
9749:
9730:
9727:
9707:
9683:
9680:
9660:
9657:
9654:
9651:
9648:
9645:
9642:
9639:
9615:
9590:
9586:
9582:
9561:
9557:
9553:
9549:
9545:
9522:
9519:
9516:
9496:
9493:
9490:
9470:
9444:
9439:
9434:
9429:
9425:
9421:
9416:
9411:
9406:
9402:
9395:
9392:
9390:
9385:
9380:
9375:
9370:
9365:
9360:
9355:
9352:
9348:
9344:
9343:
9337:
9333:
9328:
9323:
9318:
9314:
9310:
9305:
9300:
9292:
9289:
9287:
9281:
9277:
9274:
9269:
9264:
9259:
9254:
9249:
9244:
9241:
9240:
9232:
9218:
9215:
9212:
9208:
9204:
9201:
9198:
9195:
9191:
9184:
9177:
9173:
9170:
9166:
9162:
9159:
9156:
9150:
9147:
9145:
9140:
9136:
9132:
9128:
9124:
9121:
9118:
9115:
9114:
9094:
9089:
9084:
9079:
9074:
9069:
9064:
9061:
9058:
9053:
9048:
9043:
9038:
9033:
9028:
9023:
9018:
9013:
9008:
9003:
8978:
8975:
8938:
8934:
8913:
8909:
8905:
8901:
8897:
8894:
8891:
8887:
8865:
8844:
8840:
8818:
8797:
8773:
8770:
8767:
8744:
8741:
8738:
8735:
8732:
8729:
8726:
8702:
8682:
8662:
8659:
8656:
8653:
8633:
8630:
8625:
8621:
8617:
8607:tensor product
8599:Main article:
8596:
8595:Tensor product
8593:
8580:
8558:
8554:
8548:
8545:
8542:
8538:
8511:
8507:
8501:
8498:
8495:
8491:
8465:
8461:
8438:
8433:
8411:
8388:
8366:
8363:
8360:
8355:
8350:
8345:
8340:
8316:
8312:
8287:
8283:
8277:
8274:
8271:
8267:
8254:direct product
8243:direct product
8233:Direct product
8228:
8225:
8208:
8205:
8202:
8199:
8196:
8192:
8188:
8185:
8182:
8179:
8176:
8172:
8168:
8128:
8089:
8086:
8083:
8080:
8077:
8074:
8050:
8028:
8024:
8020:
8017:
8014:
8009:
8005:
8001:
7998:
7995:
7992:
7970:
7966:
7962:
7957:
7953:
7949:
7944:
7940:
7936:
7933:
7930:
7927:
7905:
7901:
7897:
7894:
7889:
7886:
7882:
7861:
7837:
7829:
7825:
7821:
7816:
7811:
7807:
7798:
7794:
7788:
7783:
7780:
7777:
7773:
7769:
7766:
7763:
7760:
7757:
7754:
7751:
7731:
7728:
7706:
7702:
7681:
7678:
7675:
7667:
7663:
7659:
7654:
7649:
7645:
7636:
7632:
7628:
7625:
7622:
7614:
7610:
7606:
7601:
7596:
7592:
7583:
7579:
7575:
7569:
7566:
7561:
7558:
7550:
7546:
7542:
7539:
7534:
7530:
7509:
7488:
7484:
7480:
7476:
7455:
7434:
7413:
7409:
7406:
7402:
7378:
7358:
7338:
7335:
7332:
7328:
7324:
7321:
7317:
7313:
7310:
7307:
7304:
7301:
7298:
7295:
7292:
7289:
7266:
7245:
7223:
7202:
7199:
7196:
7193:
7190:
7170:
7167:
7164:
7161:
7158:
7132:
7112:
7090:
7085:
7061:
7056:
7043:if and only if
7031:
7028:
7023:
7018:
7013:
7010:
7007:
7002:
6997:
6975:
6972:
6969:
6965:
6961:
6958:
6955:
6952:
6949:
6946:
6943:
6939:
6935:
6932:
6929:
6909:
6906:
6902:
6896:
6891:
6886:
6881:
6876:
6870:
6849:
6846:
6841:
6836:
6814:
6811:
6806:
6801:
6779:
6758:
6737:
6734:
6731:
6728:
6724:
6720:
6716:
6712:
6708:
6704:
6701:
6698:
6695:
6691:
6670:
6648:
6628:
6624:
6620:
6600:
6597:
6594:
6561:
6558:
6555:
6512:
6488:
6468:
6444:
6424:
6404:
6391:, or simply a
6380:
6367:) is called a
6356:
6335:
6314:
6294:
6245:
6242:
6237:
6234:
6150:
6147:
6144:
6141:
6138:
6135:
6132:
6129:
6126:
6123:
5975:, linear maps
5966:Main article:
5963:
5960:
5912:
5903:
5894:
5887:parallelepiped
5835:
5831:
5827:
5824:
5820:
5796:
5775:
5766:of the matrix
5747:
5727:
5723:
5717:
5713:
5707:
5704:
5700:
5694:
5689:
5686:
5683:
5679:
5675:
5672:
5669:
5664:
5660:
5654:
5651:
5647:
5641:
5636:
5633:
5630:
5626:
5622:
5617:
5613:
5607:
5604:
5600:
5594:
5589:
5586:
5583:
5579:
5574:
5570:
5567:
5562:
5558:
5554:
5551:
5548:
5543:
5539:
5535:
5530:
5526:
5522:
5519:
5515:
5482:
5437:
5434:
5382:-vector space
5266:is called the
5215:
4951:
4948:
4928:
4908:
4905:
4884:
4862:
4837:
4833:
4829:
4826:
4823:
4820:
4817:
4814:
4812:
4810:
4806:
4802:
4799:
4796:
4793:
4790:
4789:
4786:
4783:
4779:
4775:
4772:
4769:
4766:
4762:
4758:
4755:
4752:
4749:
4747:
4745:
4741:
4737:
4733:
4729:
4726:
4723:
4722:
4695:Main article:
4692:
4689:
4670:
4666:
4645:
4625:
4605:
4600:
4597:
4593:
4589:
4586:
4583:
4578:
4575:
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4567:
4564:
4561:
4558:
4555:
4552:
4530:
4527:
4524:
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4518:
4515:
4512:
4509:
4506:
4503:
4498:
4494:
4490:
4487:
4484:
4481:
4478:
4473:
4470:
4466:
4439:
4436:
4433:
4430:
4427:
4424:
4420:
4408:matrix product
4394:
4390:
4371:
4368:
4365:
4362:
4359:
4356:
4353:
4350:
4341:is the vector
4329:
4306:
4300:
4297:
4295:
4292:
4290:
4287:
4286:
4283:
4280:
4278:
4275:
4273:
4270:
4269:
4267:
4262:
4259:
4237:
4233:
4229:
4225:
4221:
4195:
4191:
4187:
4184:
4181:
4178:
4175:
4155:
4152:
4148:
4144:
4141:
4138:
4119:
4116:
4092:
4089:
4085:
4083:
4080:
4078:
4074:
4072:
4068:
4066:
4063:
4060:
4056:
4053:
4049:
4046:
4043:
4040:
4039:
4036:
4033:
4029:
4027:
4024:
4021:
4019:
4015:
4013:
4010:
4007:
4003:
4000:
3996:
3993:
3990:
3960:
3957:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
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3831:
3828:
3825:
3822:
3819:
3773:
3770:
3767:
3764:
3761:
3758:
3755:
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3749:
3746:
3743:
3740:
3737:
3734:
3731:
3728:
3725:
3722:
3719:
3716:
3695:
3691:
3687:
3683:
3680:
3677:
3674:
3657:Function space
3655:Main article:
3652:
3649:
3630:
3625:
3620:
3617:
3613:
3419:imaginary unit
3374:
3371:
3313:
3297:
3292:
3288:
3284:
3281:
3278:
3273:
3269:
3265:
3260:
3256:
3252:
3213:
3210:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3158:
3156:
3153:
3150:
3147:
3144:
3141:
3138:
3137:
3134:
3131:
3126:
3122:
3118:
3113:
3109:
3105:
3100:
3096:
3092:
3087:
3083:
3079:
3076:
3073:
3071:
3069:
3064:
3060:
3056:
3051:
3047:
3043:
3040:
3037:
3032:
3028:
3024:
3019:
3015:
3011:
3008:
3007:
2963:
2960:
2855:multiplication
2764:
2757:
2756:
2755:
2728:
2721:
2720:
2719:
2718:
2717:
2715:
2712:
2707:Main article:
2704:
2701:
2697:Hilbert spaces
2669:Henri Lebesgue
2561:introduced an
2531:René Descartes
2518:
2515:
2473:
2469:
2408:
2404:
2400:
2397:
2394:
2389:
2385:
2358:
2354:
2350:
2347:
2344:
2339:
2335:
2314:
2309:
2304:
2297:
2293:
2289:
2286:
2283:
2278:
2273:
2266:
2262:
2258:
2254:
2233:
2230:
2226:
2193:
2188:
2183:
2178:
2175:
2172:
2167:
2162:
2157:
2152:
2147:
2142:
2107:
2106:
2083:
2074:
2067:generating set
2023:. The span of
2015:that contains
2003:or simply the
1989:
1984:
1929:
1924:
1899:-vector space
1889:
1884:
1866:
1862:
1858:
1855:
1852:
1847:
1843:
1822:
1819:
1816:
1811:
1806:
1801:
1798:
1795:
1790:
1785:
1763:
1760:
1755:
1751:
1747:
1744:
1741:
1736:
1732:
1711:
1706:
1701:
1694:
1690:
1686:
1683:
1680:
1675:
1670:
1663:
1659:
1655:
1650:
1645:
1638:
1634:
1612:-vector space
1602:
1590:
1583:
1566:
1556:
1536:standard basis
1515:
1512:
1500:
1499:
1488:
1484:
1480:
1476:
1455:
1452:
1449:
1428:
1424:
1420:
1416:
1406:
1395:
1391:
1387:
1384:
1380:
1376:
1373:
1370:
1367:
1357:
1346:
1342:
1338:
1334:
1330:
1320:
1309:
1305:
1301:
1297:
1293:
1270:
1267:
1264:
1260:
1239:
1236:
1233:
1211:
1208:
1204:
1200:
1197:
1194:
1190:
1186:
1182:
1178:
1174:
1099:
1098:
1068:
1064:
1063:
1034:
1031:Distributivity
1027:
1026:
1001:
997:
996:
974:
970:
969:
919:
912:
911:
868:
861:
860:
841:
834:
833:
806:
799:
798:
795:
748:
747:
706:
705:
613:
610:
607:
606:
604:
603:
596:
589:
581:
578:
577:
573:
572:
567:
562:
557:
552:
547:
542:
536:
535:
534:
528:
522:
521:
518:
517:
514:
513:
510:Linear algebra
504:
503:
498:
493:
487:
486:
480:
479:
476:
475:
472:
471:
468:Lattice theory
464:
457:
456:
451:
446:
441:
436:
431:
425:
424:
418:
417:
414:
413:
404:
403:
398:
393:
388:
383:
378:
373:
368:
363:
358:
352:
351:
345:
344:
341:
340:
331:
330:
325:
320:
314:
313:
312:
307:
302:
293:
287:
281:
280:
277:
276:
261:Hilbert spaces
194:natural number
167:linear algebra
26:
9:
6:
4:
3:
2:
20080:
20069:
20068:Vector spaces
20066:
20064:
20061:
20059:
20056:
20054:
20051:
20049:
20046:
20045:
20043:
20036:
20026:
20018:
20017:
20014:
20008:
20005:
20003:
20002:Sparse matrix
20000:
19998:
19995:
19993:
19990:
19988:
19985:
19984:
19982:
19980:
19976:
19970:
19967:
19965:
19962:
19960:
19957:
19955:
19952:
19950:
19947:
19945:
19942:
19941:
19939:
19937:constructions
19936:
19932:
19926:
19925:Outermorphism
19923:
19921:
19918:
19916:
19913:
19911:
19908:
19906:
19903:
19901:
19898:
19896:
19893:
19891:
19888:
19886:
19885:Cross product
19883:
19881:
19878:
19877:
19875:
19873:
19869:
19863:
19860:
19858:
19855:
19853:
19852:Outer product
19850:
19848:
19845:
19843:
19840:
19838:
19835:
19833:
19832:Orthogonality
19830:
19829:
19827:
19825:
19821:
19815:
19812:
19810:
19809:Cramer's rule
19807:
19805:
19802:
19800:
19797:
19795:
19792:
19790:
19787:
19785:
19782:
19780:
19779:Decomposition
19777:
19775:
19772:
19771:
19769:
19767:
19763:
19758:
19748:
19745:
19743:
19740:
19738:
19735:
19733:
19730:
19728:
19725:
19723:
19720:
19718:
19715:
19713:
19710:
19708:
19705:
19703:
19700:
19698:
19695:
19693:
19690:
19688:
19685:
19683:
19680:
19678:
19675:
19673:
19670:
19668:
19665:
19663:
19660:
19658:
19655:
19654:
19652:
19648:
19642:
19639:
19637:
19634:
19633:
19630:
19626:
19619:
19614:
19612:
19607:
19605:
19600:
19599:
19596:
19588:
19584:
19583:
19578:
19574:
19573:
19562:
19558:
19554:
19550:
19546:
19540:
19536:
19532:
19528:
19519:on 2007-01-13
19515:
19511:
19507:
19503:
19499:
19494:
19489:
19485:
19481:
19474:
19469:
19466:
19460:
19456:
19455:Prentice Hall
19452:
19447:
19444:
19438:
19434:
19430:
19425:
19424:
19423:Galois Theory
19418:
19414:
19410:
19406:
19402:
19399:
19395:
19391:
19387:
19383:
19379:
19375:
19372:(in German),
19371:
19367:
19363:
19362:Schönhage, A.
19359:
19356:
19352:
19348:
19342:
19338:
19334:
19329:
19326:
19320:
19316:
19315:
19310:
19306:
19302:
19298:
19295:
19289:
19285:
19281:
19280:
19275:
19271:
19268:
19262:
19258:
19254:
19249:
19246:
19240:
19236:
19231:
19230:
19223:
19220:
19214:
19210:
19206:
19202:
19198:
19195:
19189:
19185:
19181:
19176:
19173:
19167:
19163:
19159:
19158:Fibre Bundles
19154:
19151:
19145:
19141:
19137:
19132:
19129:
19125:
19120:
19115:
19111:
19107:
19106:
19100:
19097:
19091:
19087:
19083:
19079:
19075:
19072:
19066:
19062:
19061:Prentice Hall
19058:
19054:
19050:
19047:
19041:
19037:
19033:
19028:
19025:
19021:
19017:
19011:
19007:
19003:
18999:
18995:
18992:
18988:
18984:
18980:
18976:
18972:
18971:
18965:
18962:
18956:
18952:
18948:
18944:
18940:
18937:
18931:
18927:
18923:
18919:
18915:
18912:
18906:
18902:
18898:
18894:
18890:
18887:
18883:
18879:
18873:
18869:
18865:
18858:
18853:
18850:
18846:
18842:
18838:
18834:
18831:
18827:
18823:
18817:
18813:
18809:
18805:
18801:
18798:
18792:
18787:
18786:
18780:
18776:
18772:
18771:
18761:
18757:
18753:
18749:
18745:
18739:
18731:
18727:
18723:
18719:
18714:
18710:
18706:
18705:
18699:
18696:on 2006-11-23
18695:
18691:
18690:
18685:
18681:
18677:
18676:
18671:
18667:
18664:
18658:
18654:
18653:
18647:
18644:
18638:
18634:
18630:
18622:
18621:
18616:
18612:
18608:
18607:
18602:
18598:
18595:
18591:
18586:
18581:
18577:
18573:
18572:
18567:
18562:
18558:
18554:
18550:
18545:
18541:
18537:
18533:
18529:
18528:
18523:
18519:
18516:
18512:
18508:
18504:
18500:
18497:(in French),
18496:
18495:
18487:
18483:
18479:
18478:
18467:
18463:
18459:
18455:
18451:
18447:
18443:
18437:
18433:
18429:
18425:
18421:
18418:
18412:
18408:
18403:
18399:
18395:
18391:
18385:
18381:
18376:
18373:
18369:
18365:
18359:
18354:
18349:
18345:
18340:
18337:
18331:
18327:
18323:
18319:
18315:
18312:
18306:
18302:
18298:
18297:Real analysis
18294:
18290:
18287:
18283:
18279:
18273:
18269:
18265:
18261:
18257:
18254:
18248:
18244:
18240:
18236:
18233:
18227:
18223:
18218:
18215:
18209:
18205:
18200:
18197:
18191:
18187:
18182:
18179:
18173:
18169:
18164:
18161:
18155:
18151:
18147:
18142:
18139:
18133:
18129:
18125:
18120:
18117:
18111:
18107:
18102:
18099:
18093:
18089:
18084:
18081:
18077:
18073:
18069:
18066:
18062:
18061:
18056:
18051:
18048:
18042:
18038:
18034:
18029:
18026:
18020:
18016:
18012:
18011:Integration I
18008:
18004:
18001:
17995:
17991:
17987:
17983:
17979:
17978:
17967:
17961:
17957:
17953:
17949:
17945:
17941:
17936:
17933:
17927:
17923:
17922:
17916:
17913:
17907:
17903:
17898:
17895:
17889:
17885:
17881:
17877:
17876:Roman, Steven
17873:
17869:
17868:
17862:
17859:
17853:
17849:
17845:
17844:
17838:
17835:
17829:
17825:
17821:
17817:
17814:
17810:
17806:
17800:
17796:
17792:
17788:
17784:
17781:
17775:
17771:
17767:
17763:
17758:
17755:
17749:
17745:
17744:
17738:
17734:
17729:
17726:
17720:
17717:, CRC Press,
17716:
17715:
17709:
17706:
17700:
17696:
17692:
17688:
17683:
17679:
17675:
17671:
17668:
17662:
17658:
17654:
17650:
17645:
17642:
17636:
17631:
17630:
17623:
17620:
17614:
17610:
17609:Prentice Hall
17606:
17602:
17598:
17594:
17589:
17588:
17571:
17566:
17559:
17554:
17547:
17542:
17535:
17530:
17523:
17518:
17511:
17506:
17499:
17494:
17487:
17486:Kreyszig 1991
17482:
17475:
17470:
17463:
17458:
17451:
17446:
17439:
17434:
17427:
17426:Eisenbud 1995
17422:
17415:
17410:
17404:, ch. XVII.3.
17403:
17398:
17391:
17386:
17380:, Chapter 11.
17379:
17378:Kreyszig 1999
17374:
17367:
17362:
17355:
17350:
17343:
17338:
17331:
17326:
17319:
17314:
17307:
17302:
17295:
17290:
17283:
17278:
17271:
17266:
17259:
17254:
17247:
17242:
17235:
17234:Kreyszig 1989
17230:
17223:
17222:Kreyszig 1989
17218:
17211:
17206:
17199:
17198:Bourbaki 1987
17195:
17190:
17183:
17178:
17171:
17166:
17159:
17158:Bourbaki 2004
17154:
17147:
17142:
17135:
17130:
17123:
17119:
17114:
17107:
17102:
17095:
17090:
17083:
17082:Mac Lane 1998
17078:
17071:
17066:
17059:
17054:
17047:
17042:
17035:
17030:
17023:
17018:
17011:
17006:
16999:
16994:
16987:
16982:
16975:
16970:
16963:
16958:
16951:
16946:
16939:
16934:
16927:
16922:
16915:
16910:
16903:
16898:
16891:
16886:
16879:
16874:
16867:
16862:
16855:
16850:
16843:
16838:
16831:
16826:
16819:
16814:
16807:
16802:
16795:
16790:
16783:
16778:
16771:
16767:
16762:
16755:
16754:358–359
16751:
16750:Kreyszig 2020
16746:
16739:
16735:
16734:Kreyszig 2020
16730:
16723:
16719:
16714:
16707:
16702:
16695:
16690:
16683:
16678:
16671:
16666:
16659:
16654:
16647:
16646:Hamilton 1853
16642:
16635:
16630:
16623:
16618:
16611:
16606:
16599:
16594:
16587:
16586:Bourbaki 1969
16582:
16575:
16571:
16566:
16559:
16555:
16550:
16543:
16539:
16534:
16527:
16522:
16515:
16511:
16507:
16502:
16495:
16490:
16483:
16479:
16474:
16472:
16465:, p. 92.
16464:
16459:
16452:
16447:
16441:, p. 94.
16440:
16435:
16428:
16427:Bourbaki 1998
16423:
16417:, p. 17.
16416:
16411:
16405:, p. 86.
16404:
16400:
16396:
16395:Springer 2000
16391:
16385:, p. 87.
16384:
16379:
16372:
16367:
16361:, p. 86.
16360:
16355:
16348:
16343:
16339:
16327:
16326:vector fields
16323:
16319:
16315:
16309:
16301:
16297:
16292:
16288:
16287:homeomorphism
16282:
16275:
16269:
16239:
16235:
16215:
16212:
16209:
16206:
16196:
16189:
16188:Dudley (1989)
16171:
16167:
16144:
16140:
16129:
16122:
16118:
16114:
16096:
16088:
16082:
16077:
16069:
16063:
16058:
16050:
16047:
16044:
16033:
16027:
16020:
16016:
16010:
15991:
15987:
15983:
15975:
15971:
15965:
15958:
15954:
15948:
15941:
15935:
15928:
15924:
15920:
15914:
15895:
15886:
15873:
15869:
15862:
15860:
15856:
15852:
15848:
15847:Grassmannians
15844:
15840:
15836:
15831:
15829:
15825:
15821:
15817:
15812:
15808:
15786:
15773:
15772:above section
15752:
15744:
15734:
15730:
15724:
15720:
15715:
15711:
15707:
15702:
15698:
15692:
15688:
15683:
15679:
15675:
15659:
15651:
15632:
15620:
15617:
15611:
15608:
15605:
15597:
15594:vector space
15593:
15589:
15588:affine spaces
15581:
15577:
15573:
15568:
15563:
15559:
15549:
15547:
15543:
15539:
15538:division ring
15535:
15531:
15527:
15523:
15519:
15515:
15514:
15510:
15505:
15504:abelian group
15501:
15497:
15493:
15489:
15485:
15481:
15476:
15466:
15464:
15460:
15456:
15452:
15447:
15445:
15442:
15438:
15435:
15431:
15427:
15423:
15418:
15414:
15411:
15407:
15403:
15399:
15394:
15390:
15385:
15381:
15377:
15372:
15370:
15365:
15361:
15357:
15353:
15349:
15345:
15340:
15336:
15331:
15326:
15322:
15318:
15313:
15309:
15305:
15302:
15299:, there is a
15298:
15294:
15290:
15286:
15283:a product of
15282:
15278:
15274:
15269:
15265:
15259:
15255:
15251:
15246:
15242:
15236:
15231:
15227:
15223:
15219:
15203:
15197:
15194:
15191:
15183:
15179:
15175:
15172:
15168:
15167:vector bundle
15159:
15155:
15151:
15146:
15141:
15137:
15136:Vector bundle
15122:
15120:
15102:
15092:
15087:
15077:
15074:
15069:
15059:
15054:
15039:
15023:
15018:
15008:
15003:
14974:
14964:
14959:
14944:
14940:
14924:
14917:
14901:
14896:
14886:
14883:
14880:
14875:
14865:
14860:
14846:
14830:
14807:
14801:
14791:
14786:
14784:
14783:cross product
14768:
14763:
14748:
14732:
14729:
14726:
14723:
14720:
14717:
14714:
14708:
14705:
14702:
14679:
14659:
14647:
14631:
14628:
14619:
14616:
14613:
14607:
14604:
14598:
14589:
14586:
14583:
14577:
14574:
14568:
14559:
14556:
14553:
14547:
14544:
14534:
14531:
14512:
14509:
14506:
14500:
14497:
14491:
14488:
14485:
14475:
14474:
14473:
14459:
14439:
14416:
14413:
14410:
14399:
14398:
14392:
14390:
14386:
14382:
14378:
14359:
14353:
14346:
14341:
14339:
14335:
14331:
14327:
14308:
14283:
14277:
14274:
14271:
14268:
14265:
14258:
14251:
14248:
14245:
14229:
14213:
14210:
14207:
14204:
14201:
14193:
14188:
14183:
14179:
14169:
14166:
14162:
14157:
14153:
14149:
14145:
14144:wavefunctions
14141:
14137:
14133:
14128:
14123:
14121:
14117:
14113:
14109:
14093:
14090:
14082:
14078:
14077:
14060:
14057:
14049:
14045:
14026:
14023:
14020:
14009:
13993:
13986:
13982:
13964:
13960:
13950:
13936:
13930:
13924:
13916:
13891:
13885:
13862:
13859:
13856:
13843:
13837:
13828:
13822:
13813:
13809:
13803:
13797:
13791:
13768:
13754:
13750:
13741:
13740:David Hilbert
13737:
13728:
13723:
13722:Hilbert space
13713:
13711:
13707:
13702:
13689:
13686:
13679:
13673:
13670:
13663:
13658:
13651:
13643:
13639:
13635:
13629:
13623:
13619:
13608:
13596:
13560:
13554:
13530:
13524:
13504:
13501:
13494:
13488:
13485:
13478:
13473:
13466:
13458:
13454:
13450:
13444:
13436:
13432:
13427:
13416:
13404:
13398:
13395:
13370:
13364:
13359:
13349:
13345:
13321:
13318:
13313:
13309:
13305:
13302:
13299:
13294:
13290:
13286:
13281:
13277:
13266:
13264:
13259:
13246:
13232:
13226:
13217:
13201:
13195:
13190:
13182:
13152:
13148:
13145:The space of
13143:
13130:
13124:
13121:
13115:
13107:
13101:
13098:
13091:
13078:
13072:
13058:
13053:
13048:
13043:
13035:
13025:
12998:
12995:
12986:
12973:
12970:
12965:
12962:
12958:
12954:
12949:
12945:
12941:
12936:
12933:
12929:
12921:
12917:
12911:
12908:
12905:
12901:
12897:
12892:
12882:
12852:
12849:
12841:
12838:
12834:
12830:
12824:
12821:
12816:
12813:
12809:
12799:
12784:
12755:
12752:
12749:
12746:
12726:
12720:
12717:
12709:
12693:
12690:
12668:
12665:
12661:
12638:
12634:
12613:
12609:
12605:
12602:
12599:
12596:
12593:
12590:
12585:
12582:
12578:
12574:
12571:
12568:
12563:
12560:
12556:
12552:
12547:
12544:
12540:
12535:
12531:
12526:
12499:
12496:
12474:
12470:
12460:
12447:
12441:
12438:
12426:
12423:
12417:
12411:
12399:
12395:
12384:
12380:
12375:
12370:
12365:
12328:
12322:
12319:
12303:
12299:
12288:
12280:
12236:
12233:
12230:
12227:
12217:
12203:
12181:
12177:
12174:
12169:
12165:
12161:
12158:
12155:
12150:
12146:
12142:
12137:
12133:
12128:
12124:
12111:
12096:
12092:
12080:
12078:
12077:Stefan Banach
12074:
12073:
12072:Banach spaces
12067:
12060:Banach spaces
12057:
12054:
12008:
11986:
11982:
11959:
11956:
11950:
11942:
11937:
11922:
11917:
11870:
11862:
11846:
11843:
11830:
11825:
11802:
11768:converges to
11753:
11723:
11720:
11717:
11706:
11702:
11683:
11680:
11677:
11666:
11650:
11644:
11641:
11638:
11627:
11623:
11603:
11580:
11577:
11574:
11571:
11568:
11565:
11557:
11543:
11520:
11505:
11501:
11497:
11495:
11491:
11487:
11483:
11467:
11464:
11457:
11439:
11435:
11414:
11411:
11391:
11388:
11383:
11379:
11375:
11370:
11366:
11357:
11339:
11335:
11331:
11328:
11325:
11320:
11316:
11304:
11293:
11285:
11281:
11270:
11267:
11264:
11260:
11252:
11248:
11244:
11239:
11225:
11205:
11197:
11172:
11147:
11127:
11107:
11055:
11051:
11047:
11031:
11021:
11011:
10998:
10993:
10989:
10983:
10979:
10975:
10970:
10966:
10960:
10956:
10952:
10947:
10943:
10937:
10933:
10929:
10924:
10920:
10914:
10910:
10906:
10903:
10874:
10870:
10854:
10848:
10845:
10842:
10839:
10836:
10833:
10830:
10824:
10773:
10757:
10752:
10748:
10742:
10738:
10734:
10729:
10725:
10719:
10715:
10711:
10706:
10702:
10696:
10692:
10688:
10683:
10679:
10673:
10669:
10665:
10622:
10607:
10603:
10587:
10584:
10573:
10545:
10527:
10509:
10505:
10493:
10478:
10474:
10471:
10468:
10460:
10447:
10431:
10384:
10379:
10352:
10347:
10343:
10337:
10333:
10329:
10326:
10323:
10318:
10314:
10308:
10304:
10300:
10292:
10284:
10273:
10257:
10239:
10235:
10225:
10223:
10219:
10202:
10189:
10176:
10139:
10128:
10079:
10078:inner product
10075:
10069:
10065:
10055:
10039:
10035:
10014:
9992:
9988:
9967:
9962:
9958:
9954:
9949:
9945:
9941:
9938:
9930:
9926:
9922:
9904:
9877:
9856:
9853:
9845:
9844:partial order
9840:
9837:
9831:
9827:
9823:
9812:
9810:
9794:
9783:
9772:
9769:
9758:
9747:
9728:
9725:
9705:
9697:
9681:
9678:
9658:
9655:
9649:
9646:
9643:
9640:
9637:
9630:bilinear map
9629:
9613:
9606:vector space
9605:
9584:
9551:
9536:
9520:
9517:
9514:
9494:
9491:
9488:
9468:
9442:
9437:
9427:
9419:
9414:
9404:
9393:
9391:
9378:
9368:
9363:
9350:
9331:
9326:
9316:
9308:
9303:
9290:
9288:
9275:
9267:
9257:
9252:
9230:
9213:
9202:
9199:
9193:
9182:
9171:
9160:
9157:
9148:
9146:
9130:
9119:
9116:
9092:
9087:
9077:
9072:
9062:
9059:
9056:
9051:
9041:
9036:
9026:
9021:
9011:
9006:
8992:
8976:
8973:
8965:
8957:
8953:
8949:
8936:
8903:
8892:
8842:
8795:
8787:
8771:
8768:
8765:
8758:
8742:
8736:
8733:
8730:
8727:
8724:
8716:
8700:
8680:
8660:
8657:
8654:
8651:
8631:
8628:
8623:
8619:
8615:
8608:
8602:
8592:
8578:
8556:
8552:
8546:
8543:
8540:
8536:
8527:
8524:(also called
8509:
8505:
8499:
8496:
8493:
8489:
8481:
8463:
8459:
8436:
8409:
8402:
8386:
8364:
8361:
8358:
8353:
8348:
8338:
8314:
8310:
8285:
8281:
8275:
8272:
8269:
8265:
8255:
8250:
8248:
8244:
8238:
8234:
8224:
8222:
8203:
8197:
8194:
8190:
8183:
8177:
8174:
8170:
8166:
8158:
8155:(also called
8154:
8150:
8146:
8142:
8126:
8118:
8113:
8110:
8104:
8087:
8084:
8078:
8072:
8064:
8048:
8022:
8018:
8015:
8012:
7999:
7996:
7993:
7964:
7960:
7951:
7947:
7934:
7931:
7928:
7903:
7895:
7880:
7859:
7851:
7835:
7827:
7823:
7819:
7814:
7809:
7805:
7796:
7792:
7786:
7781:
7778:
7775:
7771:
7767:
7761:
7755:
7749:
7729:
7726:
7704:
7700:
7679:
7676:
7673:
7665:
7661:
7657:
7652:
7647:
7643:
7634:
7630:
7626:
7623:
7620:
7612:
7608:
7604:
7599:
7594:
7590:
7581:
7577:
7573:
7567:
7564:
7559:
7556:
7548:
7544:
7540:
7537:
7532:
7528:
7507:
7482:
7474:
7432:
7407:
7390:
7376:
7356:
7333:
7330:
7322:
7308:
7302:
7296:
7290:
7287:
7280:
7264:
7200:
7194:
7191:
7188:
7165:
7159:
7156:
7149:
7144:
7130:
7110:
7088:
7059:
7044:
7029:
7026:
7021:
7011:
7008:
7005:
7000:
6973:
6970:
6959:
6956:
6950:
6944:
6941:
6930:
6927:
6907:
6904:
6900:
6894:
6884:
6879:
6868:
6847:
6844:
6839:
6812:
6809:
6804:
6777:
6735:
6729:
6726:
6718:
6710:
6699:
6696:
6693:
6668:
6661:
6646:
6626:
6622:
6618:
6598:
6595:
6592:
6584:
6579:
6577:
6576:
6559:
6556:
6553:
6545:
6541:
6537:
6533:
6529:
6524:
6510:
6502:
6486:
6466:
6458:
6442:
6422:
6402:
6394:
6378:
6370:
6354:
6312:
6292:
6285:
6277:
6272:
6271:
6265:
6260:
6255:
6251:
6241:
6233:
6230:
6225:
6220:
6215:
6211:
6206:
6200:
6196:
6191:
6187:
6181:
6175:
6170:
6167:, called the
6165:
6148:
6145:
6139:
6136:
6133:
6130:
6127:
6112:
6106:
6100:
6093:
6089:
6085:
6079:
6075:
6070:
6065:
6059:
6055:
6050:
6045:
6040:
6033:
6029:
6025:
6022:
6016:
6009:
6005:
5999:
5993:
5987:
5983:
5979:
5974:
5973:Endomorphisms
5969:
5959:
5957:
5953:
5949:
5944:
5938:
5934:
5933:square matrix
5928:
5923:
5911:
5902:
5893:
5888:
5883:
5879:
5876:
5872:
5868:
5863:
5858:
5852:
5833:
5825:
5809:
5773:
5765:
5761:
5745:
5725:
5721:
5715:
5711:
5705:
5702:
5698:
5692:
5687:
5684:
5681:
5677:
5673:
5670:
5667:
5662:
5658:
5652:
5649:
5645:
5639:
5634:
5631:
5628:
5624:
5620:
5615:
5611:
5605:
5602:
5598:
5592:
5587:
5584:
5581:
5577:
5572:
5560:
5556:
5552:
5549:
5546:
5541:
5537:
5533:
5528:
5524:
5517:
5503:
5497:
5480:
5471:
5465:
5460:
5452:
5447:
5443:
5433:
5430:
5424:
5418:
5412:
5406:
5402:
5398:
5392:
5386:
5380:
5376:-dimensional
5375:
5371:
5367:
5362:
5356:
5350:
5344:
5339:
5334:
5330:
5323:
5317:
5313:
5309:
5303:
5297:
5295:
5290:
5286:
5281:
5276:
5271:
5270:
5264:
5258:
5251:
5247:
5239:
5235:
5227:
5223:
5218:
5210:
5206:
5200:
5197:
5191:
5187:up (down, if
5185:
5179:
5173:
5166:
5162:
5155:
5149:
5144:
5140:
5135:
5130:
5121:
5115:
5109:
5103:
5099:
5096:
5090:
5084:
5078:
5073:
5068:
5062:
5057:
5053:
5048:
5043:
5042:identity maps
5038:
5034:
5030:
5026:
5020:
5016:
5012:
5008:
5004:
4999:
4995:
4991:
4987:
4982:
4978:
4974:
4969:
4968:
4962:
4949:
4946:
4926:
4906:
4903:
4824:
4821:
4818:
4815:
4813:
4800:
4797:
4791:
4784:
4770:
4767:
4753:
4750:
4748:
4735:
4724:
4712:
4708:
4704:
4698:
4688:
4686:
4668:
4664:
4643:
4623:
4603:
4598:
4595:
4591:
4587:
4584:
4581:
4576:
4573:
4569:
4565:
4562:
4556:
4550:
4528:
4525:
4519:
4513:
4510:
4504:
4492:
4488:
4485:
4479:
4464:
4455:
4453:
4434:
4431:
4428:
4422:
4409:
4388:
4369:
4363:
4360:
4357:
4354:
4351:
4304:
4298:
4293:
4288:
4281:
4276:
4271:
4265:
4260:
4257:
4235:
4227:
4219:
4211:
4209:
4193:
4189:
4185:
4182:
4179:
4176:
4173:
4153:
4150:
4146:
4142:
4139:
4136:
4117:
4114:
4090:
4087:
4081:
4076:
4070:
4067:
4061:
4058:
4054:
4047:
4041:
4034:
4031:
4025:
4017:
4014:
4008:
4005:
4001:
3994:
3980:
3974:
3970:
3966:
3956:
3954:
3949:
3945:
3944:integrability
3941:
3936:
3931:
3927:
3923:
3903:
3897:
3891:
3888:
3882:
3876:
3873:
3867:
3858:
3855:
3852:
3826:
3823:
3820:
3808:
3802:
3796:
3768:
3762:
3759:
3756:
3750:
3744:
3741:
3738:
3732:
3723:
3720:
3717:
3681:
3678:
3675:
3672:
3663:
3658:
3648:
3645:
3623:
3618:
3601:
3595:
3589:
3583:
3579:
3578:smaller field
3576:containing a
3574:
3569:
3565:
3560:
3558:
3557:complex plane
3552:
3548:
3541:
3538:
3534:
3529:
3524:
3518:
3512:
3506:
3500:
3493:
3489:
3485:
3481:
3477:
3473:
3469:
3465:
3458:
3454:
3450:
3446:
3442:
3438:
3434:
3430:
3426:
3420:
3415:
3409:
3403:
3399:
3394:
3390:
3384:
3380:
3370:
3368:
3362:
3356:
3352:
3346:
3339:
3334:
3331:and called a
3329:
3323:
3316:
3312:
3290:
3286:
3282:
3279:
3276:
3271:
3267:
3263:
3258:
3254:
3241:
3236:
3233:
3226:
3222:is the field
3220:
3209:
3207:
3202:
3185:
3179:
3176:
3173:
3170:
3167:
3161:
3159:
3151:
3148:
3145:
3139:
3132:
3124:
3120:
3116:
3111:
3107:
3103:
3098:
3094:
3090:
3085:
3081:
3074:
3072:
3062:
3058:
3054:
3049:
3045:
3038:
3030:
3026:
3022:
3017:
3013:
2995:
2991:
2985:
2959:
2956:
2950:
2946:
2939:
2935:
2929:
2922:
2916:
2910:
2907:
2900:
2894:
2891:
2888:
2884:is negative,
2882:
2876:
2873:
2867:
2861:
2856:
2851:
2845:
2839:
2835:
2830:
2826:
2821:
2817:
2816:parallelogram
2812:
2806:
2801:
2797:
2793:
2789:
2777:
2770:
2761:
2749:
2743:
2737:
2733:
2725:
2710:
2700:
2698:
2694:
2692:
2686:
2682:
2678:
2674:
2670:
2666:
2661:
2659:
2654:
2650:
2646:
2643:, as well as
2642:
2638:
2634:
2630:
2626:
2622:
2617:
2615:
2611:
2608:goes back to
2607:
2603:
2599:
2595:
2591:
2587:
2583:
2578:
2576:
2572:
2568:
2564:
2560:
2556:
2552:
2551:Möbius (1827)
2548:
2544:
2540:
2536:
2532:
2528:
2524:
2514:
2512:
2508:
2503:
2497:
2496:componentwise
2471:
2467:
2457:
2452:
2448:
2439:
2434:
2429:
2424:
2406:
2402:
2398:
2395:
2392:
2387:
2383:
2356:
2352:
2348:
2345:
2342:
2337:
2333:
2312:
2307:
2295:
2291:
2287:
2284:
2281:
2276:
2264:
2260:
2256:
2231:
2228:
2212:over a field
2208:of dimension
2186:
2176:
2173:
2170:
2165:
2155:
2150:
2131:
2129:
2125:
2121:
2117:
2112:
2104:
2100:
2096:
2092:
2088:
2084:
2082:
2078:
2075:
2068:
2064:
2063:
2051:
2047:
2006:
2002:
1990:
1988:
1985:
1983:
1981:
1953:
1938:
1934:
1930:
1928:
1925:
1906:
1890:
1888:
1885:
1882:
1864:
1860:
1856:
1853:
1850:
1845:
1841:
1820:
1817:
1814:
1809:
1799:
1796:
1793:
1788:
1761:
1758:
1753:
1749:
1745:
1742:
1739:
1734:
1730:
1709:
1704:
1692:
1688:
1684:
1681:
1678:
1673:
1661:
1657:
1653:
1648:
1636:
1632:
1603:
1601:
1598:
1597:
1589:
1582:
1578:
1573:
1565:
1562:
1555:
1552:
1548:
1542:
1537:
1532:
1526:
1520:
1511:
1509:
1505:
1486:
1478:
1453:
1450:
1447:
1422:
1414:
1407:
1393:
1385:
1382:
1371:
1368:
1358:
1344:
1336:
1328:
1321:
1307:
1299:
1291:
1284:
1283:
1282:
1268:
1265:
1262:
1237:
1234:
1231:
1222:
1209:
1198:
1192:
1184:
1176:
1162:
1160:
1155:
1150:
1146:
1145:abelian group
1141:
1139:
1132:
1118:
1114:
1110:
1106:
1096:
1093:
1089:
1086:
1082:
1078:
1074:
1069:
1066:
1065:
1061:
1058:
1054:
1051:
1047:
1043:
1039:
1035:
1032:
1029:
1028:
1020:
1011:
1007:
1002:
999:
998:
994:
990:
986:
983:
979:
975:
972:
971:
966:
962:
958:
952:
947:
946:
942:, called the
940:
936:
929:
925:
920:
917:
914:
913:
908:
904:
898:
894:
890:
885:
884:
880:, called the
878:
874:
869:
866:
863:
862:
858:
854:
850:
846:
842:
839:
838:Commutativity
836:
835:
831:
827:
823:
819:
815:
811:
807:
804:
803:Associativity
801:
800:
796:
793:
792:
789:
770:
764:
758:
753:
744:
741:
727:
714:
713:
708:
707:
703:
698:
694:
680:
674:
669:
665:
661:
660:
659:
657:
649:
641:
637:
629:
622:
617:
602:
597:
595:
590:
588:
583:
582:
580:
579:
571:
568:
566:
563:
561:
558:
556:
553:
551:
548:
546:
543:
541:
538:
537:
533:
530:
529:
525:
520:
519:
512:
511:
507:
506:
502:
499:
497:
494:
492:
489:
488:
483:
478:
477:
470:
469:
465:
463:
460:
459:
455:
452:
450:
447:
445:
442:
440:
437:
435:
432:
430:
427:
426:
421:
416:
415:
410:
409:
402:
399:
397:
396:Division ring
394:
392:
389:
387:
384:
382:
379:
377:
374:
372:
369:
367:
364:
362:
359:
357:
354:
353:
348:
343:
342:
337:
336:
329:
326:
324:
321:
319:
318:Abelian group
316:
315:
311:
308:
306:
303:
301:
297:
294:
292:
289:
288:
284:
279:
278:
275:
272:
271:
268:
266:
265:Banach spaces
262:
258:
257:normed spaces
254:
250:
246:
242:
238:
234:
230:
225:
223:
219:
215:
211:
207:
203:
199:
195:
191:
187:
183:
178:
176:
172:
168:
164:
161:, but also a
160:
156:
152:
148:
144:
139:
137:
133:
129:
125:
121:
117:
116:vector axioms
113:
109:
108:
103:
102:
97:
93:
89:
85:
81:
72:
68:
63:
58:
52:
46:
40:
33:
19:
20053:Group theory
20035:
19935:Vector space
19934:
19667:Vector space
19666:
19580:
19534:
19521:, retrieved
19514:the original
19483:
19479:
19450:
19422:
19417:Stewart, Ian
19408:
19373:
19369:
19335:, New York:
19332:
19313:
19278:
19255:, New York:
19252:
19228:
19207:, New York:
19204:
19179:
19157:
19135:
19109:
19103:
19081:
19056:
19031:
19001:
18974:
18968:
18946:
18921:
18896:
18863:
18844:
18807:
18784:
18729:
18708:
18702:
18694:the original
18688:
18674:
18651:
18628:
18619:
18605:
18575:
18569:
18556:
18543:
18539:
18526:
18498:
18492:
18461:
18427:
18406:
18379:
18343:
18321:
18296:
18263:
18242:
18221:
18203:
18185:
18167:
18145:
18123:
18105:
18087:
18075:
18058:
18032:
18010:
17985:
17951:
17939:
17924:, Springer,
17920:
17901:
17879:
17866:
17842:
17823:
17790:
17761:
17742:
17732:
17713:
17686:
17677:
17648:
17628:
17604:
17592:
17570:Coxeter 1987
17565:
17553:
17541:
17534:Grillet 2007
17529:
17517:
17505:
17493:
17481:
17469:
17464:, ch. XVI.8.
17457:
17452:, ch. XVI.7.
17445:
17433:
17421:
17409:
17397:
17392:, Chapter 1.
17385:
17373:
17366:Choquet 1966
17361:
17349:
17337:
17325:
17313:
17301:
17289:
17277:
17265:
17253:
17246:Choquet 1966
17241:
17229:
17217:
17205:
17189:
17177:
17165:
17153:
17141:
17129:
17122:Yoneda lemma
17118:Roman (2005)
17113:
17108:, ch. XVI.1.
17101:
17089:
17077:
17065:
17053:
17048:, ch. IV.3..
17041:
17029:
17017:
17005:
16993:
16981:
16969:
16957:
16945:
16933:
16921:
16909:
16897:
16885:
16873:
16861:
16849:
16837:
16825:
16820:, ch. VI.3..
16813:
16801:
16789:
16777:
16761:
16745:
16729:
16713:
16701:
16689:
16677:
16665:
16653:
16641:
16629:
16617:
16605:
16598:Bolzano 1804
16593:
16581:
16565:
16549:
16533:
16521:
16501:
16489:
16458:
16446:
16434:
16422:
16410:
16390:
16378:
16366:
16354:
16342:
16313:
16308:
16299:
16295:
16290:
16281:
16268:
16195:
16128:
16026:
16009:
15970:Roman (2005)
15964:
15953:affine space
15947:
15934:
15926:
15913:
15872:
15854:
15838:
15837:is known as
15834:
15832:
15827:
15819:
15815:
15810:
15806:
15732:
15728:
15722:
15718:
15713:
15709:
15700:
15696:
15690:
15686:
15681:
15677:
15673:
15587:
15585:
15579:
15575:
15572:affine plane
15558:Affine space
15534:vector space
15533:
15518:free modules
15512:
15508:
15499:
15491:
15487:
15479:
15478:
15448:
15443:
15436:
15429:
15425:
15412:
15401:
15398:vector field
15392:
15388:
15383:
15373:
15363:
15359:
15347:
15344:Möbius strip
15338:
15334:
15329:
15324:
15320:
15316:
15311:
15307:
15303:
15301:neighborhood
15296:
15292:
15291:: for every
15288:
15284:
15276:
15267:
15263:
15257:
15253:
15249:
15244:
15239:is called a
15234:
15229:
15221:
15217:
15181:
15177:
15173:
15166:
15164:
15157:
15153:
14787:
14651:
14397:Lie algebras
14395:
14393:
14342:
14337:
14333:
14329:
14323:
14124:
14080:
14074:
13951:
13913:denotes the
13735:
13733:
13703:
13260:
13144:
12987:
12461:
12081:
12070:
12069:
12066:Banach space
11938:
11704:
11619:
11354:denotes the
11251:infinite sum
11242:
11240:
11023:
10226:
10221:
10217:
10071:
9925:Riesz spaces
9841:
9821:
9818:
9627:
9603:
9533:that maps a
8963:
8961:
8606:
8604:
8528:and denoted
8479:
8253:
8251:
8246:
8242:
8240:
8114:
8108:
8102:
7391:
7145:
6582:
6580:
6573:
6572:is called a
6543:
6539:
6535:
6531:
6527:
6525:
6392:
6368:
6281:
6269:
6239:
6228:
6223:
6218:
6204:
6198:
6194:
6185:
6179:
6173:
6163:
6110:
6104:
6098:
6091:
6087:
6084:identity map
6077:
6073:
6063:
6057:
6053:
6048:
6043:
6038:
6031:
6027:
6023:
6020:
6014:
6007:
6003:
5997:
5991:
5985:
5981:
5977:
5971:
5951:
5947:
5942:
5936:
5926:
5919:
5909:
5900:
5891:
5874:
5870:
5866:
5861:
5856:
5850:
5847:
5501:
5495:
5469:
5463:
5458:
5457:
5428:
5422:
5416:
5410:
5404:
5400:
5396:
5390:
5384:
5378:
5373:
5365:
5360:
5354:
5348:
5342:
5332:
5328:
5321:
5315:
5311:
5307:
5301:
5298:
5293:
5288:
5284:
5274:
5267:
5262:
5256:
5249:
5245:
5237:
5233:
5225:
5221:
5216:
5208:
5204:
5202:Linear maps
5201:
5195:
5189:
5183:
5177:
5171:
5164:
5160:
5153:
5147:
5133:
5126:
5119:
5113:
5107:
5094:
5088:
5082:
5076:
5071:
5066:
5060:
5054:) and onto (
5046:
5036:
5032:
5028:
5024:
5018:
5014:
5010:
5006:
5003:compositions
4997:
4993:
4989:
4980:
4976:
4972:
4965:
4963:
4706:
4702:
4700:
4542:
4451:
4406:denotes the
4249:
3976:
3934:
3806:
3800:
3794:
3787:
3643:
3599:
3593:
3587:
3581:
3572:
3561:
3550:
3546:
3539:
3536:
3532:
3527:
3522:
3516:
3510:
3504:
3498:
3491:
3487:
3483:
3479:
3475:
3471:
3467:
3463:
3456:
3452:
3448:
3444:
3440:
3436:
3432:
3428:
3424:
3413:
3407:
3401:
3398:real numbers
3392:
3388:
3382:
3376:
3366:
3360:
3354:
3350:
3344:
3337:
3335:. The case
3332:
3327:
3321:
3314:
3310:
3308:of elements
3239:
3231:
3224:
3218:
3215:
3203:
2993:
2989:
2984:ordered pair
2965:
2954:
2948:
2944:
2941:. Moreover,
2937:
2933:
2927:
2920:
2914:
2908:
2905:
2898:
2895:
2889:
2886:
2880:
2874:
2871:
2865:
2859:
2854:
2853:, is called
2849:
2843:
2837:
2828:
2824:
2819:
2810:
2804:
2785:
2775:
2768:
2747:
2741:
2735:
2731:
2690:
2662:
2618:
2601:
2597:
2579:
2567:equipollence
2520:
2504:
2455:
2437:
2433:coefficients
2432:
2427:
2422:
2132:
2124:real numbers
2110:
2108:
2098:
2086:
2066:
2062:spanning set
2060:
2049:
2045:
2004:
2000:
1980:intersection
1977:
1936:
1932:
1904:
1881:coefficients
1880:
1833:The scalars
1624:of the form
1604:Given a set
1587:
1580:
1576:
1563:
1560:
1553:
1550:
1546:
1540:
1530:
1524:
1501:
1223:
1163:
1153:
1142:
1134:
1131:vector space
1130:
1116:
1108:
1105:real numbers
1102:
1094:
1091:
1087:
1084:
1080:
1076:
1072:
1059:
1056:
1052:
1049:
1045:
1041:
1037:
1017:denotes the
1009:
1005:
992:
988:
984:
981:
977:
964:
960:
956:
954:, such that
950:
943:
938:
934:
927:
923:
906:
902:
896:
892:
888:
886:, such that
881:
876:
872:
856:
852:
848:
844:
829:
825:
821:
817:
813:
809:
768:
762:
756:
749:
742:
739:
725:
710:
701:
696:
692:
678:
672:
667:
663:
655:
647:
618:
615:
570:Hopf algebra
508:
501:Vector space
500:
466:
406:
335:Group theory
333:
298: /
245:Lie algebras
226:
197:
189:
179:
140:
128:real numbers
123:
119:
115:
106:
100:
92:linear space
91:
88:vector space
87:
77:
70:
66:
61:
56:
50:
32:Vector field
18:Linear space
19915:Multivector
19880:Determinant
19837:Dot product
19682:Linear span
19314:Gravitation
19305:Thorne, Kip
18625:, reprint:
18530:(in German)
18501:: 133–181,
18318:Lang, Serge
18293:Lang, Serge
17787:Lang, Serge
17510:Atiyah 1989
17474:Spivak 1999
17318:Treves 1967
17294:Treves 1967
17282:Treves 1967
17260:, p. 34–36.
17258:Treves 1967
17194:Treves 1967
17024:, ch. 10.4.
16976:, ch. IX.4.
16890:Halmos 1974
16880:, ch. VI.6.
16718:Dorier 1995
16706:Banach 1922
16634:Dorier 1995
16610:Möbius 1827
16570:Halmos 1948
16274:Hamel basis
15957:zero vector
15528:which is a
15434:quaternions
15241:line bundle
15117:yields the
14345:polynomials
14182:Lie algebra
14156:eigenstates
14148:eigenvalues
13706:derivatives
13149:on a given
12708:zero vector
11941:functionals
10600:are called
10256:dot product
9696:composition
8422:an element
7850:derivatives
6532:vector line
6347:-vector of
6282:A nonempty
6044:eigenvector
6018:satisfying
5922:determinant
5864:linear map
5446:Determinant
4986:inverse map
4967:isomorphism
4709:. They are
3977:Systems of
3928:, or other
3792:to a field
3377:The set of
2931:is the sum
2834:real number
2790:in a fixed
2745:(blue) and
2629:linear maps
2594:quaternions
2573:is then an
2527:coordinates
2423:coordinates
2116:Hamel bases
2095:cardinality
2055:, and that
2001:linear span
1987:Linear span
1970:belongs to
883:zero vector
654:are called
555:Lie algebra
540:Associative
444:Total order
434:Semilattice
408:Ring theory
80:mathematics
20042:Categories
19949:Direct sum
19784:Invertible
19687:Linear map
19523:2017-10-25
18756:vct axioms
18416:0070542368
17579:References
17558:Meyer 2000
17546:Meyer 2000
17522:Artin 1991
17428:, ch. 1.6.
17306:Evans 1998
17184:, ch. 1.2.
17182:Naber 2003
17170:Roman 2005
17134:Rudin 1991
17094:Roman 2005
17072:, ch. 7.4.
17058:Roman 2005
17034:Roman 2005
17010:Roman 2005
16998:Roman 2005
16986:Roman 2005
16962:Roman 2005
16928:, ch. V.1.
16914:Roman 2005
16856:, ch. 7.3.
16830:Roman 2005
16796:, ch. V.1.
16784:, ch. I.1.
16768:, p.
16752:, p.
16736:, p.
16722:Moore 1995
16694:Moore 1995
16670:Peano 1888
16572:, p.
16556:, p.
16540:, p.
16538:Joshi 1989
16526:Blass 1984
16512:, p.
16494:Roman 2005
16480:, p.
16463:Brown 1991
16451:Brown 1991
16439:Brown 1991
16403:Brown 1991
16397:, p.
16383:Brown 1991
16371:Roman 2005
16359:Brown 1991
16004:from this.
15369:orientable
15150:looks like
14914:where the
14747:commutator
13584:such that
13218:, denoted
12255:given by
11883:-norm and
10602:orthogonal
8784:is called
8644:or simply
8480:direct sum
8247:direct sum
7467:such that
6575:hyperplane
6224:eigenspace
6210:eigenbasis
6192:, such as
6054:eigenvalue
5954:matrix is
5272:, denoted
5072:isomorphic
5056:surjective
4703:linear map
4697:Linear map
3940:continuity
3570:: a field
3528:isomorphic
2800:velocities
2779:are shown.
2689:spaces of
1572:orthogonal
921:For every
797:Statement
666:or simply
229:structures
186:isomorphic
149:, such as
19979:Numerical
19742:Transpose
19587:EMS Press
19510:0098-3063
19488:CiteSeerX
19390:0010-485X
19370:Computing
18515:0016-2736
18450:840278135
18398:144216834
18372:702357363
18346:, Dover,
18065:EMS Press
17524:, ch. 12.
17462:Lang 2002
17450:Lang 2002
17414:Lang 2002
17402:Lang 1993
17354:Lang 1993
17342:Lang 1993
17320:, ch. 12.
17284:, ch. 11.
17270:Lang 1983
17224:, §4.11-5
17106:Lang 2002
17046:Lang 1987
16950:Lang 1987
16938:Lang 1987
16926:Lang 1987
16902:Lang 1987
16878:Lang 1987
16866:Lang 1987
16842:Lang 1987
16818:Lang 1987
16806:Lang 1993
16794:Lang 2002
16782:Lang 1987
16766:Jain 2001
16672:, ch. IX.
16554:Heil 2011
16506:Lang 1987
16347:Lang 2002
16335:Citations
16248:Ω
16210:≠
16093:‖
16086:‖
16074:‖
16067:‖
16064:≤
16055:‖
16042:‖
15890:→
15824:nullspace
15704:(it is a
15644:↦
15615:→
15609:×
15586:Roughly,
15441:octonions
15201:→
15192:π
15093:⊗
15078:−
15060:⊗
15009:⊗
14965:⊗
14887:⊗
14884:⋯
14881:⊗
14866:⊗
14802:
14724:−
14501:−
14381:quotients
14275:−
14269:⋅
14205:⋅
14192:hyperbola
13899:¯
13851:¯
13818:Ω
13814:∫
13807:⟩
13789:⟨
13763:Ω
13708:leads to
13674:μ
13636:−
13613:Ω
13609:∫
13603:∞
13600:→
13569:Ω
13489:μ
13451:−
13421:Ω
13417:∫
13411:∞
13408:→
13368:∞
13356:‖
13342:‖
13322:…
13303:…
13241:Ω
13199:∞
13187:‖
13180:‖
13160:Ω
13102:μ
13063:Ω
13059:∫
13040:‖
13033:‖
13005:→
13002:Ω
12963:−
12955:⋅
12934:−
12902:∑
12889:‖
12873:‖
12847:→
12839:−
12814:−
12795:∞
12791:‖
12775:‖
12724:∞
12666:−
12606:…
12583:−
12572:…
12561:−
12545:−
12471:ℓ
12445:∞
12381:∑
12362:‖
12353:‖
12326:∞
12276:∞
12272:‖
12263:‖
12240:∞
12237:≤
12231:≤
12178:…
12159:…
12093:ℓ
12012:→
11987:∗
11954:→
11903:-norm on
11891:∞
11831:−
11809:∞
11806:→
11601:∞
11392:…
11329:⋯
11311:∞
11308:→
11276:∞
11261:∑
10907:−
10901:⟩
10883:⟨
10800:⟩
10782:⟨
10735:−
10663:⟩
10645:⟨
10582:⟩
10566:⟨
10528:⋅
10510:⋅
10483:∠
10475:
10461:⋅
10327:⋯
10293:⋅
10282:⟩
10266:⟨
10198:⟩
10182:⟨
10137:⟩
10121:⟨
10040:−
9963:−
9955:−
9854:≤
9826:converges
9759:⊗
9653:→
9647:×
9585:⊗
9518:⊗
9492:×
9481:from the
9428:⊗
9405:⊗
9351:⊗
9332:⊗
9309:⊗
9276:⊗
9203:⋅
9194:⊗
9172:⊗
9161:⋅
9131:⊗
9120:⋅
9078:⊗
9060:⋯
9042:⊗
9012:⊗
8964:universal
8890:↦
8769:×
8755:from the
8740:→
8734:×
8655:⊗
8620:⊗
8544:∈
8537:∐
8526:coproduct
8497:∈
8490:⨁
8401:index set
8362:∈
8273:∈
8266:∏
8198:
8191:≡
8178:
8027:′
8019:⋅
8008:′
7997:⋅
7969:′
7956:′
7943:′
7888:′
7885:′
7772:∑
7753:↦
7624:⋯
7405:↦
7331:∈
7291:
7198:→
7160:
6960:⋅
6931:⋅
6727:∈
6596:⊆
6557:−
6534:), and a
6137:⋅
6134:λ
6131:−
5823:↦
5760:summation
5746:∑
5678:∑
5671:…
5625:∑
5578:∑
5569:↦
5550:…
5080:are, via
5052:injective
4822:⋅
4801:⋅
4711:functions
4596:−
4574:−
4497:′
4472:′
4469:′
4180:−
3922:real line
3842:given by
3763:
3745:
3690:→
3280:…
2641:dimension
2633:Grassmann
2619:In 1857,
2396:…
2346:…
2285:⋯
2229:∈
2174:…
2099:dimension
2081:dimension
2050:generates
1854:…
1815:∈
1797:…
1759:∈
1743:…
1682:⋯
1522:A vector
1386:−
1369:−
1263:∈
1235:∈
1199:−
1177:−
1157:into the
565:Bialgebra
371:Near-ring
328:Lie group
296:Semigroup
220:have the
214:countably
182:dimension
163:direction
159:magnitude
20025:Category
19964:Subspace
19959:Quotient
19910:Bivector
19824:Bilinear
19766:Matrices
19641:Glossary
19561:36131259
19533:(1994).
19419:(1975),
19407:(1999),
19311:(1973),
19276:(1998),
19203:(1991),
19080:(1974),
19055:(1995),
19000:(1995),
18945:(1987),
18920:(1989),
18895:(1998),
18843:(1969),
18808:K-theory
18806:(1989),
18781:(1976),
18738:citation
18728:(1888),
18686:(1827),
18672:(1853),
18617:(1844),
18603:(1822),
18555:(1969),
18524:(1804),
18484:(1922),
18460:(1967),
18320:(1993),
18295:(1983),
18262:(1989),
18241:(1988),
18076:Topology
18074:(1966),
18009:(2004),
17984:(1987),
17974:Analysis
17950:(1993),
17878:(2005),
17822:(1999),
17789:(2002),
17676:(1948),
17603:(1991),
17476:, ch. 3.
17332:, p.190.
17308:, ch. 5.
17236:, §1.5-5
17172:, ch. 9.
16682:Guo 2021
16121:seminorm
15843:parallel
15542:spectrum
15459:Sections
15439:and the
15417:K-theory
15410:2-sphere
15356:cylinder
11626:complete
11241:In such
11046:topology
8877:the map
8786:bilinear
8399:in some
8145:category
8139:) is an
7103:lies in
6393:subspace
6036:, where
5980: :
5869: :
5758:denotes
5459:Matrices
5436:Matrices
5399: :
5310: :
5031: :
5013: :
4992: :
4975: :
4852:for all
4208:Matrices
3926:interval
2703:Examples
2649:algebras
2610:Laguerre
2590:Hamilton
2537:founded
1950:that is
1440:implies
1281:one has
1013:, where
900:for all
668:addition
401:Lie ring
366:Semiring
233:algebras
206:geometry
171:matrices
155:velocity
19636:Outline
19589:, 2001
19553:1269324
19398:9738629
19355:2044239
19128:2035388
19024:1322960
18991:2320587
18886:0763890
18830:1043170
18594:1347828
18286:0992618
17952:Algebra
17904:, CRC,
17870:, Lyryx
17824:Algebra
17813:1878556
17791:Algebra
17605:Algebra
17584:Algebra
17212:, p. 7.
16318:section
16289:from π(
15524:over a
15482:are to
15480:Modules
15469:Modules
15281:locally
15271:into a
14845:tensors
14076:closure
12029:(or to
10444:by the
9718:equals
8991:tensors
5473:matrix
5280:natural
4683:is the
4543:yields
3930:subsets
3920:is the
3555:in the
3417:is the
3235:-tuples
2878:. When
2681:algebra
2677:Hilbert
2547:Bolzano
2517:History
2486:of the
1574:basis:
1506:over a
656:scalars
648:vectors
532:Algebra
524:Algebra
429:Lattice
420:Lattice
107:scalars
101:vectors
94:) is a
84:physics
19920:Tensor
19732:Kernel
19662:Vector
19657:Scalar
19559:
19551:
19541:
19508:
19490:
19461:
19439:
19396:
19388:
19353:
19343:
19321:
19290:
19263:
19241:
19215:
19190:
19168:
19146:
19126:
19092:
19067:
19042:
19022:
19012:
18989:
18957:
18932:
18907:
18884:
18874:
18828:
18818:
18793:
18659:
18639:
18592:
18513:
18448:
18438:
18413:
18396:
18386:
18370:
18360:
18332:
18307:
18284:
18274:
18249:
18228:
18210:
18192:
18174:
18156:
18134:
18112:
18094:
18043:
18021:
17996:
17962:
17928:
17908:
17890:
17854:
17830:
17811:
17801:
17776:
17750:
17721:
17701:
17663:
17637:
17615:
17136:, p.3.
15814:where
15596:action
15582:(red).
15522:module
15475:Module
15432:, the
15224:, the
14916:degree
14532:), and
14158:. The
13875:where
13801:
13795:
13402:
13151:domain
12195:whose
11558:, for
11556:-norms
11484:. The
11297:
11291:
11247:series
11120:, and
9980:where
9822:per se
9397:
9387:
9294:
9284:
9223:
9220:
9186:
9180:
9152:
9142:
8221:groups
7148:kernel
6748:where
6660:modulo
6542:is an
6530:(also
6284:subset
6276:planes
6264:origin
6069:kernel
5907:, and
5738:where
5442:Matrix
5426:, via
5331:= dim
5294:bidual
5151:- and
5139:origin
4616:where
4410:, and
4250:where
3971:, and
3924:or an
3585:is an
3411:where
2814:, the
2796:forces
2788:arrows
2673:Banach
2621:Cayley
2586:Argand
1999:, the
1978:every
1952:closed
1722:where
1594:(red).
1504:module
794:Axiom
776:, and
752:axioms
638:and a
630:
560:Graded
491:Module
482:Module
381:Domain
300:Monoid
151:forces
19789:Minor
19774:Block
19712:Basis
19517:(PDF)
19476:(PDF)
19394:S2CID
19124:JSTOR
18987:JSTOR
18860:(PDF)
18489:(PDF)
16293:) to
15865:Notes
15859:flags
15706:coset
15530:field
15484:rings
15226:fiber
14081:basis
13334:with
12865:but
12216:-norm
11356:limit
9698:with
9535:tuple
7279:image
6536:plane
6096:. If
6052:with
5931:of a
5925:det (
5370:up to
5242:, or
3707:with
3474:) = (
3439:) = (
3431:) + (
2792:plane
2653:Peano
2543:curve
2447:tuple
2325:with
2111:Bases
2087:basis
2077:Basis
2065:or a
2059:is a
2046:spans
1895:of a
1508:field
1133:or a
987:) = (
820:) = (
621:field
526:-like
484:-like
422:-like
391:Field
349:-like
323:Magma
291:Group
285:-like
283:Group
136:field
19944:Dual
19799:Rank
19557:OCLC
19539:ISBN
19506:ISSN
19459:ISBN
19437:ISBN
19386:ISSN
19341:ISBN
19319:ISBN
19288:ISBN
19261:ISBN
19239:ISBN
19213:ISBN
19188:ISBN
19166:ISBN
19144:ISBN
19090:ISBN
19065:ISBN
19040:ISBN
19010:ISBN
18955:ISBN
18930:ISBN
18905:ISBN
18872:ISBN
18816:ISBN
18791:ISBN
18758:via
18744:link
18657:ISBN
18637:ISBN
18511:ISSN
18446:OCLC
18436:ISBN
18411:ISBN
18394:OCLC
18384:ISBN
18368:OCLC
18358:ISBN
18330:ISBN
18305:ISBN
18272:ISBN
18247:ISBN
18226:ISBN
18208:ISBN
18190:ISBN
18172:ISBN
18154:ISBN
18132:ISBN
18110:ISBN
18092:ISBN
18041:ISBN
18019:ISBN
17994:ISBN
17960:ISBN
17926:ISBN
17906:ISBN
17888:ISBN
17852:ISBN
17848:SIAM
17828:ISBN
17799:ISBN
17774:ISBN
17748:ISBN
17719:ISBN
17699:ISBN
17661:ISBN
17635:ISBN
17613:ISBN
16199:For
16034:for
16030:The
15857:and
15849:and
15726:for
15560:and
15526:ring
15449:The
15350:(by
15138:and
14989:and
14788:The
14745:the
14672:-by-
14452:and
14332:(or
14226:The
14180:and
13365:<
13196:<
12710:for
12442:<
11593:and
11492:and
11190:and
11078:and
10419:and
10220:and
10112:and
10074:norm
10066:and
10027:and
9626:and
8830:and
8693:and
8605:The
8252:The
8241:The
8235:and
7983:and
7848:the
7369:and
7146:The
7074:and
6825:and
6528:line
6457:span
6252:and
6080:· Id
5950:-by-
5920:The
5854:and
5467:-by-
5444:and
5346:and
5336:, a
5327:dim
5282:map
5117:and
5064:and
5040:are
5022:and
4919:all
4874:and
4636:and
4166:and
3804:and
3647:.
3520:and
3482:) +
3461:and
3447:) +
3405:and
3396:for
3365:(so
3358:and
2978:and
2970:and
2943:(−1)
2808:and
2772:and
2695:and
2675:and
2639:and
2600:and
2588:and
2569:. A
2533:and
2505:The
2079:and
2005:span
1774:and
1250:and
1048:) =
963:) =
959:+ (−
828:) +
780:and
766:and
676:and
356:Ring
347:Ring
263:and
243:and
212:are
153:and
130:and
122:and
86:, a
82:and
19498:doi
19378:doi
19114:doi
18979:doi
18713:doi
18580:doi
18503:doi
18432:GTM
18348:hdl
17766:doi
17691:doi
17653:doi
16738:355
16558:126
16542:450
16514:212
16399:185
15830:).
15826:of
15712:in
15708:of
15672:If
15570:An
15400:on
15306:of
15295:in
15237:= 1
15220:in
14472:):
14134:in
14083:of
13917:of
13593:lim
13392:lim
13267:not
12803:sup
12285:sup
11799:lim
11705:all
11506:in
11301:lim
11140:in
11100:in
10472:cos
10365:In
9628:any
9604:any
9572:to
9507:to
8788:if
8451:of
8175:ker
8112:).
8106:or
7257:in
7157:ker
6860:is
6395:of
6371:of
6171:of
6122:det
6046:of
5862:any
5499:to
5432:.
5420:to
5260:to
5244:𝓛(
5214:Hom
4964:An
4939:in
4896:in
4705:or
3946:or
3932:of
3760:exp
3742:sin
3724:exp
3718:sin
3679:exp
3673:sin
3466:⋅ (
3363:= 2
3340:= 1
3319:of
3243:)
2947:= −
2901:= 2
2863:by
2857:of
2820:sum
2798:or
2667:by
2584:by
2453:of
2425:of
2371:in
2069:of
2048:or
2033:If
2007:of
1935:or
1538:of
1528:in
1466:or
1021:in
948:of
812:+ (
784:in
772:in
729:in
719:in
702:sum
682:in
628:set
361:Rng
96:set
78:In
69:+ 2
20044::
19585:,
19579:,
19555:.
19549:MR
19547:.
19504:,
19496:,
19484:38
19482:,
19478:,
19457:,
19453:,
19435:,
19427:,
19392:,
19384:,
19364:;
19351:MR
19349:,
19339:,
19307:;
19303:;
19286:,
19259:,
19237:,
19186:,
19164:,
19142:,
19122:,
19110:17
19108:,
19088:,
19063:,
19038:,
19020:MR
19018:,
19008:,
18985:,
18975:86
18973:,
18953:,
18928:,
18903:,
18882:MR
18880:,
18862:,
18839:;
18826:MR
18824:,
18814:,
18777:;
18754::
18740:}}
18736:{{
18709:22
18707:,
18635:,
18590:MR
18588:,
18576:22
18574:,
18568:,
18544:13
18542:,
18509:,
18491:,
18444:.
18430:.
18392:.
18366:,
18356:,
18328:,
18303:,
18299:,
18282:MR
18280:,
18270:,
18152:,
18130:,
18126:,
18063:,
18057:,
18039:,
18017:,
17992:,
17958:,
17886:,
17850:,
17846:,
17809:MR
17807:,
17793:,
17772:,
17697:,
17659:,
17611:,
17607:,
17196:;
16770:11
16720:;
16574:12
16482:14
16470:^
16401:;
16298:×
16017:,
15927:by
15809:+
15731:∈
15721:+
15699:+
15689:∈
15511:/2
15506:)
15465:.
15457:.
15446:.
15428:,
15424::
15391:×
15362:×
15337:×
15323:→
15319:×
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2296:n
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2288:+
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2277:1
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2265:1
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2232:V
2225:v
2214:F
2210:n
2206:V
2192:)
2187:n
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2177:,
2171:,
2166:2
2161:b
2156:,
2151:1
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2141:(
2073:.
2071:W
2057:G
2053:W
2043:G
2039:G
2035:W
2031:.
2029:G
2025:G
2021:G
2017:G
2013:V
2009:G
1997:V
1993:G
1972:W
1968:W
1964:W
1960:W
1956:W
1948:V
1944:V
1940:W
1921:V
1917:G
1913:G
1909:G
1901:V
1897:F
1893:G
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1541:R
1531:R
1525:v
1487:.
1483:0
1479:=
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1427:0
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1383:=
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1372:1
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1341:0
1337:=
1333:0
1329:s
1308:,
1304:0
1300:=
1296:v
1292:0
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1238:F
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1210:.
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1203:w
1196:(
1193:+
1189:v
1185:=
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1137:F
1128:-
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1121:F
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978:a
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903:v
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873:0
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830:w
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726:v
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632:V
624:F
600:e
593:t
586:v
74:.
71:w
67:v
62:w
57:w
51:v
41:.
34:.
20:)
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