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11318:> 0. Its image consists of all sequences with first element 0, and thus its cokernel consists of the classes of sequences with identical first element. Thus, whereas its kernel has dimension 0 (it maps only the zero sequence to the zero sequence), its co-kernel has dimension 1. Since the domain and the target space are the same, the rank and the dimension of the kernel add up to the same
11581:), by rank–nullity. This gives an indication of how many solutions or how many constraints one has: if mapping from a larger space to a smaller one, the map may be onto, and thus will have degrees of freedom even without constraints. Conversely, if mapping from a smaller space to a larger one, the map cannot be onto, and thus one will have constraints even without degrees of freedom.
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An example of an unbounded, hence discontinuous, linear transformation is differentiation on the space of smooth functions equipped with the supremum norm (a function with small values can have a derivative with large values, while the derivative of 0 is 0). For a specific example,
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14331:{\displaystyle ={\begin{bmatrix}a_{1,1}&a_{1,2}&\ldots &a_{1,n}\\a_{2,1}&a_{2,2}&\ldots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{m,1}&a_{m,2}&\ldots &a_{m,n}\end{bmatrix}}}
5880:
924:. In other words, it does not matter whether the linear map is applied before (the right hand sides of the above examples) or after (the left hand sides of the examples) the operations of addition and scalar multiplication.
12414:. Linear mappings also are used as a mechanism for describing change: for example in calculus correspond to derivatives; or in relativity, used as a device to keep track of the local transformations of reference frames.
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14678:: "The main question in index theory is to provide index formulas for classes of Fredholm operators ... Index theory has become a subject on its own only after M. F. Atiyah and I. Singer published their index theorems"
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3161:
10169:{\displaystyle {\begin{aligned}\ker(f)&=\{\,\mathbf {x} \in V:f(\mathbf {x} )=\mathbf {0} \,\}\\\operatorname {im} (f)&=\{\,\mathbf {w} \in W:\mathbf {w} =f(\mathbf {x} ),\mathbf {x} \in V\,\}\end{aligned}}}
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11463:. Its image is the entire target space, and hence its co-kernel has dimension 0, but since it maps all sequences in which only the first element is non-zero to the zero sequence, its kernel has dimension 1.
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The dimension of the co-kernel and the dimension of the image (the rank) add up to the dimension of the target space. For finite dimensions, this means that the dimension of the quotient space
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No classification of linear maps could be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on the vector space.
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9260:, since the composition of two linear maps is again a linear map, and the composition of maps is always associative. This case is discussed in more detail below.
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6084:{\displaystyle f(\mathbf {v} )=f(c_{1}\mathbf {v} _{1}+\cdots +c_{n}\mathbf {v} _{n})=c_{1}f(\mathbf {v} _{1})+\cdots +c_{n}f\left(\mathbf {v} _{n}\right),}
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247:
refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that
4107:
7034:. A single linear map may be represented by many matrices. This is because the values of the elements of a matrix depend on the bases chosen.
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13533:{\textstyle A\left(\mathbf {x} _{1}+\mathbf {x} _{2}\right)=A\mathbf {x} _{1}+A\mathbf {x} _{2},\ A(c\mathbf {x} )=cA\mathbf {x} }
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6839:{\displaystyle \mathbf {M} ={\begin{pmatrix}\ \cdots &a_{1j}&\cdots \ \\&\vdots &\\&a_{mj}&\end{pmatrix}}}
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1191:{\displaystyle f(c_{1}\mathbf {u} _{1}+\cdots +c_{n}\mathbf {u} _{n})=c_{1}f(\mathbf {u} _{1})+\cdots +c_{n}f(\mathbf {u} _{n}).}
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is additive: It does not matter whether vectors are first added and then mapped or whether they are mapped and finally added:
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8129:{\displaystyle \mathbf {A} ={\begin{pmatrix}\cos 2\theta &\sin 2\theta \\\sin 2\theta &-\cos 2\theta \end{pmatrix}}}
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9263:
Given again the finite-dimensional case, if bases have been chosen, then the composition of linear maps corresponds to the
12394:
does not, so differentiation is not continuous at 0 (and by a variation of this argument, it is not continuous anywhere).
5581:
13692:
2540:) with a fixed integration starting point defines a linear map from the space of all real-valued integrable functions on
7876:{\displaystyle \mathbf {A} ={\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{pmatrix}}}
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2012:
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is homogeneous: It does not matter whether a vector is first scaled and then mapped or first mapped and then scaled:
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defines a linear map from the space of all differentiable functions to the space of all functions. It also defines a
100:
78:
13223:{\textstyle \Lambda (\alpha \mathbf {x} +\beta \mathbf {y} )=\alpha \Lambda \mathbf {x} +\beta \Lambda \mathbf {y} }
12138:
4414:{\displaystyle F\left(c_{1}s_{1}+\cdots c_{n}s_{n}\right)=c_{1}f\left(s_{1}\right)+\cdots +c_{n}f\left(s_{n}\right)}
71:
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9271:, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.
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If a linear map is only composed of rotation, reflection, and/or uniform scaling, then the linear map is a
6965:
2704:(which is in fact a function, and as such an element of a vector space) is linear, as for random variables
990:
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11374:), but in the infinite-dimensional case it cannot be inferred that the kernel and the co-kernel of an
1943:
1914:
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15975:
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15319:. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:
15267:. International Series in Pure and Applied Mathematics. Vol. 25 (First ed.). New York, NY:
10564:
9491:. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of
9060:
6377:{\displaystyle f\left(\mathbf {v} _{j}\right)=a_{1j}\mathbf {w} _{1}+\cdots +a_{mj}\mathbf {w} _{m}.}
5490:. This is useful because it allows concrete calculations. Matrices yield examples of linear maps: if
5251:
602:
16457:
11052:
that the solutions must satisfy, and its dimension is the maximal number of independent constraints.
4648:
16559:
16477:
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15760:
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12403:
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11319:
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7887:
7516:. The equivalent method would be the "longer" method going clockwise from the same point such that
6889:
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1388:
487:
65:
14664:
12929:
11162:
8360:{\displaystyle \mathbf {A} ={\begin{pmatrix}1&-\sin \theta \\0&\cos \theta \end{pmatrix}}}
5782:
2565:
2543:
2523:{\displaystyle \int _{u}^{v}\left(af(x)+bg(x)\right)dx=a\int _{u}^{v}f(x)dx+b\int _{u}^{v}g(x)dx.}
2349:
2322:{\displaystyle {\frac {d}{dx}}\left(af(x)+bg(x)\right)=a{\frac {df(x)}{dx}}+b{\frac {dg(x)}{dx}}.}
1277:
1248:
31:
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13059:{\displaystyle \Lambda ^{-1}(\{0\})=\{\mathbf {x} \in X:\Lambda \mathbf {x} =0\}={N}(\Lambda )}
12422:
12418:
12410:, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a
12095:
11573:
For a transformation between finite-dimensional vector spaces, this is just the difference dim(
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7720:
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3539:
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3037:
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2336:
1865:
82:
14064:{\displaystyle A\mathbf {x} _{j}=\sum _{i=1}^{m}a_{i,j}\mathbf {y} _{i}\quad (1\leq j\leq n).}
12851:
12367:, for example, when the domain is finite-dimensional. An infinite-dimensional domain may have
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4545:
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15295:. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). New York: McGraw–Hill.
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179:
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2701:
1514:
755:
147:
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9872:
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1613:
without modification, and to any right-module upon reversing of the scalar multiplication.
860:
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183:
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8:
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9257:
9253:
8928:
8208:{\displaystyle \mathbf {A} ={\begin{pmatrix}2&0\\0&2\end{pmatrix}}=2\mathbf {I} }
3293:
Often, a linear map is constructed by defining it on a subset of a vector space and then
2170:
1754:
1674:
1567:
1362:
658:
342:
294:
212:
191:
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9377:
together with addition, composition and scalar multiplication as defined above forms an
8433:{\displaystyle \mathbf {A} ={\begin{pmatrix}k&0\\0&{\frac {1}{k}}\end{pmatrix}}}
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14916:
12492:
11563:{\displaystyle \operatorname {ind} (f):=\dim(\ker(f))-\dim(\operatorname {coker} (f)),}
11042:
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9516:
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7017:
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1894:
1845:
1596:
1540:
1502:{\displaystyle f(\mathbf {0} _{V})=f(0\mathbf {v} )=0f(\mathbf {v} )=\mathbf {0} _{W}.}
1228:
1208:
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182:. The same names and the same definition are also used for the more general case of
16564:
16239:
16206:
16191:
16073:
15942:
15803:
15707:
15560:
15357:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
12556:
Is called a 'linear mapping' or 'linear transformation' or 'linear operator' from
12452:
12364:
11471:
For a linear operator with finite-dimensional kernel and co-kernel, one may define
10751:
10731:
10711:
10691:
10671:
10651:
10570:
10441:
10435:
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9984:
9932:
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9701:
9681:
9522:
9494:
9474:
9446:
9384:
9328:
9139:
9119:
9099:
8948:
8883:{\displaystyle (f_{1}+f_{2})(\mathbf {x} )=f_{1}(\mathbf {x} )+f_{2}(\mathbf {x} )}
8657:
7245:
7223:
7203:
7106:
7084:
7064:
7044:
3005:
2985:
491:
30:"Linear transformation" redirects here. For fractional linear transformations, see
14563:
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16482:
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16034:
15862:
15655:
15615:
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15463:
15239:
12478:
12099:
11597:
10210:
9268:
8369:
5875:{\displaystyle \mathbf {v} =c_{1}\mathbf {v} _{1}+\cdots +c_{n}\mathbf {v} _{n}.}
5623:
5141:
2166:
2162:
928:
594:
539:
443:
347:
175:
45:
12363:. A linear operator on a normed linear space is continuous if and only if it is
11049:
37:"Linear Operators" redirects here. For the textbook by Dunford and Schwarz, see
16574:
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16128:
16113:
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15523:
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11151:
is the freedom in a solution – while the cokernel may be expressed via the map
10931:
9402:
2697:
2537:
338:
118:
16519:
2155:
is a linear map. This result is not necessarily true for complex normed space.
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15783:
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15402:
15372:
15223:
15082:
14623:, 0.2.3 Vector spaces associated with a matrix or linear transformation, p. 6
12446:
12082:
11222:
An example illustrating the infinite-dimensional case is afforded by the map
9604:
8217:
1970:
15429:
15338:
11139:, 0), (one degree of freedom). The kernel may be expressed as the subspace (
8506:{\displaystyle \mathbf {A} ={\begin{pmatrix}0&0\\0&1\end{pmatrix}}.}
8023:{\displaystyle \mathbf {A} ={\begin{pmatrix}-1&0\\0&1\end{pmatrix}}}
7953:{\displaystyle \mathbf {A} ={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}
7782:{\displaystyle \mathbf {A} ={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}
4194:{\displaystyle 0=c_{1}f\left(s_{1}\right)+\cdots +c_{n}f\left(s_{n}\right).}
851:{\displaystyle f(\mathbf {u} +\mathbf {v} )=f(\mathbf {u} )+f(\mathbf {v} )}
16579:
16383:
16368:
16333:
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16166:
15833:
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15672:
15565:
15437:
15310:
15286:
15256:
12888:
12434:
12352:
12105:
11664:
11378:
have the same dimension (0 ≠ 1). The reverse situation obtains for the map
11375:
10898:{\displaystyle \operatorname {coker} (f):=W/f(V)=W/\operatorname {im} (f).}
9512:
9468:
9322:
9284:
9280:
8274:{\displaystyle \mathbf {A} ={\begin{pmatrix}1&m\\0&1\end{pmatrix}}}
2097:
782:
171:
2588:
of the differentiable functions by the linear space of constant functions.
2342:
is a linear map from the space of all real-valued integrable functions on
16467:
16441:
16363:
16052:
15991:
15813:
15778:
15735:
15580:
15129:
12081:
is said to be a scaling transformation or scalar multiplication map; see
11973:
11866:
11827:
11068:) is the dimension of the target space minus the dimension of the image.
11000:{\displaystyle 0\to \ker(f)\to V\to W\to \operatorname {coker} (f)\to 0.}
9751:
9464:
3302:
3156:{\textstyle f(\mathbf {a} +\mathbf {b} )=f(\mathbf {a} )+f(\mathbf {b} )}
1888:
288:
114:
15842:
15231:
12926:
is a subspace (or a convex set, or a balanced set) the same is true of
12884:
12058:
11990:
11839:
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2158:
1622:
A prototypical example that gives linear maps their name is a function
14071:
It is convenient to represent these numbers in a rectangular array of
11611:
4887:{\displaystyle \operatorname {span} \{(1,0),(0,1)\}=\mathbb {R} ^{2}.}
4752:
can be linearly extended from the linearly independent set of vectors
16323:
16274:
15640:
14778:. Then each of the following four properties implies the other three:
13873:{\textstyle \left\{\mathbf {y} _{1},\ldots ,\mathbf {y} _{m}\right\}}
13813:{\textstyle \left\{\mathbf {x} _{1},\ldots ,\mathbf {x} _{n}\right\}}
12619:{\textstyle a(\mathbf {u} +\mathbf {v} )=a\mathbf {u} +a\mathbf {v} }
12037:
11982:
11676:
11658:
10286:
8742:
7705:
7037:
The matrices of a linear transformation can be represented visually:
6645:{\displaystyle {\begin{pmatrix}a_{1j}\\\vdots \\a_{mj}\end{pmatrix}}}
1681:
198:
16353:
16338:
15808:
11846:
10815:
10777:
10381:{\displaystyle \dim(\ker(f))+\dim(\operatorname {im} (f))=\dim(V).}
8646:
5212:
2860:
2533:
2332:
2178:
2093:
598:
6482:, then we can conveniently use it to compute the vector output of
16047:
16009:
15492:
2584:. Without a fixed starting point, the antiderivative maps to the
1557:
viewed as a one-dimensional vector space over itself is called a
13591:{\textstyle \mathbf {x} ,\mathbf {x} _{1},\mathbf {x} _{2}\in X}
11977:
if it is both left- and right-invertible. This is equivalent to
16373:
15965:
15818:
13741:
13739:
12793:
whose proofs are so easy that we omit them; it is assumed that
12455: – Linear transformation between topological vector spaces
12322:
11570:
namely the degrees of freedom minus the number of constraints.
6158:{\displaystyle f(\mathbf {v} _{1}),\ldots ,f(\mathbf {v} _{n})}
4574:
into any vector space has a linear extension to a (linear) map
1716:
centered in the origin of a vector space is a linear map (here
12472: – Linear map from a vector space to its field of scalars
11123:= 0 (one constraint), and in that case the solution space is (
9405:). The multiplicative identity element of this algebra is the
9047:{\textstyle (\alpha f)(\mathbf {x} )=\alpha (f(\mathbf {x} ))}
12359:. If its domain and codomain are the same, it will then be a
11439:{\textstyle \left\{a_{n}\right\}\mapsto \left\{c_{n}\right\}}
11283:{\textstyle \left\{a_{n}\right\}\mapsto \left\{b_{n}\right\}}
9918:
6690:
as defined above. To define it more clearly, for some column
6214:{\displaystyle \{\mathbf {w} _{1},\ldots ,\mathbf {w} _{m}\}}
5678:{\displaystyle \{\mathbf {v} _{1},\ldots ,\mathbf {v} _{n}\}}
5446:
is defined for each vector space, then every linear map from
2562:
to the space of all real-valued, differentiable functions on
13736:
11869:
or right-cancellable, which is to say, for any vector space
7324:
The relationship between matrices in a linear transformation
3277:{\textstyle f(\lambda \mathbf {a} )=\lambda f(\mathbf {a} )}
11717:
or left-cancellable, which is to say, for any vector space
980:{\textstyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}\in V}
15934:
14452:{\textstyle \{\mathbf {y} _{1},\ldots ,\mathbf {y} _{m}\}}
13693:"terminology - What does 'linear' mean in Linear Algebra?"
12704:{\textstyle a(\lambda \mathbf {u} )=\lambda a\mathbf {u} }
1666:{\displaystyle f:\mathbb {R} \to \mathbb {R} :x\mapsto cx}
13726:
13724:
2053:
vector spaces can be represented in this manner; see the
13709:
11831:
if any of the following equivalent conditions are true:
11668:
if any of the following equivalent conditions are true:
12483:
Pages displaying short descriptions of redirect targets
12481: – Distance-preserving mathematical transformation
12474:
Pages displaying short descriptions of redirect targets
11009:
These can be interpreted thus: given a linear equation
5019:{\displaystyle (x,y)=x(1,0)+y(0,1)\in \mathbb {R} ^{2}}
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of a linear map, when defined, is again a linear map.
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be a linear functional on a topological vector space
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Therefore, linear maps are said to be 1-co- 1-contra-
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of the space it transforms vector coordinates as =
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is an object of study, with a major result being the
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5615:{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}}
612:
12417:Another application of these transformations is in
11612:
Algebraic classifications of linear transformations
10688:are finite-dimensional, bases have been chosen and
2919:{\textstyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}
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4930:{\displaystyle F:\mathbb {R} ^{2}\to \mathbb {R} }
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2631:are finite-dimensional vector spaces over a field
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1352:{\textstyle f(\mathbf {0} _{V})=\mathbf {0} _{W}.}
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329:
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9267:, the addition of linear maps corresponds to the
5191:even guarantees that when this linear functional
2042:{\displaystyle A\mathbf {x} \in \mathbb {R} ^{m}}
910:{\displaystyle f(c\mathbf {u} )=cf(\mathbf {u} )}
863:of degree 1 / operation of scalar multiplication
16592:
11322:as the rank and the dimension of the co-kernel (
10748:are equal to the rank and nullity of the matrix
4097:{\displaystyle 0=c_{1}s_{1}+\cdots +c_{n}s_{n},}
2002:{\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}
1205:Denoting the zero elements of the vector spaces
10782:A subtler invariant of a linear transformation
1564:These statements generalize to any left-module
15470:. Mineola, New York: Dover Publications, Inc.
15108:Bronshtein, I. N.; Semendyayev, K. A. (2004).
11045:in the space of solutions, if it is not empty;
10911:notion to the kernel: just as the kernel is a
9603:. Since the automorphisms are precisely those
8523:
1709:{\textstyle \mathbf {v} \mapsto c\mathbf {v} }
588:rotation and reflection linear transformations
15950:
15508:
15345:
12402:A specific application of linear maps is for
8528:The composition of linear maps is linear: if
7699:
7328:Such that starting in the bottom left corner
5571:{\displaystyle f(\mathbf {x} )=A\mathbf {x} }
4600:{\displaystyle \;\operatorname {span} S\to Y}
4264:{\displaystyle F:\operatorname {span} S\to Y}
3898:{\displaystyle F:\operatorname {span} S\to Y}
1746:{\textstyle \mathbf {x} \mapsto \mathbf {0} }
1413:in the equation for homogeneity of degree 1:
15155:Horn, Roger A.; Johnson, Charles R. (2013).
14446:
14410:
13036:
13005:
12996:
12990:
12761:Here are some properties of linear mappings
12548:be two real vector spaces. A mapping a from
12290:{\displaystyle \left=B^{-1}AB\left=A'\left.}
11931:, which is to say there exists a linear map
11779:, which is to say there exists a linear map
10159:
10104:
10075:
10034:
6208:
6172:
5672:
5636:
4863:
4827:
4801:
4765:
4542:is linearly independent then every function
3606:) and takes its values from the codomain of
778:the following two conditions are satisfied:
15468:Modern Methods in Topological Vector Spaces
15154:
15114:(4th ed.). New York: Springer-Verlag.
14620:
13259:{\textstyle \mathbf {x} ,\mathbf {y} \in X}
12655:{\textstyle \mathbf {u} ,\mathbf {v} \in V}
5733:is uniquely determined by the coefficients
929:the associativity of the addition operation
745:{\textstyle \mathbf {u} ,\mathbf {v} \in V}
601:of vector spaces, and they form a category
15957:
15943:
15515:
15501:
12443: – Conjugate homogeneous additive map
12299:Therefore, the matrix in the new basis is
12135:Substituting this in the first expression
11989:being both epic and monic, and so being a
11584:The index of an operator is precisely the
11041:) = 0, and its dimension is the number of
9919:Kernel, image and the rank–nullity theorem
9607:which possess inverses under composition,
7689:{\textstyle P^{-1}AP\left_{B'}=\left_{B'}}
5342:) then there exists a linear extension to
4581:
2982:is a linear map. This function scales the
2643:, then the function that maps linear maps
582:. Linear maps can often be represented as
15379:
13745:
10427:{\textstyle \dim(\operatorname {im} (f))}
10158:
10107:
10074:
10037:
5787:
5602:
5587:
5234:
5006:
4923:
4909:
4871:
4659:
4625:
3810:then it has a linear extension to all of
2906:
2891:
2570:
2548:
2354:
2029:
1989:
1949:
1920:
1753:between two vector spaces (over the same
1644:
1636:
1198:Thus a linear map is one which preserves
101:Learn how and when to remove this message
16314:Covariance and contravariance of vectors
15462:
15385:Handbook of Analysis and Its Foundations
15203:
15183:A (Terse) Introduction to Linear Algebra
13730:
13715:
12449: – Special type of Boolean function
12120:. As vectors change with the inverse of
11367:{\textstyle \aleph _{0}+0=\aleph _{0}+1}
8637:of all vector spaces over a given field
7370:and looking for the bottom right corner
7319:
6406:is entirely determined by the values of
5108:{\displaystyle F(x,y)=x(-1)+y(2)=-x+2y.}
4022:{\displaystyle s_{1},\ldots ,s_{n}\in S}
3729:is guaranteed to exist if (and only if)
518:to either a plane through the origin in
64:This article includes a list of general
27:Mathematical function, in linear algebra
14743:{\textstyle \Lambda \mathbf {x} \neq 0}
9433:{\textstyle \operatorname {id} :V\to V}
5189:Hahn–Banach dominated extension theorem
3803:{\displaystyle \operatorname {span} S,}
317:), or it can be used to emphasize that
14:
16593:
15906:Comparison of linear algebra libraries
15412:An introduction to Functional Analysis
15409:
15128:
14662:
14650:
11071:As a simple example, consider the map
9187:{\textstyle \operatorname {Hom} (V,W)}
6097:is entirely determined by the vectors
4469:{\displaystyle n,c_{1},\ldots ,c_{n},}
2173:, that is, a linear map with the same
1035:{\textstyle c_{1},\ldots ,c_{n}\in K,}
15938:
15496:
15309:
15285:
15255:
15081:
14687:
14646:
14644:
14631:
14629:
14596:
13757:
13671:
13391:
13106:
12756:
12528:
10612:{\textstyle \operatorname {null} (f)}
10483:{\textstyle \operatorname {rank} (f)}
9862:{\textstyle \operatorname {GL} (n,K)}
9381:with identity element over the field
7007:{\displaystyle a_{1j},\cdots ,a_{mj}}
6241:. Then we can represent each vector
2049:. Conversely, any linear map between
15321:McGraw-Hill Science/Engineering/Math
15269:McGraw-Hill Science/Engineering/Math
15230:
14560:is spanned by the column vectors of
9895:invertible matrices with entries in
9743:{\textstyle \operatorname {End} (V)}
9668:{\textstyle \operatorname {End} (V)}
9632:{\textstyle \operatorname {Aut} (V)}
9564:{\textstyle \operatorname {Aut} (V)}
9370:{\textstyle \operatorname {End} (V)}
9345:; the set of all such endomorphisms
9245:{\textstyle \operatorname {End} (V)}
7509:{\textstyle A'\left_{B'}=\left_{B'}}
3288:
3002:component of a vector by the factor
2689:(below) is a linear map, and even a
291:vector spaces (not necessarily with
50:
15292:Principles of Mathematical Analysis
14521:are therefore sometimes called the
13355:{\textstyle \Lambda (\mathbf {x} )}
12386:converges to 0, but its derivative
12128:) its inverse transformation is =
11119:) to have a solution, we must have
10258:{\textstyle \operatorname {im} (f)}
9596:{\textstyle \operatorname {GL} (V)}
7712:linear maps are described by 2 × 2
7425:, one would left-multiply—that is,
6955:{\displaystyle f(\mathbf {v} _{j})}
6741:{\displaystyle f(\mathbf {v} _{j})}
6683:{\displaystyle f(\mathbf {v} _{j})}
6272:{\displaystyle f(\mathbf {v} _{j})}
5772:{\displaystyle c_{1},\ldots ,c_{n}}
5241:{\displaystyle p:X\to \mathbb {R} }
4515:{\displaystyle s_{1},\ldots ,s_{n}}
3970:{\displaystyle c_{1},\ldots ,c_{n}}
3761:is a linear map. In particular, if
2863:of a random variable is not linear.
1764:on any module is a linear operator.
24:
16177:Tensors in curvilinear coordinates
15522:
15436:
14878:
14849:
14818:
14789:
14726:
14702:
14641:
14636:Katznelson & Katznelson (2008)
14626:
14608:
13674:, p. 14. Linear mappings of
13369:
13338:
13313:
13212:
13198:
13164:
13128:
13081:
13050:
13022:
12975:
12934:
12898:
12855:
12768:
12089:
11981:being both one-to-one and onto (a
11627:denote vector spaces over a field
11349:
11330:
9066:
8945:is an element of the ground field
5144:of a real or complex vector space
4807:{\displaystyle S:=\{(1,0),(0,1)\}}
4638:{\displaystyle X=\mathbb {R} ^{2}}
341:, which is a common convention in
70:it lacks sufficient corresponding
25:
16627:
15387:. San Diego, CA: Academic Press.
15181:; Katznelson, Yonatan R. (2008).
14891:is bounded in some neighbourhood
13678:onto its scalar field are called
13324:{\textstyle \Lambda \mathbf {x} }
12745:Bronshtein & Semendyayev 2004
12732:{\displaystyle \mathbf {u} \in V}
9136:itself forms a vector space over
6436:. If we put these values into an
5726:{\displaystyle \mathbf {v} \in V}
5164:has a linear extension to all of
4233:exists then the linear extension
2685:matrices in the way described in
920:Thus, a linear map is said to be
613:Definition and first consequences
174:that preserves the operations of
15919:
15918:
15896:Basic Linear Algebra Subprograms
15654:
15134:Finite-Dimensional Vector Spaces
14758:
14730:
14501:
14436:
14415:
14383:
14026:
13974:
13855:
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13774:
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13548:
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12587:
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10137:
10123:
10109:
10070:
10059:
10039:
9194:. Furthermore, in the case that
9089:{\textstyle {\mathcal {L}}(V,W)}
9034:
9011:
8873:
8849:
8825:
8633:. It follows from this that the
8457:
8380:
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8228:
8201:
8151:
8044:
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7904:
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7733:
7725:by 90 degrees counterclockwise:
7662:
7626:
7529:
7482:
7446:
7391:
7341:
6939:
6756:
6725:
6710:that corresponds to the mapping
6667:
6361:
6327:
6295:
6256:
6198:
6177:
6142:
6112:
6093:which implies that the function
6064:
6023:
5989:
5958:
5930:
5859:
5828:
5809:
5713:
5662:
5641:
5564:
5550:
3867:can be extended to a linear map
3267:
3247:
3167:
3146:
3129:
3112:
3104:
3027:
2869:
2020:
1980:
1962:{\displaystyle \mathbb {R} ^{m}}
1933:{\displaystyle \mathbb {R} ^{n}}
1739:
1731:
1702:
1691:
1486:
1474:
1454:
1428:
1393:
1336:
1318:
1283:
1254:
1172:
1132:
1098:
1067:
961:
940:
900:
880:
841:
824:
807:
799:
732:
724:
55:
15794:Seven-dimensional cross product
15075:
14681:
14656:
14614:
14602:
14590:
14036:
13751:
13109:, p. 14. Suppose now that
12397:
12314:the matrix of the given basis.
12104:Given a linear map which is an
11648:
10728:, then the rank and nullity of
9275:Endomorphisms and automorphisms
8518:conformal linear transformation
8032:through a line making an angle
6886:. In other words, every column
5295:{\displaystyle |f(m)|\leq p(m)}
657:be vector spaces over the same
15053:
15047:
15038:
15032:
14907:
14852:
14846:
14821:
14815:
14771:{\textstyle \mathbf {x} \in X}
14573:
14567:
14538:
14532:
14514:{\textstyle A\mathbf {x} _{j}}
14476:
14470:
14396:{\textstyle A\mathbf {x} _{j}}
14105:
14099:
14055:
14037:
13916:
13904:
13685:
13665:
13645:
13637:
13513:
13502:
13385:
13349:
13341:
13189:
13167:
13137:
13100:
13053:
13047:
12999:
12987:
12952:
12946:
12907:
12901:
12777:
12750:
12684:
12673:
12591:
12575:
12534:
12505:
12369:discontinuous linear operators
12189:{\displaystyle B\left=AB\left}
12014:If, for some positive integer
11963:
11811:
11554:
11551:
11545:
11536:
11524:
11521:
11515:
11506:
11494:
11488:
11415:
11259:
11190:
11184:
11181:
11178:
11166:
11131:) or equivalently stated, (0,
11048:the co-kernel is the space of
10991:
10988:
10982:
10973:
10967:
10961:
10958:
10952:
10943:
10889:
10883:
10863:
10857:
10840:
10834:
10798:
10635:
10629:
10606:
10600:
10550:
10547:
10541:
10532:
10506:
10500:
10477:
10471:
10421:
10418:
10412:
10403:
10372:
10366:
10354:
10351:
10345:
10336:
10324:
10321:
10315:
10306:
10252:
10246:
10196:
10190:
10141:
10133:
10094:
10088:
10063:
10055:
10024:
10018:
9953:
9856:
9844:
9737:
9731:
9662:
9656:
9626:
9620:
9590:
9584:
9558:
9552:
9424:
9364:
9358:
9305:
9239:
9233:
9181:
9169:
9083:
9071:
9041:
9038:
9030:
9024:
9015:
9007:
9004:
8995:
8909:
8877:
8869:
8853:
8845:
8829:
8821:
8818:
8792:
8725:
8686:
8617:
8576:
8544:
6949:
6934:
6735:
6720:
6677:
6662:
6266:
6251:
6152:
6137:
6122:
6107:
6033:
6018:
5999:
5943:
5934:
5926:
5900:
5597:
5554:
5546:
5289:
5283:
5273:
5269:
5263:
5256:
5230:
5081:
5075:
5066:
5057:
5048:
5036:
4998:
4986:
4977:
4965:
4956:
4944:
4919:
4860:
4848:
4842:
4830:
4798:
4786:
4780:
4768:
4736:
4733:
4721:
4695:
4692:
4680:
4666:{\displaystyle Y=\mathbb {R} }
4591:
4558:
4255:
4217:
3889:
3851:
3745:
3567:
3561:
3552:
3546:
3477:
3461:if it exists, is a linear map
3361:
3271:
3263:
3251:
3240:
3217:
3202:
3196:
3184:
3150:
3142:
3133:
3125:
3116:
3100:
3077:
3062:
3056:
3044:
2969:
2954:
2948:
2936:
2901:
2846:
2840:
2828:
2819:
2796:
2790:
2781:
2775:
2766:
2754:
2659:
2508:
2502:
2469:
2463:
2425:
2419:
2407:
2401:
2302:
2296:
2267:
2261:
2238:
2232:
2220:
2214:
2116:
2110:
2076:
1813:
1777:
1735:
1695:
1654:
1640:
1521:
1478:
1470:
1458:
1447:
1438:
1423:
1406:{\textstyle \mathbf {v} \in V}
1328:
1313:
1303:respectively, it follows that
1182:
1167:
1142:
1127:
1108:
1052:
1042:the following equality holds:
931:denoted as +, for any vectors
904:
896:
884:
873:
845:
837:
828:
820:
811:
795:
696:
586:, and simple examples include
154:
13:
1:
16230:Exterior covariant derivative
16162:Tensor (intrinsic definition)
15187:American Mathematical Society
14551:. With this terminology, the
14338:Observe that the coordinates
13602:. Note that one often writes
13306:. Note that one often writes
12958:{\textstyle \Lambda ^{-1}(B)}
12437: – Z-module homomorphism
12406:, such as those performed in
12328:
11196:{\textstyle (a,b)\mapsto (a)}
10708:is represented by the matrix
9220:, this vector space, denoted
8741:are linear, then so is their
8592:are linear, then so is their
6917:{\displaystyle j=1,\ldots ,n}
4607:(the converse is also true).
3689:-valued) linear extension of
3403:{\displaystyle S\subseteq X.}
2686:
2054:
1677:is a line through the origin.
1296:{\textstyle \mathbf {0} _{W}}
1267:{\textstyle \mathbf {0} _{V}}
16255:Raising and lowering indices
15636:Eigenvalues and eigenvectors
15442:An Introduction to Manifolds
15349:; Wolff, Manfred P. (1999).
15238:(Third ed.), New York:
13929:determines a set of numbers
13651:{\textstyle A(\mathbf {x} )}
13146:{\textstyle \Lambda :X\to Y}
12786:{\textstyle \Lambda :X\to Y}
12459:Cauchy's functional equation
12421:of nested-loop code, and in
12032:, is identically zero, then
11873:and any pair of linear maps
11721:and any pair of linear maps
9804:. The automorphism group of
8626:{\textstyle g\circ f:V\to Z}
5794:{\displaystyle \mathbb {R} }
4894:The unique linear extension
2577:{\displaystyle \mathbb {R} }
2555:{\displaystyle \mathbb {R} }
2361:{\displaystyle \mathbb {R} }
7:
16493:Gluon field strength tensor
15964:
15206:Elements of operator theory
14670:Encyclopedia of Mathematics
14403:(with respect to the basis
13888:, respectively. Then every
13880:are bases of vector spaces
12511:"Linear transformations of
12461: – Functional equation
12428:
11606:Atiyah–Singer index theorem
11219:to there being a solution.
11025:the kernel is the space of
10771:
8778:{\displaystyle f_{1}+f_{2}}
8524:Vector space of linear maps
7716:. These are some examples:
7014:are the elements of column
6924:has a corresponding vector
5387:
5211:is dominated by some given
4707:{\displaystyle (1,0)\to -1}
2635:, of respective dimensions
1793:{\textstyle x\mapsto x^{2}}
1616:
235:, a linear map is called a
117:, and more specifically in
10:
16632:
16304:Cartan formalism (physics)
16124:Penrose graphical notation
15161:Cambridge University Press
15062:{\displaystyle F(s)=f(s).}
13697:Mathematics Stack Exchange
13620:{\textstyle A\mathbf {x} }
13119:over the same scalar field
12465:Continuous linear operator
12361:continuous linear operator
12335:Continuous linear operator
12332:
12093:
12010:is an endomorphism, then:
11588:of the 2-term complex 0 →
10775:
10556:{\textstyle \dim(\ker(f))}
9922:
9278:
7700:Examples in two dimensions
5391:
5362:that is also dominated by
4745:{\displaystyle (0,1)\to 2}
3781:has a linear extension to
3223:{\textstyle f(x,y)=(2x,y)}
3083:{\textstyle f(x,y)=(2x,y)}
2975:{\textstyle f(x,y)=(2x,y)}
1911:defines a linear map from
1803:For real numbers, the map
1767:For real numbers, the map
490:); for example, it maps a
402:always maps the origin of
43:
36:
29:
16616:Transformation (function)
16510:
16450:
16399:
16392:
16284:
16215:
16152:
16096:
16043:
15990:
15983:
15976:Glossary of tensor theory
15972:
15914:
15876:
15832:
15769:
15721:
15663:
15652:
15548:
15530:
15351:Topological Vector Spaces
15204:Kubrusly, Carlos (2001).
15087:Linear Algebra Done Right
13394:, p. 206. A mapping
12870:{\textstyle \Lambda 0=0.}
12404:geometric transformations
12349:topological vector spaces
12124:(vectors coordinates are
9969:is linear, we define the
9784:matrices with entries in
8734:{\textstyle f_{2}:V\to W}
8695:{\textstyle f_{1}:V\to W}
6455:{\displaystyle m\times n}
5529:{\displaystyle m\times n}
4201:If a linear extension of
3573:{\displaystyle F(s)=f(s)}
2169:(a linear operator is a
1881:{\displaystyle m\times n}
1835:is not linear (but is an
1828:{\textstyle x\mapsto x+1}
466:onto linear subspaces in
135:vector space homomorphism
16560:Gregorio Ricci-Curbastro
16432:Riemann curvature tensor
16139:Van der Waerden notation
15410:Swartz, Charles (1992).
14858:{\textstyle N(\Lambda )}
14827:{\textstyle N(\Lambda )}
14663:Nistor, Victor (2001) ,
13922:{\textstyle A\in L(X,Y)}
12967:In particular, the set:
12913:{\textstyle \Lambda (A)}
12498:
12339:Discontinuous linear map
12321:objects, or type (1, 1)
11466:
11099:). Then for an equation
10289:formula is known as the
9289:A linear transformation
8553:{\displaystyle f:V\to W}
8218:horizontal shear mapping
8143:by 2 in all directions:
7558:is left-multiplied with
5486:can be represented by a
4567:{\displaystyle f:S\to Y}
4226:{\displaystyle f:S\to Y}
3905:if and only if whenever
3860:{\displaystyle f:S\to Y}
3754:{\displaystyle f:S\to Y}
3649:is a vector subspace of
3486:{\displaystyle F:X\to Y}
3370:{\displaystyle f:S\to Y}
3305:of the domain. Suppose
785:/ operation of addition
705:{\displaystyle f:V\to W}
562:, or just the origin in
351:has the same meaning as
44:Not to be confused with
16530:Elwin Bruno Christoffel
16463:Angular momentum tensor
16134:Tetrad (index notation)
16104:Abstract index notation
15414:. New York: M. Dekker.
15111:Handbook of Mathematics
14621:Horn & Johnson 2013
12838:{\textstyle B\subset Y}
12812:{\textstyle A\subset X}
9925:Kernel (linear algebra)
7551:{\textstyle \left_{B'}}
7418:{\textstyle \left_{B'}}
7363:{\textstyle \left_{B'}}
7242:Transition matrix from
7175:Transition matrix from
5578:describes a linear map
3381:defined on some subset
716:if for any two vectors
85:more precise citations.
39:Linear Operators (book)
16606:Functions and mappings
16344:Levi-Civita connection
15621:Row and column vectors
15208:. Boston: Birkhäuser.
15063:
15016:
14996:
14979:is defined at a point
14973:
14953:
14927:
14885:
14859:
14828:
14796:
14772:
14744:
14709:
14580:
14545:
14515:
14483:
14453:
14397:
14365:
14332:
14065:
14007:
13956:
13923:
13874:
13814:
13760:, p. 210 Suppose
13652:
13621:
13592:
13534:
13376:
13356:
13325:
13300:
13280:
13260:
13224:
13147:
13088:
13060:
12959:
12914:
12891:) the same is true of
12871:
12839:
12813:
12787:
12733:
12705:
12656:
12620:
12423:parallelizing compiler
12419:compiler optimizations
12291:
12190:
12096:Basis (linear algebra)
11564:
11440:
11368:
11284:
11197:
11001:
10930:Formally, one has the
10899:
10821:, which is defined as
10808:
10762:
10742:
10722:
10702:
10682:
10662:
10642:
10613:
10581:
10557:
10513:
10484:
10452:
10428:
10382:
10279:
10259:
10227:
10203:
10170:
9995:
9963:
9909:
9889:
9888:{\textstyle n\times n}
9863:
9818:
9798:
9778:
9777:{\textstyle n\times n}
9744:
9712:
9692:
9669:
9633:
9597:
9565:
9533:
9505:
9485:
9457:
9434:
9395:
9371:
9339:
9315:
9246:
9214:
9188:
9150:
9130:
9110:
9090:
9048:
8982:
8959:
8939:
8919:
8884:
8785:, which is defined by
8779:
8735:
8696:
8627:
8586:
8554:
8507:
8434:
8361:
8275:
8209:
8130:
8024:
7954:
7877:
7783:
7690:
7588:
7552:
7510:
7419:
7364:
7325:
7311:
7281:
7256:
7234:
7214:
7194:
7167:
7142:
7117:
7095:
7075:
7055:
7028:
7008:
6956:
6918:
6880:
6860:
6840:
6742:
6704:
6684:
6646:
6576:
6556:
6536:
6516:
6496:
6476:
6456:
6430:
6429:{\displaystyle a_{ij}}
6400:
6378:
6273:
6235:
6215:
6159:
6085:
5910:
5876:
5795:
5773:
5727:
5699:
5679:
5616:
5572:
5530:
5504:
5480:
5460:
5432:
5412:
5379:
5356:
5336:
5316:
5296:
5242:
5205:
5181:
5158:
5134:
5117:Every (scalar-valued)
5109:
5020:
4937:is the map that sends
4931:
4888:
4808:
4746:
4708:
4667:
4639:
4601:
4568:
4536:
4516:
4470:
4415:
4265:
4227:
4195:
4098:
4029:are vectors such that
4023:
3971:
3925:
3924:{\displaystyle n>0}
3899:
3861:
3827:
3804:
3775:
3755:
3723:
3703:
3683:
3663:
3643:
3623:
3600:
3599:{\displaystyle s\in S}
3574:
3530:
3507:
3487:
3454:
3431:
3404:
3371:
3345:are vector spaces and
3339:
3319:
3297:extending by linearity
3278:
3224:
3157:
3084:
3016:
2996:
2976:
2920:
2853:
2803:
2738:
2718:
2669:
2625:
2605:
2578:
2556:
2524:
2362:
2323:
2149:
2129:
2086:
2043:
2003:
1963:
1934:
1905:
1882:
1856:
1829:
1794:
1747:
1710:
1667:
1607:
1587:
1551:
1531:
1530:{\displaystyle V\to K}
1503:
1407:
1379:
1353:
1297:
1268:
1239:
1219:
1192:
1036:
981:
911:
852:
772:
771:{\displaystyle c\in K}
746:
706:
674:
651:
631:
597:, linear maps are the
576:
556:
542:through the origin in
532:
512:
480:
460:
436:
416:
396:
376:
331:
311:
281:
261:
229:
164:
163:{\displaystyle V\to W}
137:, or in some contexts
16570:Jan Arnoldus Schouten
16525:Augustin-Louis Cauchy
16005:Differential geometry
15626:Row and column spaces
15571:Scalar multiplication
15064:
15017:
14997:
14974:
14954:
14928:
14886:
14884:{\textstyle \Lambda }
14860:
14829:
14797:
14795:{\textstyle \Lambda }
14773:
14745:
14710:
14708:{\textstyle \Lambda }
14581:
14546:
14516:
14484:
14454:
14398:
14366:
14333:
14066:
13987:
13957:
13924:
13875:
13815:
13653:
13622:
13593:
13535:
13408:linear transformation
13377:
13375:{\textstyle \Lambda }
13357:
13326:
13301:
13281:
13261:
13225:
13148:
13089:
13087:{\textstyle \Lambda }
13061:
12960:
12915:
12872:
12840:
12814:
12788:
12734:
12706:
12657:
12621:
12412:transformation matrix
12345:linear transformation
12292:
12191:
12077:is some scalar, then
11565:
11441:
11369:
11285:
11198:
11002:
10900:
10809:
10807:{\textstyle f:V\to W}
10763:
10743:
10723:
10703:
10683:
10663:
10643:
10614:
10582:
10558:
10514:
10512:{\textstyle \rho (f)}
10485:
10453:
10429:
10383:
10280:
10260:
10228:
10204:
10171:
9996:
9964:
9962:{\textstyle f:V\to W}
9910:
9890:
9864:
9819:
9799:
9779:
9745:
9713:
9698:has finite dimension
9693:
9670:
9634:
9598:
9566:
9534:
9506:
9486:
9458:
9435:
9401:(and in particular a
9396:
9372:
9340:
9316:
9314:{\textstyle f:V\to V}
9265:matrix multiplication
9247:
9215:
9189:
9151:
9131:
9111:
9091:
9049:
8983:
8981:{\textstyle \alpha f}
8960:
8940:
8920:
8918:{\textstyle f:V\to W}
8885:
8780:
8736:
8697:
8628:
8587:
8585:{\textstyle g:W\to Z}
8555:
8508:
8435:
8362:
8276:
8210:
8131:
8025:
7955:
7878:
7784:
7691:
7589:
7587:{\textstyle P^{-1}AP}
7553:
7511:
7420:
7365:
7323:
7312:
7282:
7257:
7235:
7215:
7195:
7168:
7143:
7118:
7096:
7076:
7056:
7029:
7009:
6957:
6919:
6881:
6861:
6841:
6743:
6705:
6685:
6647:
6577:
6557:
6537:
6517:
6497:
6477:
6457:
6431:
6401:
6379:
6274:
6236:
6216:
6160:
6086:
5911:
5909:{\textstyle f:V\to W}
5877:
5796:
5774:
5728:
5700:
5680:
5617:
5573:
5531:
5505:
5481:
5461:
5433:
5413:
5394:Transformation matrix
5380:
5357:
5337:
5317:
5297:
5243:
5206:
5182:
5159:
5135:
5110:
5021:
4932:
4889:
4809:
4747:
4709:
4668:
4640:
4602:
4569:
4537:
4517:
4471:
4416:
4266:
4228:
4196:
4099:
4024:
3972:
3926:
3900:
3862:
3828:
3805:
3776:
3756:
3724:
3704:
3684:
3664:
3644:
3624:
3601:
3575:
3531:
3508:
3488:
3455:
3432:
3405:
3372:
3340:
3320:
3279:
3225:
3158:
3085:
3017:
2997:
2977:
2921:
2854:
2804:
2802:{\displaystyle E=E+E}
2739:
2719:
2670:
2668:{\textstyle f:V\to W}
2626:
2606:
2579:
2557:
2525:
2363:
2324:
2150:
2130:
2087:
2085:{\textstyle f:V\to W}
2044:
2009:to the column vector
2004:
1964:
1935:
1906:
1883:
1857:
1837:affine transformation
1830:
1795:
1748:
1711:
1668:
1608:
1588:
1552:
1532:
1504:
1408:
1380:
1354:
1298:
1269:
1240:
1220:
1193:
1037:
982:
912:
853:
773:
747:
707:
675:
652:
632:
577:
557:
533:
513:
486:(possibly of a lower
481:
461:
442:. Moreover, it maps
437:
417:
397:
377:
345:. Sometimes the term
332:
312:
282:
262:
239:. Sometimes the term
230:
197:If a linear map is a
180:scalar multiplication
165:
131:linear transformation
32:Möbius transformation
16545:Carl Friedrich Gauss
16478:stress–energy tensor
16473:Cauchy stress tensor
16225:Covariant derivative
16187:Antisymmetric tensor
16119:Multi-index notation
15761:Gram–Schmidt process
15713:Gaussian elimination
15130:Halmos, Paul Richard
15026:
15006:
14983:
14963:
14943:
14917:
14875:
14840:
14809:
14786:
14754:
14723:
14699:
14564:
14529:
14493:
14467:
14407:
14375:
14364:{\textstyle a_{i,j}}
14342:
14096:
13966:
13955:{\textstyle a_{i,j}}
13933:
13892:
13824:
13764:
13631:
13606:
13544:
13414:
13402:into a vector space
13366:
13335:
13310:
13290:
13279:{\textstyle \alpha }
13270:
13234:
13161:
13125:
13078:
12971:
12930:
12895:
12883:is a subspace (or a
12852:
12823:
12797:
12765:
12715:
12667:
12630:
12569:
12488:Category of matrices
12200:
12139:
11985:of sets) or also to
11586:Euler characteristic
11479:
11394:
11326:
11238:
11163:
10937:
10825:
10786:
10752:
10732:
10712:
10692:
10672:
10652:
10641:{\textstyle \nu (f)}
10623:
10591:
10571:
10523:
10494:
10462:
10442:
10394:
10297:
10291:rank–nullity theorem
10269:
10237:
10217:
10202:{\textstyle \ker(f)}
10181:
10005:
9985:
9941:
9899:
9873:
9835:
9830:general linear group
9808:
9788:
9762:
9722:
9702:
9682:
9647:
9611:
9575:
9543:
9539:which is denoted by
9523:
9495:
9475:
9447:
9412:
9385:
9349:
9329:
9293:
9224:
9198:
9160:
9156:, sometimes denoted
9140:
9120:
9100:
9096:of linear maps from
9061:
8992:
8969:
8949:
8938:{\textstyle \alpha }
8929:
8897:
8789:
8749:
8706:
8667:
8599:
8564:
8532:
8453:
8376:
8295:
8224:
8147:
8040:
7970:
7900:
7799:
7729:
7598:
7562:
7520:
7429:
7374:
7332:
7291:
7266:
7246:
7224:
7204:
7179:
7152:
7127:
7107:
7085:
7065:
7045:
7018:
6966:
6928:
6890:
6870:
6850:
6752:
6714:
6694:
6656:
6586:
6566:
6546:
6526:
6506:
6486:
6466:
6440:
6410:
6390:
6283:
6245:
6225:
6169:
6101:
5920:
5888:
5805:
5783:
5737:
5709:
5705:. Then every vector
5689:
5633:
5582:
5540:
5514:
5494:
5470:
5450:
5442:vector spaces and a
5422:
5402:
5366:
5346:
5326:
5306:
5252:
5218:
5195:
5168:
5148:
5124:
5030:
4941:
4898:
4818:
4756:
4718:
4677:
4673:then the assignment
4649:
4614:
4578:
4546:
4526:
4480:
4425:
4275:
4237:
4205:
4108:
4033:
3981:
3935:
3909:
3871:
3839:
3814:
3785:
3765:
3733:
3713:
3693:
3673:
3653:
3633:
3610:
3584:
3540:
3520:
3497:
3465:
3441:
3421:
3385:
3349:
3329:
3309:
3234:
3178:
3094:
3038:
3006:
2986:
2930:
2880:
2852:{\displaystyle E=aE}
2813:
2748:
2728:
2708:
2647:
2615:
2595:
2566:
2544:
2372:
2350:
2185:
2165:on the space of all
2139:
2104:
2064:
2013:
1976:
1944:
1915:
1895:
1866:
1846:
1807:
1771:
1727:
1687:
1680:More generally, any
1626:
1597:
1586:{\textstyle {}_{R}M}
1568:
1541:
1515:
1417:
1389:
1363:
1307:
1278:
1249:
1229:
1209:
1046:
991:
935:
922:operation preserving
867:
789:
756:
720:
684:
664:
641:
621:
566:
546:
522:
502:
470:
450:
426:
406:
386:
366:
321:
295:
271:
251:
213:
209:. In the case where
201:then it is called a
148:
16422:Nonmetricity tensor
16277:(2nd-order tensors)
16245:Hodge star operator
16235:Exterior derivative
16084:Transport phenomena
16069:Continuum mechanics
16025:Multilinear algebra
15891:Numerical stability
15771:Multilinear algebra
15746:Inner product space
15596:Linear independence
15347:Schaefer, Helmut H.
15316:Functional Analysis
15263:Functional Analysis
15179:Katznelson, Yitzhak
15159:(Second ed.).
14079:columns, called an
13748:, pp. 277–280.
13299:{\textstyle \beta }
10919:the co-kernel is a
10434:is also called the
9929:Image (mathematics)
9756:associative algebra
9443:An endomorphism of
9379:associative algebra
9258:composition of maps
9254:associative algebra
7310:{\textstyle P^{-1}}
6386:Thus, the function
4814:to a linear map on
2498:
2459:
2389:
2171:linear endomorphism
2128:{\textstyle f(0)=0}
1378:{\displaystyle c=0}
1200:linear combinations
607:the one of matrices
593:In the language of
343:functional analysis
310:{\displaystyle V=W}
237:linear endomorphism
228:{\displaystyle V=W}
192:Module homomorphism
16555:Tullio Levi-Civita
16498:Metric tensor (GR)
16412:Levi-Civita symbol
16265:Tensor contraction
16079:General relativity
16015:Euclidean geometry
15601:Linear combination
15083:Axler, Sheldon Jay
15059:
15012:
14995:{\displaystyle s,}
14992:
14969:
14949:
14923:
14881:
14855:
14824:
14792:
14768:
14740:
14705:
14576:
14541:
14511:
14479:
14449:
14393:
14361:
14328:
14322:
14061:
13952:
13919:
13870:
13810:
13680:linear functionals
13648:
13617:
13588:
13530:
13398:of a vector space
13372:
13352:
13321:
13296:
13276:
13256:
13220:
13143:
13117:are vector spaces
13084:
13056:
12955:
12910:
12867:
12835:
12809:
12783:
12729:
12701:
12652:
12616:
12493:Quasilinearization
12287:
12186:
11602:Fredholm operators
11560:
11436:
11364:
11280:
11203:: given a vector (
11193:
11043:degrees of freedom
10997:
10895:
10804:
10758:
10738:
10718:
10698:
10678:
10658:
10638:
10609:
10577:
10553:
10509:
10480:
10448:
10424:
10378:
10275:
10255:
10223:
10199:
10166:
10164:
9991:
9959:
9905:
9885:
9859:
9814:
9794:
9774:
9740:
9708:
9688:
9665:
9629:
9593:
9561:
9529:
9517:automorphism group
9501:
9481:
9453:
9430:
9391:
9367:
9335:
9311:
9242:
9210:
9184:
9146:
9126:
9106:
9086:
9054:, is also linear.
9044:
8978:
8955:
8935:
8915:
8880:
8775:
8731:
8692:
8623:
8582:
8550:
8503:
8494:
8430:
8424:
8357:
8351:
8271:
8265:
8205:
8188:
8126:
8120:
8020:
8014:
7950:
7944:
7873:
7867:
7795:counterclockwise:
7779:
7773:
7686:
7584:
7548:
7506:
7415:
7360:
7326:
7307:
7277:
7252:
7230:
7210:
7190:
7163:
7138:
7113:
7091:
7071:
7051:
7024:
7004:
6962:whose coordinates
6952:
6914:
6876:
6856:
6836:
6830:
6738:
6700:
6680:
6642:
6636:
6572:
6552:
6532:
6512:
6502:for any vector in
6492:
6472:
6452:
6426:
6396:
6374:
6269:
6231:
6211:
6155:
6081:
5906:
5872:
5791:
5769:
5723:
5695:
5675:
5612:
5568:
5526:
5500:
5476:
5456:
5440:finite-dimensional
5428:
5408:
5378:{\displaystyle p.}
5375:
5352:
5332:
5312:
5292:
5238:
5201:
5180:{\displaystyle X.}
5177:
5154:
5130:
5105:
5016:
4927:
4884:
4804:
4742:
4704:
4663:
4635:
4597:
4564:
4532:
4512:
4466:
4411:
4261:
4223:
4191:
4094:
4019:
3967:
3921:
3895:
3857:
3826:{\displaystyle X.}
3823:
3800:
3771:
3751:
3719:
3699:
3679:
3659:
3639:
3622:{\displaystyle f.}
3619:
3596:
3570:
3526:
3503:
3483:
3453:{\displaystyle X,}
3450:
3427:
3400:
3367:
3335:
3315:
3274:
3220:
3153:
3080:
3012:
2992:
2972:
2916:
2849:
2799:
2734:
2714:
2691:linear isomorphism
2665:
2621:
2601:
2574:
2552:
2520:
2484:
2445:
2375:
2358:
2319:
2145:
2125:
2082:
2051:finite-dimensional
2039:
1999:
1959:
1930:
1901:
1878:
1852:
1825:
1790:
1743:
1706:
1663:
1603:
1583:
1547:
1527:
1499:
1403:
1375:
1349:
1293:
1264:
1235:
1215:
1188:
1032:
977:
907:
848:
768:
742:
702:
670:
647:
627:
572:
552:
528:
508:
476:
456:
432:
412:
392:
372:
362:A linear map from
327:
307:
277:
257:
225:
205:linear isomorphism
160:
16588:
16587:
16550:Hermann Grassmann
16506:
16505:
16458:Moment of inertia
16319:Differential form
16294:Affine connection
16109:Einstein notation
16092:
16091:
16020:Exterior calculus
16000:Coordinate system
15932:
15931:
15799:Geometric algebra
15756:Kronecker product
15591:Linear projection
15576:Vector projection
15477:978-0-486-49353-4
15455:978-0-8218-4419-9
15421:978-0-8247-8643-4
15394:978-0-12-622760-4
15364:978-1-4612-7155-0
15330:978-0-07-054236-5
15302:978-0-07-054235-8
15215:978-1-4757-3328-0
15196:978-0-8218-4419-9
15170:978-0-521-83940-2
15100:978-3-319-11079-0
15015:{\displaystyle F}
14972:{\displaystyle f}
14952:{\displaystyle f}
14926:{\displaystyle F}
13718:, pp. 21–26.
13498:
13066:is a subspace of
12519:are often called
12470:Linear functional
12408:computer graphics
11971:is said to be an
11842:as a map of sets.
11645:be a linear map.
10285:. The following
10265:is a subspace of
8420:
8287:axis by an angle
8036:with the origin:
7027:{\displaystyle j}
6879:{\displaystyle f}
6866:is the matrix of
6859:{\displaystyle M}
6799:
6773:
6703:{\displaystyle j}
6652:corresponding to
6575:{\displaystyle M}
6555:{\displaystyle j}
6535:{\displaystyle M}
6515:{\displaystyle V}
6495:{\displaystyle f}
6475:{\displaystyle M}
6399:{\displaystyle f}
6234:{\displaystyle W}
5916:is a linear map,
5698:{\displaystyle V}
5503:{\displaystyle A}
5479:{\displaystyle W}
5459:{\displaystyle V}
5431:{\displaystyle W}
5411:{\displaystyle V}
5355:{\displaystyle X}
5335:{\displaystyle f}
5322:in the domain of
5315:{\displaystyle m}
5204:{\displaystyle f}
5157:{\displaystyle X}
5133:{\displaystyle f}
5119:linear functional
4535:{\displaystyle S}
4104:then necessarily
3977:are scalars, and
3774:{\displaystyle f}
3722:{\displaystyle X}
3702:{\displaystyle f}
3682:{\displaystyle Y}
3662:{\displaystyle X}
3642:{\displaystyle S}
3529:{\displaystyle f}
3506:{\displaystyle X}
3430:{\displaystyle f}
3338:{\displaystyle Y}
3318:{\displaystyle X}
3289:Linear extensions
2737:{\displaystyle Y}
2717:{\displaystyle X}
2624:{\displaystyle W}
2604:{\displaystyle V}
2314:
2279:
2201:
2148:{\displaystyle f}
1904:{\displaystyle A}
1855:{\displaystyle A}
1606:{\displaystyle R}
1559:linear functional
1550:{\displaystyle K}
1238:{\displaystyle W}
1218:{\displaystyle V}
673:{\displaystyle K}
650:{\displaystyle W}
630:{\displaystyle V}
575:{\displaystyle W}
555:{\displaystyle W}
531:{\displaystyle W}
511:{\displaystyle V}
479:{\displaystyle W}
459:{\displaystyle V}
435:{\displaystyle W}
422:to the origin of
415:{\displaystyle V}
395:{\displaystyle W}
375:{\displaystyle V}
330:{\displaystyle V}
280:{\displaystyle W}
260:{\displaystyle V}
111:
110:
103:
16:(Redirected from
16623:
16611:Linear operators
16601:Abstract algebra
16565:Bernhard Riemann
16397:
16396:
16240:Exterior product
16207:Two-point tensor
16192:Symmetric tensor
16074:Electromagnetism
15988:
15987:
15959:
15952:
15945:
15936:
15935:
15922:
15921:
15804:Exterior algebra
15741:Hadamard product
15658:
15646:Linear equations
15517:
15510:
15503:
15494:
15493:
15489:
15464:Wilansky, Albert
15459:
15444:(2nd ed.).
15433:
15406:
15376:
15342:
15306:
15282:
15266:
15252:
15227:
15200:
15174:
15151:
15136:(2nd ed.).
15125:
15104:
15089:(3rd ed.).
15069:
15068:
15066:
15065:
15060:
15021:
15019:
15018:
15013:
15001:
14999:
14998:
14993:
14978:
14976:
14975:
14970:
14958:
14956:
14955:
14950:
14932:
14930:
14929:
14924:
14911:
14899:
14894:
14890:
14888:
14887:
14882:
14868:
14865:is not dense in
14864:
14862:
14861:
14856:
14833:
14831:
14830:
14825:
14801:
14799:
14798:
14793:
14777:
14775:
14774:
14769:
14761:
14749:
14747:
14746:
14741:
14733:
14718:
14714:
14712:
14711:
14706:
14685:
14679:
14677:
14660:
14654:
14648:
14639:
14633:
14624:
14618:
14612:
14606:
14600:
14594:
14588:
14585:
14583:
14582:
14577:
14558:
14550:
14548:
14547:
14542:
14520:
14518:
14517:
14512:
14510:
14509:
14504:
14488:
14486:
14485:
14480:
14459:) appear in the
14458:
14456:
14455:
14450:
14445:
14444:
14439:
14424:
14423:
14418:
14402:
14400:
14399:
14394:
14392:
14391:
14386:
14370:
14368:
14367:
14362:
14360:
14359:
14337:
14335:
14334:
14329:
14327:
14326:
14319:
14318:
14296:
14295:
14278:
14277:
14236:
14235:
14213:
14212:
14195:
14194:
14175:
14174:
14152:
14151:
14134:
14133:
14088:
14082:
14078:
14074:
14070:
14068:
14067:
14062:
14035:
14034:
14029:
14023:
14022:
14006:
14001:
13983:
13982:
13977:
13961:
13959:
13958:
13953:
13951:
13950:
13928:
13926:
13925:
13920:
13887:
13883:
13879:
13877:
13876:
13871:
13869:
13865:
13864:
13863:
13858:
13843:
13842:
13837:
13819:
13817:
13816:
13811:
13809:
13805:
13804:
13803:
13798:
13783:
13782:
13777:
13755:
13749:
13743:
13734:
13728:
13719:
13713:
13707:
13706:
13704:
13703:
13689:
13683:
13677:
13669:
13663:
13661:
13657:
13655:
13654:
13649:
13644:
13626:
13624:
13623:
13618:
13616:
13601:
13598:and all scalars
13597:
13595:
13594:
13589:
13581:
13580:
13575:
13566:
13565:
13560:
13551:
13539:
13537:
13536:
13531:
13529:
13512:
13496:
13492:
13491:
13486:
13474:
13473:
13468:
13456:
13452:
13451:
13450:
13445:
13436:
13435:
13430:
13406:is said to be a
13405:
13401:
13397:
13389:
13383:
13381:
13379:
13378:
13373:
13361:
13359:
13358:
13353:
13348:
13330:
13328:
13327:
13322:
13320:
13305:
13303:
13302:
13297:
13285:
13283:
13282:
13277:
13266:and all scalars
13265:
13263:
13262:
13257:
13249:
13241:
13229:
13227:
13226:
13221:
13219:
13205:
13188:
13177:
13152:
13150:
13149:
13144:
13116:
13112:
13104:
13098:
13093:
13091:
13090:
13085:
13069:
13065:
13063:
13062:
13057:
13046:
13029:
13012:
12986:
12985:
12964:
12962:
12961:
12956:
12945:
12944:
12925:
12919:
12917:
12916:
12911:
12882:
12876:
12874:
12873:
12868:
12844:
12842:
12841:
12836:
12818:
12816:
12815:
12810:
12792:
12790:
12789:
12784:
12754:
12748:
12742:
12738:
12736:
12735:
12730:
12722:
12710:
12708:
12707:
12702:
12700:
12683:
12661:
12659:
12658:
12653:
12645:
12637:
12625:
12623:
12622:
12617:
12615:
12604:
12590:
12582:
12563:
12559:
12555:
12551:
12547:
12543:
12538:
12532:
12526:
12521:linear operators
12518:
12514:
12509:
12484:
12475:
12453:Bounded operator
12393:
12385:
12296:
12294:
12293:
12288:
12283:
12279:
12267:
12256:
12252:
12234:
12233:
12218:
12214:
12195:
12193:
12192:
12187:
12185:
12181:
12160:
12156:
12108:whose matrix is
12080:
12076:
12072:
12056:
12052:
12035:
12031:
12025:
12021:
12017:
12009:
11988:
11980:
11970:
11958:
11950:
11944:
11929:right-invertible
11926:
11920:
11910:
11900:
11886:
11872:
11864:
11859:
11837:
11818:
11806:
11798:
11792:
11774:
11768:
11758:
11748:
11734:
11720:
11712:
11707:
11698:
11674:
11655:
11644:
11630:
11626:
11622:
11569:
11567:
11566:
11561:
11445:
11443:
11442:
11437:
11435:
11431:
11430:
11414:
11410:
11409:
11373:
11371:
11370:
11365:
11357:
11356:
11338:
11337:
11289:
11287:
11286:
11281:
11279:
11275:
11274:
11258:
11254:
11253:
11211:), the value of
11202:
11200:
11199:
11194:
11006:
11004:
11003:
10998:
10904:
10902:
10901:
10896:
10876:
10853:
10813:
10811:
10810:
10805:
10768:, respectively.
10767:
10765:
10764:
10759:
10747:
10745:
10744:
10739:
10727:
10725:
10724:
10719:
10707:
10705:
10704:
10699:
10687:
10685:
10684:
10679:
10667:
10665:
10664:
10659:
10647:
10645:
10644:
10639:
10618:
10616:
10615:
10610:
10586:
10584:
10583:
10578:
10562:
10560:
10559:
10554:
10518:
10516:
10515:
10510:
10490:, or sometimes,
10489:
10487:
10486:
10481:
10457:
10455:
10454:
10449:
10433:
10431:
10430:
10425:
10387:
10385:
10384:
10379:
10284:
10282:
10281:
10276:
10264:
10262:
10261:
10256:
10232:
10230:
10229:
10224:
10208:
10206:
10205:
10200:
10175:
10173:
10172:
10167:
10165:
10151:
10140:
10126:
10112:
10073:
10062:
10042:
10000:
9998:
9997:
9992:
9968:
9966:
9965:
9960:
9933:Rank of a matrix
9914:
9912:
9911:
9906:
9894:
9892:
9891:
9886:
9868:
9866:
9865:
9860:
9823:
9821:
9820:
9815:
9803:
9801:
9800:
9795:
9783:
9781:
9780:
9775:
9749:
9747:
9746:
9741:
9717:
9715:
9714:
9709:
9697:
9695:
9694:
9689:
9674:
9672:
9671:
9666:
9639:is the group of
9638:
9636:
9635:
9630:
9602:
9600:
9599:
9594:
9570:
9568:
9567:
9562:
9538:
9536:
9535:
9530:
9510:
9508:
9507:
9502:
9490:
9488:
9487:
9482:
9463:that is also an
9462:
9460:
9459:
9454:
9439:
9437:
9436:
9431:
9400:
9398:
9397:
9392:
9376:
9374:
9373:
9368:
9344:
9342:
9341:
9336:
9320:
9318:
9317:
9312:
9251:
9249:
9248:
9243:
9219:
9217:
9216:
9213:{\textstyle V=W}
9211:
9193:
9191:
9190:
9185:
9155:
9153:
9152:
9147:
9135:
9133:
9132:
9127:
9115:
9113:
9112:
9107:
9095:
9093:
9092:
9087:
9070:
9069:
9053:
9051:
9050:
9045:
9037:
9014:
8987:
8985:
8984:
8979:
8964:
8962:
8961:
8956:
8944:
8942:
8941:
8936:
8924:
8922:
8921:
8916:
8889:
8887:
8886:
8881:
8876:
8868:
8867:
8852:
8844:
8843:
8828:
8817:
8816:
8804:
8803:
8784:
8782:
8781:
8776:
8774:
8773:
8761:
8760:
8740:
8738:
8737:
8732:
8718:
8717:
8701:
8699:
8698:
8693:
8679:
8678:
8645:-linear maps as
8641:, together with
8632:
8630:
8629:
8624:
8591:
8589:
8588:
8583:
8559:
8557:
8556:
8551:
8512:
8510:
8509:
8504:
8499:
8498:
8460:
8439:
8437:
8436:
8431:
8429:
8428:
8421:
8413:
8383:
8366:
8364:
8363:
8358:
8356:
8355:
8302:
8280:
8278:
8277:
8272:
8270:
8269:
8231:
8214:
8212:
8211:
8206:
8204:
8193:
8192:
8154:
8135:
8133:
8132:
8127:
8125:
8124:
8047:
8029:
8027:
8026:
8021:
8019:
8018:
7977:
7959:
7957:
7956:
7951:
7949:
7948:
7907:
7882:
7880:
7879:
7874:
7872:
7871:
7806:
7788:
7786:
7785:
7780:
7778:
7777:
7736:
7695:
7693:
7692:
7687:
7685:
7684:
7683:
7674:
7670:
7669:
7665:
7644:
7643:
7642:
7633:
7629:
7613:
7612:
7593:
7591:
7590:
7585:
7577:
7576:
7557:
7555:
7554:
7549:
7547:
7546:
7545:
7536:
7532:
7515:
7513:
7512:
7507:
7505:
7504:
7503:
7494:
7490:
7489:
7485:
7464:
7463:
7462:
7453:
7449:
7439:
7424:
7422:
7421:
7416:
7414:
7413:
7412:
7403:
7399:
7398:
7394:
7369:
7367:
7366:
7361:
7359:
7358:
7357:
7348:
7344:
7316:
7314:
7313:
7308:
7306:
7305:
7286:
7284:
7283:
7278:
7276:
7261:
7259:
7258:
7253:
7239:
7237:
7236:
7231:
7219:
7217:
7216:
7211:
7199:
7197:
7196:
7191:
7189:
7172:
7170:
7169:
7164:
7162:
7147:
7145:
7144:
7139:
7137:
7122:
7120:
7119:
7114:
7100:
7098:
7097:
7092:
7080:
7078:
7077:
7072:
7060:
7058:
7057:
7052:
7033:
7031:
7030:
7025:
7013:
7011:
7010:
7005:
7003:
7002:
6981:
6980:
6961:
6959:
6958:
6953:
6948:
6947:
6942:
6923:
6921:
6920:
6915:
6885:
6883:
6882:
6877:
6865:
6863:
6862:
6857:
6845:
6843:
6842:
6837:
6835:
6834:
6828:
6826:
6825:
6812:
6809:
6803:
6797:
6791:
6790:
6771:
6759:
6747:
6745:
6744:
6739:
6734:
6733:
6728:
6709:
6707:
6706:
6701:
6689:
6687:
6686:
6681:
6676:
6675:
6670:
6651:
6649:
6648:
6643:
6641:
6640:
6633:
6632:
6609:
6608:
6581:
6579:
6578:
6573:
6561:
6559:
6558:
6553:
6541:
6539:
6538:
6533:
6521:
6519:
6518:
6513:
6501:
6499:
6498:
6493:
6481:
6479:
6478:
6473:
6461:
6459:
6458:
6453:
6435:
6433:
6432:
6427:
6425:
6424:
6405:
6403:
6402:
6397:
6383:
6381:
6380:
6375:
6370:
6369:
6364:
6358:
6357:
6336:
6335:
6330:
6324:
6323:
6308:
6304:
6303:
6298:
6278:
6276:
6275:
6270:
6265:
6264:
6259:
6240:
6238:
6237:
6232:
6220:
6218:
6217:
6212:
6207:
6206:
6201:
6186:
6185:
6180:
6164:
6162:
6161:
6156:
6151:
6150:
6145:
6121:
6120:
6115:
6090:
6088:
6087:
6082:
6077:
6073:
6072:
6067:
6054:
6053:
6032:
6031:
6026:
6014:
6013:
5998:
5997:
5992:
5986:
5985:
5967:
5966:
5961:
5955:
5954:
5933:
5915:
5913:
5912:
5907:
5881:
5879:
5878:
5873:
5868:
5867:
5862:
5856:
5855:
5837:
5836:
5831:
5825:
5824:
5812:
5800:
5798:
5797:
5792:
5790:
5778:
5776:
5775:
5770:
5768:
5767:
5749:
5748:
5732:
5730:
5729:
5724:
5716:
5704:
5702:
5701:
5696:
5684:
5682:
5681:
5676:
5671:
5670:
5665:
5650:
5649:
5644:
5621:
5619:
5618:
5613:
5611:
5610:
5605:
5596:
5595:
5590:
5577:
5575:
5574:
5569:
5567:
5553:
5535:
5533:
5532:
5527:
5509:
5507:
5506:
5501:
5485:
5483:
5482:
5477:
5465:
5463:
5462:
5457:
5437:
5435:
5434:
5429:
5417:
5415:
5414:
5409:
5384:
5382:
5381:
5376:
5361:
5359:
5358:
5353:
5341:
5339:
5338:
5333:
5321:
5319:
5318:
5313:
5301:
5299:
5298:
5293:
5276:
5259:
5247:
5245:
5244:
5239:
5237:
5210:
5208:
5207:
5202:
5186:
5184:
5183:
5178:
5163:
5161:
5160:
5155:
5139:
5137:
5136:
5131:
5114:
5112:
5111:
5106:
5025:
5023:
5022:
5017:
5015:
5014:
5009:
4936:
4934:
4933:
4928:
4926:
4918:
4917:
4912:
4893:
4891:
4890:
4885:
4880:
4879:
4874:
4813:
4811:
4810:
4805:
4751:
4749:
4748:
4743:
4713:
4711:
4710:
4705:
4672:
4670:
4669:
4664:
4662:
4644:
4642:
4641:
4636:
4634:
4633:
4628:
4610:For example, if
4606:
4604:
4603:
4598:
4573:
4571:
4570:
4565:
4541:
4539:
4538:
4533:
4521:
4519:
4518:
4513:
4511:
4510:
4492:
4491:
4475:
4473:
4472:
4467:
4462:
4461:
4443:
4442:
4420:
4418:
4417:
4412:
4410:
4406:
4405:
4389:
4388:
4370:
4366:
4365:
4349:
4348:
4336:
4332:
4331:
4330:
4321:
4320:
4305:
4304:
4295:
4294:
4270:
4268:
4267:
4262:
4232:
4230:
4229:
4224:
4200:
4198:
4197:
4192:
4187:
4183:
4182:
4166:
4165:
4147:
4143:
4142:
4126:
4125:
4103:
4101:
4100:
4095:
4090:
4089:
4080:
4079:
4061:
4060:
4051:
4050:
4028:
4026:
4025:
4020:
4012:
4011:
3993:
3992:
3976:
3974:
3973:
3968:
3966:
3965:
3947:
3946:
3930:
3928:
3927:
3922:
3904:
3902:
3901:
3896:
3866:
3864:
3863:
3858:
3832:
3830:
3829:
3824:
3809:
3807:
3806:
3801:
3780:
3778:
3777:
3772:
3760:
3758:
3757:
3752:
3728:
3726:
3725:
3720:
3708:
3706:
3705:
3700:
3688:
3686:
3685:
3680:
3668:
3666:
3665:
3660:
3648:
3646:
3645:
3640:
3629:When the subset
3628:
3626:
3625:
3620:
3605:
3603:
3602:
3597:
3579:
3577:
3576:
3571:
3535:
3533:
3532:
3527:
3512:
3510:
3509:
3504:
3492:
3490:
3489:
3484:
3459:
3457:
3456:
3451:
3436:
3434:
3433:
3428:
3416:
3415:
3414:linear extension
3409:
3407:
3406:
3401:
3376:
3374:
3373:
3368:
3344:
3342:
3341:
3336:
3324:
3322:
3321:
3316:
3299:
3298:
3283:
3281:
3280:
3275:
3270:
3250:
3229:
3227:
3226:
3221:
3171:
3162:
3160:
3159:
3154:
3149:
3132:
3115:
3107:
3089:
3087:
3086:
3081:
3031:
3021:
3019:
3018:
3013:
3001:
2999:
2998:
2993:
2981:
2979:
2978:
2973:
2925:
2923:
2922:
2917:
2915:
2914:
2909:
2900:
2899:
2894:
2873:
2858:
2856:
2855:
2850:
2808:
2806:
2805:
2800:
2743:
2741:
2740:
2735:
2723:
2721:
2720:
2715:
2684:
2674:
2672:
2671:
2666:
2642:
2638:
2634:
2630:
2628:
2627:
2622:
2610:
2608:
2607:
2602:
2583:
2581:
2580:
2575:
2573:
2561:
2559:
2558:
2553:
2551:
2529:
2527:
2526:
2521:
2497:
2492:
2458:
2453:
2432:
2428:
2388:
2383:
2367:
2365:
2364:
2359:
2357:
2345:
2341:
2328:
2326:
2325:
2320:
2315:
2313:
2305:
2288:
2280:
2278:
2270:
2253:
2245:
2241:
2202:
2200:
2189:
2167:smooth functions
2154:
2152:
2151:
2146:
2134:
2132:
2131:
2126:
2091:
2089:
2088:
2083:
2048:
2046:
2045:
2040:
2038:
2037:
2032:
2023:
2008:
2006:
2005:
2000:
1998:
1997:
1992:
1983:
1968:
1966:
1965:
1960:
1958:
1957:
1952:
1939:
1937:
1936:
1931:
1929:
1928:
1923:
1910:
1908:
1907:
1902:
1887:
1885:
1884:
1879:
1861:
1859:
1858:
1853:
1834:
1832:
1831:
1826:
1799:
1797:
1796:
1791:
1789:
1788:
1752:
1750:
1749:
1744:
1742:
1734:
1719:
1715:
1713:
1712:
1707:
1705:
1694:
1672:
1670:
1669:
1664:
1647:
1639:
1612:
1610:
1609:
1604:
1592:
1590:
1589:
1584:
1579:
1578:
1573:
1556:
1554:
1553:
1548:
1536:
1534:
1533:
1528:
1508:
1506:
1505:
1500:
1495:
1494:
1489:
1477:
1457:
1437:
1436:
1431:
1412:
1410:
1409:
1404:
1396:
1384:
1382:
1381:
1376:
1358:
1356:
1355:
1350:
1345:
1344:
1339:
1327:
1326:
1321:
1302:
1300:
1299:
1294:
1292:
1291:
1286:
1273:
1271:
1270:
1265:
1263:
1262:
1257:
1244:
1242:
1241:
1236:
1224:
1222:
1221:
1216:
1197:
1195:
1194:
1189:
1181:
1180:
1175:
1163:
1162:
1141:
1140:
1135:
1123:
1122:
1107:
1106:
1101:
1095:
1094:
1076:
1075:
1070:
1064:
1063:
1041:
1039:
1038:
1033:
1022:
1021:
1003:
1002:
986:
984:
983:
978:
970:
969:
964:
949:
948:
943:
916:
914:
913:
908:
903:
883:
857:
855:
854:
849:
844:
827:
810:
802:
777:
775:
774:
769:
751:
749:
748:
743:
735:
727:
712:is said to be a
711:
709:
708:
703:
679:
677:
676:
671:
656:
654:
653:
648:
636:
634:
633:
628:
581:
579:
578:
573:
561:
559:
558:
553:
537:
535:
534:
529:
517:
515:
514:
509:
485:
483:
482:
477:
465:
463:
462:
457:
444:linear subspaces
441:
439:
438:
433:
421:
419:
418:
413:
401:
399:
398:
393:
381:
379:
378:
373:
336:
334:
333:
328:
316:
314:
313:
308:
286:
284:
283:
278:
266:
264:
263:
258:
245:
244:
234:
232:
231:
226:
207:
206:
169:
167:
166:
161:
106:
99:
95:
92:
86:
81:this article by
72:inline citations
59:
58:
51:
21:
16631:
16630:
16626:
16625:
16624:
16622:
16621:
16620:
16591:
16590:
16589:
16584:
16535:Albert Einstein
16502:
16483:Einstein tensor
16446:
16427:Ricci curvature
16407:Kronecker delta
16393:Notable tensors
16388:
16309:Connection form
16286:
16280:
16211:
16197:Tensor operator
16154:
16148:
16088:
16064:Computer vision
16057:
16039:
16035:Tensor calculus
15979:
15968:
15963:
15933:
15928:
15910:
15872:
15828:
15765:
15717:
15659:
15650:
15616:Change of basis
15606:Multilinear map
15544:
15526:
15521:
15478:
15456:
15422:
15395:
15381:Schechter, Eric
15365:
15331:
15303:
15279:
15250:
15240:Springer-Verlag
15216:
15197:
15171:
15157:Matrix Analysis
15148:
15122:
15101:
15078:
15073:
15072:
15027:
15024:
15023:
15007:
15004:
15003:
14984:
14981:
14980:
14964:
14961:
14960:
14944:
14941:
14940:
14918:
14915:
14914:
14912:
14908:
14903:
14902:
14898:
14892:
14876:
14873:
14872:
14866:
14841:
14838:
14837:
14810:
14807:
14806:
14805:The null space
14787:
14784:
14783:
14757:
14755:
14752:
14751:
14729:
14724:
14721:
14720:
14716:
14700:
14697:
14696:
14686:
14682:
14661:
14657:
14649:
14642:
14634:
14627:
14619:
14615:
14607:
14603:
14595:
14591:
14565:
14562:
14561:
14556:
14530:
14527:
14526:
14505:
14500:
14499:
14494:
14491:
14490:
14468:
14465:
14464:
14440:
14435:
14434:
14419:
14414:
14413:
14408:
14405:
14404:
14387:
14382:
14381:
14376:
14373:
14372:
14349:
14345:
14343:
14340:
14339:
14321:
14320:
14308:
14304:
14302:
14297:
14285:
14281:
14279:
14267:
14263:
14260:
14259:
14254:
14249:
14244:
14238:
14237:
14225:
14221:
14219:
14214:
14202:
14198:
14196:
14184:
14180:
14177:
14176:
14164:
14160:
14158:
14153:
14141:
14137:
14135:
14123:
14119:
14112:
14111:
14097:
14094:
14093:
14086:
14080:
14076:
14072:
14030:
14025:
14024:
14012:
14008:
14002:
13991:
13978:
13973:
13972:
13967:
13964:
13963:
13940:
13936:
13934:
13931:
13930:
13893:
13890:
13889:
13885:
13881:
13859:
13854:
13853:
13838:
13833:
13832:
13831:
13827:
13825:
13822:
13821:
13799:
13794:
13793:
13778:
13773:
13772:
13771:
13767:
13765:
13762:
13761:
13756:
13752:
13744:
13737:
13729:
13722:
13714:
13710:
13701:
13699:
13691:
13690:
13686:
13675:
13670:
13666:
13659:
13640:
13632:
13629:
13628:
13612:
13607:
13604:
13603:
13599:
13576:
13571:
13570:
13561:
13556:
13555:
13547:
13545:
13542:
13541:
13525:
13508:
13487:
13482:
13481:
13469:
13464:
13463:
13446:
13441:
13440:
13431:
13426:
13425:
13424:
13420:
13415:
13412:
13411:
13403:
13399:
13395:
13390:
13386:
13367:
13364:
13363:
13344:
13336:
13333:
13332:
13316:
13311:
13308:
13307:
13291:
13288:
13287:
13271:
13268:
13267:
13245:
13237:
13235:
13232:
13231:
13215:
13201:
13184:
13173:
13162:
13159:
13158:
13126:
13123:
13122:
13114:
13110:
13105:
13101:
13097:
13079:
13076:
13075:
13067:
13042:
13025:
13008:
12978:
12974:
12972:
12969:
12968:
12937:
12933:
12931:
12928:
12927:
12923:
12896:
12893:
12892:
12880:
12853:
12850:
12849:
12824:
12821:
12820:
12798:
12795:
12794:
12766:
12763:
12762:
12760:
12755:
12751:
12740:
12718:
12716:
12713:
12712:
12696:
12679:
12668:
12665:
12664:
12663:
12641:
12633:
12631:
12628:
12627:
12611:
12600:
12586:
12578:
12570:
12567:
12566:
12565:
12561:
12557:
12553:
12549:
12545:
12541:
12539:
12535:
12524:
12516:
12512:
12510:
12506:
12501:
12482:
12479:Linear isometry
12473:
12431:
12400:
12387:
12376:
12341:
12333:Main articles:
12331:
12272:
12268:
12260:
12245:
12241:
12226:
12222:
12207:
12203:
12201:
12198:
12197:
12174:
12170:
12149:
12145:
12140:
12137:
12136:
12112:, in the basis
12102:
12100:Change of basis
12094:Main articles:
12092:
12090:Change of basis
12078:
12074:
12064:
12054:
12044:
12033:
12027:
12023:
12022:-th iterate of
12019:
12015:
11997:
11986:
11978:
11968:
11966:
11956:
11946:
11932:
11924:
11912:
11902:
11901:, the equation
11888:
11874:
11870:
11862:
11857:
11845:
11835:
11816:
11814:
11804:
11794:
11780:
11777:left-invertible
11772:
11760:
11750:
11749:, the equation
11736:
11722:
11718:
11710:
11701:
11696:
11686:
11672:
11653:
11651:
11632:
11628:
11624:
11620:
11614:
11600:, the index of
11598:operator theory
11480:
11477:
11476:
11469:
11462:
11451:
11426:
11422:
11418:
11405:
11401:
11397:
11395:
11392:
11391:
11352:
11348:
11333:
11329:
11327:
11324:
11323:
11312:
11306:
11296:
11270:
11266:
11262:
11249:
11245:
11241:
11239:
11236:
11235:
11164:
11161:
11160:
11147:: the value of
10938:
10935:
10934:
10872:
10849:
10826:
10823:
10822:
10787:
10784:
10783:
10780:
10774:
10753:
10750:
10749:
10733:
10730:
10729:
10713:
10710:
10709:
10693:
10690:
10689:
10673:
10670:
10669:
10653:
10650:
10649:
10624:
10621:
10620:
10592:
10589:
10588:
10587:and written as
10572:
10569:
10568:
10524:
10521:
10520:
10495:
10492:
10491:
10463:
10460:
10459:
10458:and written as
10443:
10440:
10439:
10395:
10392:
10391:
10298:
10295:
10294:
10270:
10267:
10266:
10238:
10235:
10234:
10218:
10215:
10214:
10182:
10179:
10178:
10163:
10162:
10147:
10136:
10122:
10108:
10097:
10079:
10078:
10069:
10058:
10038:
10027:
10008:
10006:
10003:
10002:
9986:
9983:
9982:
9942:
9939:
9938:
9935:
9923:Main articles:
9921:
9900:
9897:
9896:
9874:
9871:
9870:
9836:
9833:
9832:
9809:
9806:
9805:
9789:
9786:
9785:
9763:
9760:
9759:
9723:
9720:
9719:
9703:
9700:
9699:
9683:
9680:
9679:
9648:
9645:
9644:
9612:
9609:
9608:
9576:
9573:
9572:
9544:
9541:
9540:
9524:
9521:
9520:
9496:
9493:
9492:
9476:
9473:
9472:
9448:
9445:
9444:
9413:
9410:
9409:
9386:
9383:
9382:
9350:
9347:
9346:
9330:
9327:
9326:
9294:
9291:
9290:
9287:
9279:Main articles:
9277:
9269:matrix addition
9225:
9222:
9221:
9199:
9196:
9195:
9161:
9158:
9157:
9141:
9138:
9137:
9121:
9118:
9117:
9101:
9098:
9097:
9065:
9064:
9062:
9059:
9058:
9033:
9010:
8993:
8990:
8989:
8970:
8967:
8966:
8965:, then the map
8950:
8947:
8946:
8930:
8927:
8926:
8898:
8895:
8894:
8872:
8863:
8859:
8848:
8839:
8835:
8824:
8812:
8808:
8799:
8795:
8790:
8787:
8786:
8769:
8765:
8756:
8752:
8750:
8747:
8746:
8713:
8709:
8707:
8704:
8703:
8674:
8670:
8668:
8665:
8664:
8600:
8597:
8596:
8565:
8562:
8561:
8533:
8530:
8529:
8526:
8493:
8492:
8487:
8481:
8480:
8475:
8465:
8464:
8456:
8454:
8451:
8450:
8423:
8422:
8412:
8410:
8404:
8403:
8398:
8388:
8387:
8379:
8377:
8374:
8373:
8370:squeeze mapping
8350:
8349:
8338:
8332:
8331:
8317:
8307:
8306:
8298:
8296:
8293:
8292:
8264:
8263:
8258:
8252:
8251:
8246:
8236:
8235:
8227:
8225:
8222:
8221:
8200:
8187:
8186:
8181:
8175:
8174:
8169:
8159:
8158:
8150:
8148:
8145:
8144:
8119:
8118:
8101:
8086:
8085:
8071:
8052:
8051:
8043:
8041:
8038:
8037:
8013:
8012:
8007:
8001:
8000:
7995:
7982:
7981:
7973:
7971:
7968:
7967:
7943:
7942:
7934:
7928:
7927:
7922:
7912:
7911:
7903:
7901:
7898:
7897:
7866:
7865:
7854:
7842:
7841:
7827:
7811:
7810:
7802:
7800:
7797:
7796:
7772:
7771:
7766:
7760:
7759:
7751:
7741:
7740:
7732:
7730:
7727:
7726:
7702:
7676:
7675:
7661:
7657:
7653:
7649:
7648:
7635:
7634:
7625:
7621:
7620:
7605:
7601:
7599:
7596:
7595:
7569:
7565:
7563:
7560:
7559:
7538:
7537:
7528:
7524:
7523:
7521:
7518:
7517:
7496:
7495:
7481:
7477:
7473:
7469:
7468:
7455:
7454:
7445:
7441:
7440:
7432:
7430:
7427:
7426:
7405:
7404:
7390:
7386:
7382:
7378:
7377:
7375:
7372:
7371:
7350:
7349:
7340:
7336:
7335:
7333:
7330:
7329:
7298:
7294:
7292:
7289:
7288:
7280:{\textstyle B'}
7269:
7267:
7264:
7263:
7247:
7244:
7243:
7225:
7222:
7221:
7205:
7202:
7201:
7193:{\textstyle B'}
7182:
7180:
7177:
7176:
7166:{\textstyle A'}
7155:
7153:
7150:
7149:
7141:{\textstyle B'}
7130:
7128:
7125:
7124:
7108:
7105:
7104:
7086:
7083:
7082:
7066:
7063:
7062:
7046:
7043:
7042:
7019:
7016:
7015:
6995:
6991:
6973:
6969:
6967:
6964:
6963:
6943:
6938:
6937:
6929:
6926:
6925:
6891:
6888:
6887:
6871:
6868:
6867:
6851:
6848:
6847:
6829:
6827:
6818:
6814:
6810:
6808:
6801:
6800:
6792:
6783:
6779:
6777:
6764:
6763:
6755:
6753:
6750:
6749:
6729:
6724:
6723:
6715:
6712:
6711:
6695:
6692:
6691:
6671:
6666:
6665:
6657:
6654:
6653:
6635:
6634:
6625:
6621:
6618:
6617:
6611:
6610:
6601:
6597:
6590:
6589:
6587:
6584:
6583:
6567:
6564:
6563:
6547:
6544:
6543:
6542:, every column
6527:
6524:
6523:
6507:
6504:
6503:
6487:
6484:
6483:
6467:
6464:
6463:
6441:
6438:
6437:
6417:
6413:
6411:
6408:
6407:
6391:
6388:
6387:
6365:
6360:
6359:
6350:
6346:
6331:
6326:
6325:
6316:
6312:
6299:
6294:
6293:
6289:
6284:
6281:
6280:
6260:
6255:
6254:
6246:
6243:
6242:
6226:
6223:
6222:
6221:be a basis for
6202:
6197:
6196:
6181:
6176:
6175:
6170:
6167:
6166:
6146:
6141:
6140:
6116:
6111:
6110:
6102:
6099:
6098:
6068:
6063:
6062:
6058:
6049:
6045:
6027:
6022:
6021:
6009:
6005:
5993:
5988:
5987:
5981:
5977:
5962:
5957:
5956:
5950:
5946:
5929:
5921:
5918:
5917:
5889:
5886:
5885:
5863:
5858:
5857:
5851:
5847:
5832:
5827:
5826:
5820:
5816:
5808:
5806:
5803:
5802:
5786:
5784:
5781:
5780:
5763:
5759:
5744:
5740:
5738:
5735:
5734:
5712:
5710:
5707:
5706:
5690:
5687:
5686:
5685:be a basis for
5666:
5661:
5660:
5645:
5640:
5639:
5634:
5631:
5630:
5624:Euclidean space
5606:
5601:
5600:
5591:
5586:
5585:
5583:
5580:
5579:
5563:
5549:
5541:
5538:
5537:
5515:
5512:
5511:
5495:
5492:
5491:
5471:
5468:
5467:
5451:
5448:
5447:
5423:
5420:
5419:
5403:
5400:
5399:
5396:
5390:
5367:
5364:
5363:
5347:
5344:
5343:
5327:
5324:
5323:
5307:
5304:
5303:
5272:
5255:
5253:
5250:
5249:
5233:
5219:
5216:
5215:
5196:
5193:
5192:
5169:
5166:
5165:
5149:
5146:
5145:
5142:vector subspace
5125:
5122:
5121:
5031:
5028:
5027:
5010:
5005:
5004:
4942:
4939:
4938:
4922:
4913:
4908:
4907:
4899:
4896:
4895:
4875:
4870:
4869:
4819:
4816:
4815:
4757:
4754:
4753:
4719:
4716:
4715:
4678:
4675:
4674:
4658:
4650:
4647:
4646:
4629:
4624:
4623:
4615:
4612:
4611:
4579:
4576:
4575:
4547:
4544:
4543:
4527:
4524:
4523:
4506:
4502:
4487:
4483:
4481:
4478:
4477:
4457:
4453:
4438:
4434:
4426:
4423:
4422:
4401:
4397:
4393:
4384:
4380:
4361:
4357:
4353:
4344:
4340:
4326:
4322:
4316:
4312:
4300:
4296:
4290:
4286:
4285:
4281:
4276:
4273:
4272:
4238:
4235:
4234:
4206:
4203:
4202:
4178:
4174:
4170:
4161:
4157:
4138:
4134:
4130:
4121:
4117:
4109:
4106:
4105:
4085:
4081:
4075:
4071:
4056:
4052:
4046:
4042:
4034:
4031:
4030:
4007:
4003:
3988:
3984:
3982:
3979:
3978:
3961:
3957:
3942:
3938:
3936:
3933:
3932:
3931:is an integer,
3910:
3907:
3906:
3872:
3869:
3868:
3840:
3837:
3836:
3815:
3812:
3811:
3786:
3783:
3782:
3766:
3763:
3762:
3734:
3731:
3730:
3714:
3711:
3710:
3694:
3691:
3690:
3674:
3671:
3670:
3654:
3651:
3650:
3634:
3631:
3630:
3611:
3608:
3607:
3585:
3582:
3581:
3541:
3538:
3537:
3521:
3518:
3517:
3498:
3495:
3494:
3466:
3463:
3462:
3442:
3439:
3438:
3422:
3419:
3418:
3413:
3412:
3386:
3383:
3382:
3350:
3347:
3346:
3330:
3327:
3326:
3310:
3307:
3306:
3296:
3295:
3291:
3284:
3266:
3246:
3235:
3232:
3231:
3179:
3176:
3175:
3172:
3163:
3145:
3128:
3111:
3103:
3095:
3092:
3091:
3039:
3036:
3035:
3032:
3023:
3007:
3004:
3003:
2987:
2984:
2983:
2931:
2928:
2927:
2910:
2905:
2904:
2895:
2890:
2889:
2881:
2878:
2877:
2874:
2814:
2811:
2810:
2749:
2746:
2745:
2729:
2726:
2725:
2709:
2706:
2705:
2702:random variable
2687:§ Matrices
2676:
2648:
2645:
2644:
2640:
2636:
2632:
2616:
2613:
2612:
2596:
2593:
2592:
2569:
2567:
2564:
2563:
2547:
2545:
2542:
2541:
2493:
2488:
2454:
2449:
2394:
2390:
2384:
2379:
2373:
2370:
2369:
2353:
2351:
2348:
2347:
2343:
2339:
2306:
2289:
2287:
2271:
2254:
2252:
2207:
2203:
2193:
2188:
2186:
2183:
2182:
2163:linear operator
2159:Differentiation
2140:
2137:
2136:
2105:
2102:
2101:
2065:
2062:
2061:
2055:§ Matrices
2033:
2028:
2027:
2019:
2014:
2011:
2010:
1993:
1988:
1987:
1979:
1977:
1974:
1973:
1953:
1948:
1947:
1945:
1942:
1941:
1924:
1919:
1918:
1916:
1913:
1912:
1896:
1893:
1892:
1867:
1864:
1863:
1847:
1844:
1843:
1808:
1805:
1804:
1784:
1780:
1772:
1769:
1768:
1738:
1730:
1728:
1725:
1724:
1717:
1701:
1690:
1688:
1685:
1684:
1673:, of which the
1643:
1635:
1627:
1624:
1623:
1619:
1598:
1595:
1594:
1574:
1572:
1571:
1569:
1566:
1565:
1542:
1539:
1538:
1516:
1513:
1512:
1490:
1485:
1484:
1473:
1453:
1432:
1427:
1426:
1418:
1415:
1414:
1392:
1390:
1387:
1386:
1364:
1361:
1360:
1340:
1335:
1334:
1322:
1317:
1316:
1308:
1305:
1304:
1287:
1282:
1281:
1279:
1276:
1275:
1258:
1253:
1252:
1250:
1247:
1246:
1230:
1227:
1226:
1210:
1207:
1206:
1176:
1171:
1170:
1158:
1154:
1136:
1131:
1130:
1118:
1114:
1102:
1097:
1096:
1090:
1086:
1071:
1066:
1065:
1059:
1055:
1047:
1044:
1043:
1017:
1013:
998:
994:
992:
989:
988:
965:
960:
959:
944:
939:
938:
936:
933:
932:
899:
879:
868:
865:
864:
840:
823:
806:
798:
790:
787:
786:
757:
754:
753:
752:and any scalar
731:
723:
721:
718:
717:
685:
682:
681:
665:
662:
661:
642:
639:
638:
622:
619:
618:
615:
595:category theory
567:
564:
563:
547:
544:
543:
523:
520:
519:
503:
500:
499:
471:
468:
467:
451:
448:
447:
427:
424:
423:
407:
404:
403:
387:
384:
383:
367:
364:
363:
348:linear function
322:
319:
318:
296:
293:
292:
272:
269:
268:
252:
249:
248:
243:linear operator
242:
241:
214:
211:
210:
204:
203:
176:vector addition
149:
146:
145:
139:linear function
125:(also called a
107:
96:
90:
87:
77:Please help to
76:
60:
56:
49:
46:linear function
42:
35:
28:
23:
22:
18:Linear operator
15:
12:
11:
5:
16629:
16619:
16618:
16613:
16608:
16603:
16586:
16585:
16583:
16582:
16577:
16575:Woldemar Voigt
16572:
16567:
16562:
16557:
16552:
16547:
16542:
16540:Leonhard Euler
16537:
16532:
16527:
16522:
16516:
16514:
16512:Mathematicians
16508:
16507:
16504:
16503:
16501:
16500:
16495:
16490:
16485:
16480:
16475:
16470:
16465:
16460:
16454:
16452:
16448:
16447:
16445:
16444:
16439:
16437:Torsion tensor
16434:
16429:
16424:
16419:
16414:
16409:
16403:
16401:
16394:
16390:
16389:
16387:
16386:
16381:
16376:
16371:
16366:
16361:
16356:
16351:
16346:
16341:
16336:
16331:
16326:
16321:
16316:
16311:
16306:
16301:
16296:
16290:
16288:
16282:
16281:
16279:
16278:
16272:
16270:Tensor product
16267:
16262:
16260:Symmetrization
16257:
16252:
16250:Lie derivative
16247:
16242:
16237:
16232:
16227:
16221:
16219:
16213:
16212:
16210:
16209:
16204:
16199:
16194:
16189:
16184:
16179:
16174:
16172:Tensor density
16169:
16164:
16158:
16156:
16150:
16149:
16147:
16146:
16144:Voigt notation
16141:
16136:
16131:
16129:Ricci calculus
16126:
16121:
16116:
16114:Index notation
16111:
16106:
16100:
16098:
16094:
16093:
16090:
16089:
16087:
16086:
16081:
16076:
16071:
16066:
16060:
16058:
16056:
16055:
16050:
16044:
16041:
16040:
16038:
16037:
16032:
16030:Tensor algebra
16027:
16022:
16017:
16012:
16010:Dyadic algebra
16007:
16002:
15996:
15994:
15985:
15981:
15980:
15973:
15970:
15969:
15962:
15961:
15954:
15947:
15939:
15930:
15929:
15927:
15926:
15915:
15912:
15911:
15909:
15908:
15903:
15898:
15893:
15888:
15886:Floating-point
15882:
15880:
15874:
15873:
15871:
15870:
15868:Tensor product
15865:
15860:
15855:
15853:Function space
15850:
15845:
15839:
15837:
15830:
15829:
15827:
15826:
15821:
15816:
15811:
15806:
15801:
15796:
15791:
15789:Triple product
15786:
15781:
15775:
15773:
15767:
15766:
15764:
15763:
15758:
15753:
15748:
15743:
15738:
15733:
15727:
15725:
15719:
15718:
15716:
15715:
15710:
15705:
15703:Transformation
15700:
15695:
15693:Multiplication
15690:
15685:
15680:
15675:
15669:
15667:
15661:
15660:
15653:
15651:
15649:
15648:
15643:
15638:
15633:
15628:
15623:
15618:
15613:
15608:
15603:
15598:
15593:
15588:
15583:
15578:
15573:
15568:
15563:
15558:
15552:
15550:
15549:Basic concepts
15546:
15545:
15543:
15542:
15537:
15531:
15528:
15527:
15524:Linear algebra
15520:
15519:
15512:
15505:
15497:
15491:
15490:
15476:
15460:
15454:
15434:
15420:
15407:
15393:
15377:
15363:
15343:
15329:
15307:
15301:
15283:
15277:
15253:
15248:
15236:Linear Algebra
15228:
15214:
15201:
15195:
15175:
15169:
15152:
15146:
15126:
15120:
15105:
15099:
15077:
15074:
15071:
15070:
15058:
15055:
15052:
15049:
15046:
15043:
15040:
15037:
15034:
15031:
15011:
14991:
14988:
14968:
14948:
14937:
14922:
14905:
14904:
14901:
14900:
14897:
14896:
14880:
14870:
14854:
14851:
14848:
14845:
14835:
14823:
14820:
14817:
14814:
14803:
14791:
14780:
14767:
14764:
14760:
14739:
14736:
14732:
14728:
14704:
14680:
14665:"Index theory"
14655:
14640:
14638:p. 52, § 2.5.1
14625:
14613:
14611:, p. 19, § 3.1
14601:
14589:
14575:
14572:
14569:
14540:
14537:
14534:
14523:column vectors
14508:
14503:
14498:
14489:. The vectors
14478:
14475:
14472:
14448:
14443:
14438:
14433:
14430:
14427:
14422:
14417:
14412:
14390:
14385:
14380:
14371:of the vector
14358:
14355:
14352:
14348:
14325:
14317:
14314:
14311:
14307:
14303:
14301:
14298:
14294:
14291:
14288:
14284:
14280:
14276:
14273:
14270:
14266:
14262:
14261:
14258:
14255:
14253:
14250:
14248:
14245:
14243:
14240:
14239:
14234:
14231:
14228:
14224:
14220:
14218:
14215:
14211:
14208:
14205:
14201:
14197:
14193:
14190:
14187:
14183:
14179:
14178:
14173:
14170:
14167:
14163:
14159:
14157:
14154:
14150:
14147:
14144:
14140:
14136:
14132:
14129:
14126:
14122:
14118:
14117:
14115:
14110:
14107:
14104:
14101:
14060:
14057:
14054:
14051:
14048:
14045:
14042:
14039:
14033:
14028:
14021:
14018:
14015:
14011:
14005:
14000:
13997:
13994:
13990:
13986:
13981:
13976:
13971:
13949:
13946:
13943:
13939:
13918:
13915:
13912:
13909:
13906:
13903:
13900:
13897:
13868:
13862:
13857:
13852:
13849:
13846:
13841:
13836:
13830:
13808:
13802:
13797:
13792:
13789:
13786:
13781:
13776:
13770:
13750:
13746:Schechter 1996
13735:
13720:
13708:
13684:
13664:
13647:
13643:
13639:
13636:
13615:
13611:
13587:
13584:
13579:
13574:
13569:
13564:
13559:
13554:
13550:
13528:
13524:
13521:
13518:
13515:
13511:
13507:
13504:
13501:
13495:
13490:
13485:
13480:
13477:
13472:
13467:
13462:
13459:
13455:
13449:
13444:
13439:
13434:
13429:
13423:
13419:
13384:
13371:
13351:
13347:
13343:
13340:
13331:, rather than
13319:
13315:
13295:
13275:
13255:
13252:
13248:
13244:
13240:
13218:
13214:
13211:
13208:
13204:
13200:
13197:
13194:
13191:
13187:
13183:
13180:
13176:
13172:
13169:
13166:
13153:is said to be
13142:
13139:
13136:
13133:
13130:
13099:
13096:
13095:
13083:
13055:
13052:
13049:
13045:
13041:
13038:
13035:
13032:
13028:
13024:
13021:
13018:
13015:
13011:
13007:
13004:
13001:
12998:
12995:
12992:
12989:
12984:
12981:
12977:
12965:
12954:
12951:
12948:
12943:
12940:
12936:
12920:
12909:
12906:
12903:
12900:
12877:
12866:
12863:
12860:
12857:
12846:
12834:
12831:
12828:
12808:
12805:
12802:
12782:
12779:
12776:
12773:
12770:
12749:
12728:
12725:
12721:
12699:
12695:
12692:
12689:
12686:
12682:
12678:
12675:
12672:
12651:
12648:
12644:
12640:
12636:
12614:
12610:
12607:
12603:
12599:
12596:
12593:
12589:
12585:
12581:
12577:
12574:
12533:
12503:
12502:
12500:
12497:
12496:
12495:
12490:
12485:
12476:
12467:
12462:
12456:
12450:
12444:
12441:Antilinear map
12438:
12430:
12427:
12399:
12396:
12351:, for example
12330:
12327:
12286:
12282:
12278:
12275:
12271:
12266:
12263:
12259:
12255:
12251:
12248:
12244:
12240:
12237:
12232:
12229:
12225:
12221:
12217:
12213:
12210:
12206:
12184:
12180:
12177:
12173:
12169:
12166:
12163:
12159:
12155:
12152:
12148:
12144:
12091:
12088:
12087:
12086:
12061:
12057:is said to be
12041:
12036:is said to be
11965:
11962:
11961:
11960:
11922:
11860:
11853:
11843:
11819:is said to be
11813:
11810:
11809:
11808:
11770:
11708:
11699:
11692:
11684:
11656:is said to be
11650:
11647:
11613:
11610:
11559:
11556:
11553:
11550:
11547:
11544:
11541:
11538:
11535:
11532:
11529:
11526:
11523:
11520:
11517:
11514:
11511:
11508:
11505:
11502:
11499:
11496:
11493:
11490:
11487:
11484:
11468:
11465:
11457:
11449:
11434:
11429:
11425:
11421:
11417:
11413:
11408:
11404:
11400:
11363:
11360:
11355:
11351:
11347:
11344:
11341:
11336:
11332:
11310:
11301:
11294:
11278:
11273:
11269:
11265:
11261:
11257:
11252:
11248:
11244:
11192:
11189:
11186:
11183:
11180:
11177:
11174:
11171:
11168:
11054:
11053:
11046:
10996:
10993:
10990:
10987:
10984:
10981:
10978:
10975:
10972:
10969:
10966:
10963:
10960:
10957:
10954:
10951:
10948:
10945:
10942:
10932:exact sequence
10894:
10891:
10888:
10885:
10882:
10879:
10875:
10871:
10868:
10865:
10862:
10859:
10856:
10852:
10848:
10845:
10842:
10839:
10836:
10833:
10830:
10803:
10800:
10797:
10794:
10791:
10776:Main article:
10773:
10770:
10761:{\textstyle A}
10757:
10741:{\textstyle f}
10737:
10721:{\textstyle A}
10717:
10701:{\textstyle f}
10697:
10681:{\textstyle W}
10677:
10661:{\textstyle V}
10657:
10637:
10634:
10631:
10628:
10608:
10605:
10602:
10599:
10596:
10580:{\textstyle f}
10576:
10563:is called the
10552:
10549:
10546:
10543:
10540:
10537:
10534:
10531:
10528:
10508:
10505:
10502:
10499:
10479:
10476:
10473:
10470:
10467:
10451:{\textstyle f}
10447:
10423:
10420:
10417:
10414:
10411:
10408:
10405:
10402:
10399:
10377:
10374:
10371:
10368:
10365:
10362:
10359:
10356:
10353:
10350:
10347:
10344:
10341:
10338:
10335:
10332:
10329:
10326:
10323:
10320:
10317:
10314:
10311:
10308:
10305:
10302:
10278:{\textstyle W}
10274:
10254:
10251:
10248:
10245:
10242:
10226:{\textstyle V}
10222:
10198:
10195:
10192:
10189:
10186:
10161:
10157:
10154:
10150:
10146:
10143:
10139:
10135:
10132:
10129:
10125:
10121:
10118:
10115:
10111:
10106:
10103:
10100:
10098:
10096:
10093:
10090:
10087:
10084:
10081:
10080:
10077:
10072:
10068:
10065:
10061:
10057:
10054:
10051:
10048:
10045:
10041:
10036:
10033:
10030:
10028:
10026:
10023:
10020:
10017:
10014:
10011:
10010:
9994:{\textstyle f}
9990:
9958:
9955:
9952:
9949:
9946:
9920:
9917:
9908:{\textstyle K}
9904:
9884:
9881:
9878:
9858:
9855:
9852:
9849:
9846:
9843:
9840:
9817:{\textstyle V}
9813:
9797:{\textstyle K}
9793:
9773:
9770:
9767:
9739:
9736:
9733:
9730:
9727:
9711:{\textstyle n}
9707:
9691:{\textstyle V}
9687:
9664:
9661:
9658:
9655:
9652:
9628:
9625:
9622:
9619:
9616:
9592:
9589:
9586:
9583:
9580:
9560:
9557:
9554:
9551:
9548:
9532:{\textstyle V}
9528:
9504:{\textstyle V}
9500:
9484:{\textstyle V}
9480:
9456:{\textstyle V}
9452:
9429:
9426:
9423:
9420:
9417:
9394:{\textstyle K}
9390:
9366:
9363:
9360:
9357:
9354:
9338:{\textstyle V}
9334:
9310:
9307:
9304:
9301:
9298:
9276:
9273:
9241:
9238:
9235:
9232:
9229:
9209:
9206:
9203:
9183:
9180:
9177:
9174:
9171:
9168:
9165:
9149:{\textstyle K}
9145:
9129:{\textstyle W}
9125:
9109:{\textstyle V}
9105:
9085:
9082:
9079:
9076:
9073:
9068:
9043:
9040:
9036:
9032:
9029:
9026:
9023:
9020:
9017:
9013:
9009:
9006:
9003:
9000:
8997:
8977:
8974:
8958:{\textstyle K}
8954:
8934:
8925:is linear and
8914:
8911:
8908:
8905:
8902:
8879:
8875:
8871:
8866:
8862:
8858:
8855:
8851:
8847:
8842:
8838:
8834:
8831:
8827:
8823:
8820:
8815:
8811:
8807:
8802:
8798:
8794:
8772:
8768:
8764:
8759:
8755:
8730:
8727:
8724:
8721:
8716:
8712:
8691:
8688:
8685:
8682:
8677:
8673:
8622:
8619:
8616:
8613:
8610:
8607:
8604:
8581:
8578:
8575:
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8569:
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8546:
8543:
8540:
8537:
8525:
8522:
8514:
8513:
8502:
8497:
8491:
8488:
8486:
8483:
8482:
8479:
8476:
8474:
8471:
8470:
8468:
8463:
8459:
8440:
8427:
8419:
8416:
8411:
8409:
8406:
8405:
8402:
8399:
8397:
8394:
8393:
8391:
8386:
8382:
8367:
8354:
8348:
8345:
8342:
8339:
8337:
8334:
8333:
8330:
8327:
8324:
8321:
8318:
8316:
8313:
8312:
8310:
8305:
8301:
8281:
8268:
8262:
8259:
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8254:
8253:
8250:
8247:
8245:
8242:
8241:
8239:
8234:
8230:
8215:
8203:
8199:
8196:
8191:
8185:
8182:
8180:
8177:
8176:
8173:
8170:
8168:
8165:
8164:
8162:
8157:
8153:
8138:
8137:
8136:
8123:
8117:
8114:
8111:
8108:
8105:
8102:
8100:
8097:
8094:
8091:
8088:
8087:
8084:
8081:
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8070:
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8064:
8061:
8058:
8057:
8055:
8050:
8046:
8030:
8017:
8011:
8008:
8006:
8003:
8002:
7999:
7996:
7994:
7991:
7988:
7987:
7985:
7980:
7976:
7960:
7947:
7941:
7938:
7935:
7933:
7930:
7929:
7926:
7923:
7921:
7918:
7917:
7915:
7910:
7906:
7885:
7884:
7883:
7870:
7864:
7861:
7858:
7855:
7853:
7850:
7847:
7844:
7843:
7840:
7837:
7834:
7831:
7828:
7826:
7823:
7820:
7817:
7816:
7814:
7809:
7805:
7789:
7776:
7770:
7767:
7765:
7762:
7761:
7758:
7755:
7752:
7750:
7747:
7746:
7744:
7739:
7735:
7701:
7698:
7682:
7679:
7673:
7668:
7664:
7660:
7656:
7652:
7647:
7641:
7638:
7632:
7628:
7624:
7619:
7616:
7611:
7608:
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7583:
7580:
7575:
7572:
7568:
7544:
7541:
7535:
7531:
7527:
7502:
7499:
7493:
7488:
7484:
7480:
7476:
7472:
7467:
7461:
7458:
7452:
7448:
7444:
7438:
7435:
7411:
7408:
7402:
7397:
7393:
7389:
7385:
7381:
7356:
7353:
7347:
7343:
7339:
7318:
7317:
7304:
7301:
7297:
7275:
7272:
7255:{\textstyle B}
7251:
7240:
7233:{\textstyle P}
7229:
7213:{\textstyle B}
7209:
7188:
7185:
7173:
7161:
7158:
7136:
7133:
7116:{\textstyle T}
7112:
7101:
7094:{\textstyle A}
7090:
7074:{\textstyle B}
7070:
7054:{\textstyle T}
7050:
7023:
7001:
6998:
6994:
6990:
6987:
6984:
6979:
6976:
6972:
6951:
6946:
6941:
6936:
6933:
6913:
6910:
6907:
6904:
6901:
6898:
6895:
6875:
6855:
6833:
6824:
6821:
6817:
6813:
6811:
6807:
6804:
6802:
6796:
6793:
6789:
6786:
6782:
6778:
6776:
6770:
6769:
6767:
6762:
6758:
6737:
6732:
6727:
6722:
6719:
6699:
6679:
6674:
6669:
6664:
6661:
6639:
6631:
6628:
6624:
6620:
6619:
6616:
6613:
6612:
6607:
6604:
6600:
6596:
6595:
6593:
6571:
6551:
6531:
6511:
6491:
6471:
6451:
6448:
6445:
6423:
6420:
6416:
6395:
6373:
6368:
6363:
6356:
6353:
6349:
6345:
6342:
6339:
6334:
6329:
6322:
6319:
6315:
6311:
6307:
6302:
6297:
6292:
6288:
6268:
6263:
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6250:
6230:
6210:
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6184:
6179:
6174:
6154:
6149:
6144:
6139:
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6130:
6127:
6124:
6119:
6114:
6109:
6106:
6080:
6076:
6071:
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6061:
6057:
6052:
6048:
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6038:
6035:
6030:
6025:
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6017:
6012:
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6004:
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5991:
5984:
5980:
5976:
5973:
5970:
5965:
5960:
5953:
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5945:
5942:
5939:
5936:
5932:
5928:
5925:
5905:
5902:
5899:
5896:
5893:
5871:
5866:
5861:
5854:
5850:
5846:
5843:
5840:
5835:
5830:
5823:
5819:
5815:
5811:
5789:
5766:
5762:
5758:
5755:
5752:
5747:
5743:
5722:
5719:
5715:
5694:
5674:
5669:
5664:
5659:
5656:
5653:
5648:
5643:
5638:
5609:
5604:
5599:
5594:
5589:
5566:
5562:
5559:
5556:
5552:
5548:
5545:
5525:
5522:
5519:
5499:
5475:
5455:
5427:
5407:
5392:Main article:
5389:
5386:
5374:
5371:
5351:
5331:
5311:
5302:holds for all
5291:
5288:
5285:
5282:
5279:
5275:
5271:
5268:
5265:
5262:
5258:
5248:(meaning that
5236:
5232:
5229:
5226:
5223:
5200:
5176:
5173:
5153:
5129:
5104:
5101:
5098:
5095:
5092:
5089:
5086:
5083:
5080:
5077:
5074:
5071:
5068:
5065:
5062:
5059:
5056:
5053:
5050:
5047:
5044:
5041:
5038:
5035:
5013:
5008:
5003:
5000:
4997:
4994:
4991:
4988:
4985:
4982:
4979:
4976:
4973:
4970:
4967:
4964:
4961:
4958:
4955:
4952:
4949:
4946:
4925:
4921:
4916:
4911:
4906:
4903:
4883:
4878:
4873:
4868:
4865:
4862:
4859:
4856:
4853:
4850:
4847:
4844:
4841:
4838:
4835:
4832:
4829:
4826:
4823:
4803:
4800:
4797:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4773:
4770:
4767:
4764:
4761:
4741:
4738:
4735:
4732:
4729:
4726:
4723:
4703:
4700:
4697:
4694:
4691:
4688:
4685:
4682:
4661:
4657:
4654:
4632:
4627:
4622:
4619:
4596:
4593:
4590:
4587:
4584:
4563:
4560:
4557:
4554:
4551:
4531:
4522:as above. If
4509:
4505:
4501:
4498:
4495:
4490:
4486:
4465:
4460:
4456:
4452:
4449:
4446:
4441:
4437:
4433:
4430:
4421:holds for all
4409:
4404:
4400:
4396:
4392:
4387:
4383:
4379:
4376:
4373:
4369:
4364:
4360:
4356:
4352:
4347:
4343:
4339:
4335:
4329:
4325:
4319:
4315:
4311:
4308:
4303:
4299:
4293:
4289:
4284:
4280:
4271:is unique and
4260:
4257:
4254:
4251:
4248:
4245:
4242:
4222:
4219:
4216:
4213:
4210:
4190:
4186:
4181:
4177:
4173:
4169:
4164:
4160:
4156:
4153:
4150:
4146:
4141:
4137:
4133:
4129:
4124:
4120:
4116:
4113:
4093:
4088:
4084:
4078:
4074:
4070:
4067:
4064:
4059:
4055:
4049:
4045:
4041:
4038:
4018:
4015:
4010:
4006:
4002:
3999:
3996:
3991:
3987:
3964:
3960:
3956:
3953:
3950:
3945:
3941:
3920:
3917:
3914:
3894:
3891:
3888:
3885:
3882:
3879:
3876:
3856:
3853:
3850:
3847:
3844:
3822:
3819:
3799:
3796:
3793:
3790:
3770:
3750:
3747:
3744:
3741:
3738:
3718:
3698:
3678:
3658:
3638:
3618:
3615:
3595:
3592:
3589:
3569:
3566:
3563:
3560:
3557:
3554:
3551:
3548:
3545:
3536:(meaning that
3525:
3502:
3482:
3479:
3476:
3473:
3470:
3449:
3446:
3426:
3399:
3396:
3393:
3390:
3366:
3363:
3360:
3357:
3354:
3334:
3314:
3300:
3290:
3287:
3286:
3285:
3273:
3269:
3265:
3262:
3259:
3256:
3253:
3249:
3245:
3242:
3239:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3173:
3166:
3164:
3152:
3148:
3144:
3141:
3138:
3135:
3131:
3127:
3124:
3121:
3118:
3114:
3110:
3106:
3102:
3099:
3079:
3076:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3033:
3026:
3024:
3015:{\textstyle 2}
3011:
2995:{\textstyle x}
2991:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2941:
2938:
2935:
2913:
2908:
2903:
2898:
2893:
2888:
2885:
2875:
2868:
2865:
2864:
2848:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2733:
2713:
2698:expected value
2694:
2664:
2661:
2658:
2655:
2652:
2620:
2600:
2589:
2586:quotient space
2572:
2550:
2538:antiderivative
2532:An indefinite
2530:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2496:
2491:
2487:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2457:
2452:
2448:
2444:
2441:
2438:
2435:
2431:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2393:
2387:
2382:
2378:
2356:
2329:
2318:
2312:
2309:
2304:
2301:
2298:
2295:
2292:
2286:
2283:
2277:
2274:
2269:
2266:
2263:
2260:
2257:
2251:
2248:
2244:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2206:
2199:
2196:
2192:
2156:
2144:
2124:
2121:
2118:
2115:
2112:
2109:
2081:
2078:
2075:
2072:
2069:
2058:
2036:
2031:
2026:
2022:
2018:
1996:
1991:
1986:
1982:
1956:
1951:
1927:
1922:
1900:
1877:
1874:
1871:
1851:
1840:
1824:
1821:
1818:
1815:
1812:
1801:
1800:is not linear.
1787:
1783:
1779:
1776:
1765:
1758:
1741:
1737:
1733:
1721:
1704:
1700:
1697:
1693:
1678:
1662:
1659:
1656:
1653:
1650:
1646:
1642:
1638:
1634:
1631:
1618:
1615:
1602:
1582:
1577:
1546:
1526:
1523:
1520:
1498:
1493:
1488:
1483:
1480:
1476:
1472:
1469:
1466:
1463:
1460:
1456:
1452:
1449:
1446:
1443:
1440:
1435:
1430:
1425:
1422:
1402:
1399:
1395:
1374:
1371:
1368:
1348:
1343:
1338:
1333:
1330:
1325:
1320:
1315:
1312:
1290:
1285:
1261:
1256:
1234:
1214:
1187:
1184:
1179:
1174:
1169:
1166:
1161:
1157:
1153:
1150:
1147:
1144:
1139:
1134:
1129:
1126:
1121:
1117:
1113:
1110:
1105:
1100:
1093:
1089:
1085:
1082:
1079:
1074:
1069:
1062:
1058:
1054:
1051:
1031:
1028:
1025:
1020:
1016:
1012:
1009:
1006:
1001:
997:
976:
973:
968:
963:
958:
955:
952:
947:
942:
918:
917:
906:
902:
898:
895:
892:
889:
886:
882:
878:
875:
872:
858:
847:
843:
839:
836:
833:
830:
826:
822:
819:
816:
813:
809:
805:
801:
797:
794:
767:
764:
761:
741:
738:
734:
730:
726:
701:
698:
695:
692:
689:
680:. A function
669:
646:
626:
614:
611:
571:
551:
527:
507:
475:
455:
431:
411:
391:
371:
339:function space
326:
306:
303:
300:
276:
256:
224:
221:
218:
159:
156:
153:
127:linear mapping
119:linear algebra
109:
108:
63:
61:
54:
26:
9:
6:
4:
3:
2:
16628:
16617:
16614:
16612:
16609:
16607:
16604:
16602:
16599:
16598:
16596:
16581:
16578:
16576:
16573:
16571:
16568:
16566:
16563:
16561:
16558:
16556:
16553:
16551:
16548:
16546:
16543:
16541:
16538:
16536:
16533:
16531:
16528:
16526:
16523:
16521:
16518:
16517:
16515:
16513:
16509:
16499:
16496:
16494:
16491:
16489:
16486:
16484:
16481:
16479:
16476:
16474:
16471:
16469:
16466:
16464:
16461:
16459:
16456:
16455:
16453:
16449:
16443:
16440:
16438:
16435:
16433:
16430:
16428:
16425:
16423:
16420:
16418:
16417:Metric tensor
16415:
16413:
16410:
16408:
16405:
16404:
16402:
16398:
16395:
16391:
16385:
16382:
16380:
16377:
16375:
16372:
16370:
16367:
16365:
16362:
16360:
16357:
16355:
16352:
16350:
16347:
16345:
16342:
16340:
16337:
16335:
16332:
16330:
16329:Exterior form
16327:
16325:
16322:
16320:
16317:
16315:
16312:
16310:
16307:
16305:
16302:
16300:
16297:
16295:
16292:
16291:
16289:
16283:
16276:
16273:
16271:
16268:
16266:
16263:
16261:
16258:
16256:
16253:
16251:
16248:
16246:
16243:
16241:
16238:
16236:
16233:
16231:
16228:
16226:
16223:
16222:
16220:
16218:
16214:
16208:
16205:
16203:
16202:Tensor bundle
16200:
16198:
16195:
16193:
16190:
16188:
16185:
16183:
16180:
16178:
16175:
16173:
16170:
16168:
16165:
16163:
16160:
16159:
16157:
16151:
16145:
16142:
16140:
16137:
16135:
16132:
16130:
16127:
16125:
16122:
16120:
16117:
16115:
16112:
16110:
16107:
16105:
16102:
16101:
16099:
16095:
16085:
16082:
16080:
16077:
16075:
16072:
16070:
16067:
16065:
16062:
16061:
16059:
16054:
16051:
16049:
16046:
16045:
16042:
16036:
16033:
16031:
16028:
16026:
16023:
16021:
16018:
16016:
16013:
16011:
16008:
16006:
16003:
16001:
15998:
15997:
15995:
15993:
15989:
15986:
15982:
15978:
15977:
15971:
15967:
15960:
15955:
15953:
15948:
15946:
15941:
15940:
15937:
15925:
15917:
15916:
15913:
15907:
15904:
15902:
15901:Sparse matrix
15899:
15897:
15894:
15892:
15889:
15887:
15884:
15883:
15881:
15879:
15875:
15869:
15866:
15864:
15861:
15859:
15856:
15854:
15851:
15849:
15846:
15844:
15841:
15840:
15838:
15836:constructions
15835:
15831:
15825:
15824:Outermorphism
15822:
15820:
15817:
15815:
15812:
15810:
15807:
15805:
15802:
15800:
15797:
15795:
15792:
15790:
15787:
15785:
15784:Cross product
15782:
15780:
15777:
15776:
15774:
15772:
15768:
15762:
15759:
15757:
15754:
15752:
15751:Outer product
15749:
15747:
15744:
15742:
15739:
15737:
15734:
15732:
15731:Orthogonality
15729:
15728:
15726:
15724:
15720:
15714:
15711:
15709:
15708:Cramer's rule
15706:
15704:
15701:
15699:
15696:
15694:
15691:
15689:
15686:
15684:
15681:
15679:
15678:Decomposition
15676:
15674:
15671:
15670:
15668:
15666:
15662:
15657:
15647:
15644:
15642:
15639:
15637:
15634:
15632:
15629:
15627:
15624:
15622:
15619:
15617:
15614:
15612:
15609:
15607:
15604:
15602:
15599:
15597:
15594:
15592:
15589:
15587:
15584:
15582:
15579:
15577:
15574:
15572:
15569:
15567:
15564:
15562:
15559:
15557:
15554:
15553:
15551:
15547:
15541:
15538:
15536:
15533:
15532:
15529:
15525:
15518:
15513:
15511:
15506:
15504:
15499:
15498:
15495:
15487:
15483:
15479:
15473:
15469:
15465:
15461:
15457:
15451:
15447:
15443:
15439:
15438:Tu, Loring W.
15435:
15431:
15427:
15423:
15417:
15413:
15408:
15404:
15400:
15396:
15390:
15386:
15382:
15378:
15374:
15370:
15366:
15360:
15356:
15352:
15348:
15344:
15340:
15336:
15332:
15326:
15322:
15318:
15317:
15312:
15311:Rudin, Walter
15308:
15304:
15298:
15294:
15293:
15288:
15287:Rudin, Walter
15284:
15280:
15278:9780070542259
15274:
15270:
15265:
15264:
15258:
15257:Rudin, Walter
15254:
15251:
15249:0-387-96412-6
15245:
15241:
15237:
15233:
15229:
15225:
15221:
15217:
15211:
15207:
15202:
15198:
15192:
15188:
15184:
15180:
15176:
15172:
15166:
15162:
15158:
15153:
15149:
15147:0-387-90093-4
15143:
15139:
15135:
15131:
15127:
15123:
15121:3-540-43491-7
15117:
15113:
15112:
15106:
15102:
15096:
15092:
15088:
15084:
15080:
15079:
15056:
15050:
15044:
15041:
15035:
15029:
15009:
14989:
14986:
14966:
14946:
14938:
14935:
14920:
14910:
14906:
14871:
14843:
14836:
14812:
14804:
14802:is continuous
14782:
14781:
14779:
14765:
14762:
14737:
14734:
14693:
14690:, p. 15
14689:
14684:
14676:
14672:
14671:
14666:
14659:
14652:
14651:Halmos (1974)
14647:
14645:
14637:
14632:
14630:
14622:
14617:
14610:
14605:
14598:
14593:
14586:
14579:{\textstyle }
14570:
14554:
14544:{\textstyle }
14535:
14524:
14506:
14496:
14482:{\textstyle }
14473:
14462:
14441:
14431:
14428:
14425:
14420:
14388:
14378:
14356:
14353:
14350:
14346:
14323:
14315:
14312:
14309:
14305:
14299:
14292:
14289:
14286:
14282:
14274:
14271:
14268:
14264:
14256:
14251:
14246:
14241:
14232:
14229:
14226:
14222:
14216:
14209:
14206:
14203:
14199:
14191:
14188:
14185:
14181:
14171:
14168:
14165:
14161:
14155:
14148:
14145:
14142:
14138:
14130:
14127:
14124:
14120:
14113:
14108:
14102:
14091:
14085:
14058:
14052:
14049:
14046:
14043:
14040:
14031:
14019:
14016:
14013:
14009:
14003:
13998:
13995:
13992:
13988:
13984:
13979:
13969:
13947:
13944:
13941:
13937:
13913:
13910:
13907:
13901:
13898:
13895:
13866:
13860:
13850:
13847:
13844:
13839:
13828:
13806:
13800:
13790:
13787:
13784:
13779:
13768:
13759:
13754:
13747:
13742:
13740:
13733:, p. 57.
13732:
13731:Kubrusly 2001
13727:
13725:
13717:
13716:Wilansky 2013
13712:
13698:
13694:
13688:
13681:
13673:
13668:
13634:
13609:
13585:
13582:
13577:
13567:
13562:
13552:
13522:
13519:
13516:
13505:
13499:
13493:
13488:
13478:
13475:
13470:
13460:
13457:
13453:
13447:
13437:
13432:
13421:
13417:
13409:
13393:
13388:
13293:
13273:
13253:
13250:
13242:
13209:
13206:
13195:
13192:
13181:
13178:
13170:
13156:
13140:
13134:
13131:
13120:
13108:
13103:
13073:
13070:, called the
13043:
13039:
13033:
13030:
13019:
13016:
13013:
13002:
12993:
12982:
12979:
12966:
12949:
12941:
12938:
12921:
12904:
12890:
12886:
12878:
12864:
12861:
12858:
12848:
12847:
12832:
12829:
12826:
12806:
12803:
12800:
12780:
12774:
12771:
12758:
12753:
12747:, p. 316
12746:
12739:and all real
12726:
12723:
12693:
12690:
12687:
12676:
12670:
12649:
12646:
12638:
12608:
12605:
12597:
12594:
12583:
12572:
12537:
12531:, p. 207
12530:
12522:
12508:
12504:
12494:
12491:
12489:
12486:
12480:
12477:
12471:
12468:
12466:
12463:
12460:
12457:
12454:
12451:
12448:
12447:Bent function
12445:
12442:
12439:
12436:
12433:
12432:
12426:
12424:
12420:
12415:
12413:
12409:
12405:
12395:
12391:
12384:
12380:
12372:
12370:
12366:
12362:
12358:
12354:
12353:normed spaces
12350:
12346:
12340:
12336:
12326:
12324:
12320:
12315:
12313:
12309:
12306:
12302:
12297:
12284:
12280:
12276:
12273:
12269:
12264:
12261:
12257:
12253:
12249:
12246:
12242:
12238:
12235:
12230:
12227:
12223:
12219:
12215:
12211:
12208:
12204:
12182:
12178:
12175:
12171:
12167:
12164:
12161:
12157:
12153:
12150:
12146:
12142:
12133:
12131:
12127:
12126:contravariant
12123:
12119:
12115:
12111:
12107:
12101:
12097:
12084:
12083:scalar matrix
12071:
12067:
12062:
12060:
12051:
12047:
12042:
12039:
12030:
12013:
12012:
12011:
12008:
12004:
12000:
11994:
11992:
11984:
11976:
11975:
11954:
11949:
11943:
11939:
11935:
11930:
11923:
11919:
11915:
11909:
11905:
11899:
11895:
11891:
11885:
11881:
11877:
11868:
11861:
11856:
11851:
11848:
11844:
11841:
11834:
11833:
11832:
11830:
11829:
11824:
11823:
11802:
11797:
11791:
11787:
11783:
11778:
11771:
11767:
11763:
11757:
11753:
11747:
11743:
11739:
11733:
11729:
11725:
11716:
11709:
11705:
11700:
11695:
11690:
11685:
11682:
11678:
11671:
11670:
11669:
11667:
11666:
11661:
11660:
11646:
11643:
11639:
11635:
11617:
11609:
11607:
11603:
11599:
11595:
11591:
11587:
11582:
11580:
11576:
11571:
11557:
11548:
11542:
11539:
11533:
11530:
11527:
11518:
11512:
11509:
11503:
11500:
11497:
11491:
11485:
11482:
11474:
11464:
11460:
11456:
11452:
11432:
11427:
11423:
11419:
11411:
11406:
11402:
11398:
11389:
11385:
11381:
11377:
11361:
11358:
11353:
11345:
11342:
11339:
11334:
11321:
11317:
11313:
11304:
11300:
11293:
11276:
11271:
11267:
11263:
11255:
11250:
11246:
11242:
11233:
11229:
11225:
11220:
11218:
11214:
11210:
11206:
11187:
11175:
11172:
11169:
11158:
11154:
11150:
11146:
11142:
11138:
11134:
11130:
11126:
11122:
11118:
11114:
11110:
11106:
11102:
11098:
11094:
11090:
11086:
11082:
11078:
11074:
11069:
11067:
11063:
11059:
11051:
11047:
11044:
11040:
11036:
11032:
11028:
11024:
11023:
11022:
11020:
11016:
11012:
11007:
10994:
10985:
10979:
10976:
10970:
10964:
10955:
10949:
10946:
10940:
10933:
10929:
10925:
10923:
10918:
10915:space of the
10914:
10910:
10905:
10892:
10886:
10880:
10877:
10873:
10869:
10866:
10860:
10854:
10850:
10846:
10843:
10837:
10831:
10828:
10820:
10818:
10801:
10795:
10792:
10789:
10779:
10769:
10755:
10735:
10715:
10695:
10675:
10655:
10632:
10626:
10603:
10597:
10594:
10574:
10566:
10544:
10538:
10535:
10529:
10526:
10519:; the number
10503:
10497:
10474:
10468:
10465:
10445:
10437:
10415:
10409:
10406:
10400:
10397:
10388:
10375:
10369:
10363:
10360:
10357:
10348:
10342:
10339:
10333:
10330:
10327:
10318:
10312:
10309:
10303:
10300:
10292:
10288:
10272:
10249:
10243:
10240:
10220:
10212:
10193:
10187:
10184:
10176:
10155:
10152:
10144:
10130:
10127:
10119:
10116:
10113:
10101:
10099:
10091:
10085:
10082:
10066:
10052:
10049:
10046:
10043:
10031:
10029:
10021:
10015:
10012:
9988:
9980:
9976:
9972:
9956:
9950:
9947:
9944:
9934:
9930:
9926:
9916:
9902:
9882:
9879:
9876:
9853:
9850:
9847:
9841:
9838:
9831:
9827:
9811:
9791:
9771:
9768:
9765:
9757:
9753:
9734:
9728:
9725:
9705:
9685:
9676:
9659:
9653:
9650:
9642:
9623:
9617:
9614:
9606:
9605:endomorphisms
9587:
9581:
9578:
9555:
9549:
9546:
9526:
9518:
9514:
9498:
9478:
9470:
9467:is called an
9466:
9450:
9441:
9427:
9421:
9418:
9415:
9408:
9404:
9388:
9380:
9361:
9355:
9352:
9332:
9324:
9308:
9302:
9299:
9296:
9286:
9282:
9272:
9270:
9266:
9261:
9259:
9255:
9236:
9230:
9227:
9207:
9204:
9201:
9178:
9175:
9172:
9166:
9163:
9143:
9123:
9103:
9080:
9077:
9074:
9057:Thus the set
9055:
9027:
9021:
9018:
9001:
8998:
8988:, defined by
8975:
8972:
8952:
8932:
8912:
8906:
8903:
8900:
8891:
8864:
8860:
8856:
8840:
8836:
8832:
8813:
8809:
8805:
8800:
8796:
8770:
8766:
8762:
8757:
8753:
8744:
8728:
8722:
8719:
8714:
8710:
8689:
8683:
8680:
8675:
8671:
8661:
8659:
8654:
8652:
8648:
8644:
8640:
8636:
8620:
8614:
8611:
8608:
8605:
8602:
8595:
8579:
8573:
8570:
8567:
8547:
8541:
8538:
8535:
8521:
8519:
8500:
8495:
8489:
8484:
8477:
8472:
8466:
8461:
8448:
8444:
8441:
8425:
8417:
8414:
8407:
8400:
8395:
8389:
8384:
8371:
8368:
8352:
8346:
8343:
8340:
8335:
8328:
8325:
8322:
8319:
8314:
8308:
8303:
8290:
8286:
8282:
8266:
8260:
8255:
8248:
8243:
8237:
8232:
8219:
8216:
8197:
8194:
8189:
8183:
8178:
8171:
8166:
8160:
8155:
8142:
8139:
8121:
8115:
8112:
8109:
8106:
8103:
8098:
8095:
8092:
8089:
8082:
8079:
8076:
8073:
8068:
8065:
8062:
8059:
8053:
8048:
8035:
8031:
8015:
8009:
8004:
7997:
7992:
7989:
7983:
7978:
7965:
7961:
7945:
7939:
7936:
7931:
7924:
7919:
7913:
7908:
7895:
7891:
7890:
7889:
7886:
7868:
7862:
7859:
7856:
7851:
7848:
7845:
7838:
7835:
7832:
7829:
7824:
7821:
7818:
7812:
7807:
7794:
7790:
7774:
7768:
7763:
7756:
7753:
7748:
7742:
7737:
7724:
7723:
7722:
7719:
7718:
7717:
7715:
7711:
7707:
7697:
7680:
7677:
7671:
7666:
7658:
7654:
7650:
7645:
7639:
7636:
7630:
7622:
7617:
7614:
7609:
7606:
7602:
7581:
7578:
7573:
7570:
7566:
7542:
7539:
7533:
7525:
7500:
7497:
7491:
7486:
7478:
7474:
7470:
7465:
7459:
7456:
7450:
7442:
7436:
7433:
7409:
7406:
7400:
7395:
7387:
7383:
7379:
7354:
7351:
7345:
7337:
7322:
7302:
7299:
7295:
7273:
7270:
7249:
7241:
7227:
7207:
7186:
7183:
7174:
7159:
7156:
7134:
7131:
7110:
7102:
7088:
7068:
7048:
7040:
7039:
7038:
7035:
7021:
6999:
6996:
6992:
6988:
6985:
6982:
6977:
6974:
6970:
6944:
6931:
6911:
6908:
6905:
6902:
6899:
6896:
6893:
6873:
6853:
6831:
6822:
6819:
6815:
6805:
6794:
6787:
6784:
6780:
6774:
6765:
6760:
6730:
6717:
6697:
6672:
6659:
6637:
6629:
6626:
6622:
6614:
6605:
6602:
6598:
6591:
6569:
6549:
6529:
6509:
6489:
6469:
6449:
6446:
6443:
6421:
6418:
6414:
6393:
6384:
6371:
6366:
6354:
6351:
6347:
6343:
6340:
6337:
6332:
6320:
6317:
6313:
6309:
6305:
6300:
6290:
6286:
6261:
6248:
6228:
6203:
6193:
6190:
6187:
6182:
6147:
6134:
6131:
6128:
6125:
6117:
6104:
6096:
6091:
6078:
6074:
6069:
6059:
6055:
6050:
6046:
6042:
6039:
6036:
6028:
6015:
6010:
6006:
6002:
5994:
5982:
5978:
5974:
5971:
5968:
5963:
5951:
5947:
5940:
5937:
5923:
5903:
5897:
5894:
5891:
5882:
5869:
5864:
5852:
5848:
5844:
5841:
5838:
5833:
5821:
5817:
5813:
5779:in the field
5764:
5760:
5756:
5753:
5750:
5745:
5741:
5720:
5717:
5692:
5667:
5657:
5654:
5651:
5646:
5627:
5625:
5607:
5592:
5560:
5557:
5543:
5536:matrix, then
5523:
5520:
5517:
5497:
5489:
5473:
5453:
5445:
5441:
5425:
5405:
5395:
5385:
5372:
5369:
5349:
5329:
5309:
5286:
5280:
5277:
5266:
5260:
5227:
5224:
5221:
5214:
5198:
5190:
5174:
5171:
5151:
5143:
5140:defined on a
5127:
5120:
5115:
5102:
5099:
5096:
5093:
5090:
5087:
5084:
5078:
5072:
5069:
5063:
5060:
5054:
5051:
5045:
5042:
5039:
5033:
5011:
5001:
4995:
4992:
4989:
4983:
4980:
4974:
4971:
4968:
4962:
4959:
4953:
4950:
4947:
4914:
4904:
4901:
4881:
4876:
4866:
4857:
4854:
4851:
4845:
4839:
4836:
4833:
4824:
4821:
4795:
4792:
4789:
4783:
4777:
4774:
4771:
4762:
4759:
4739:
4730:
4727:
4724:
4701:
4698:
4689:
4686:
4683:
4655:
4652:
4630:
4620:
4617:
4608:
4594:
4588:
4585:
4582:
4561:
4555:
4552:
4549:
4529:
4507:
4503:
4499:
4496:
4493:
4488:
4484:
4463:
4458:
4454:
4450:
4447:
4444:
4439:
4435:
4431:
4428:
4407:
4402:
4398:
4394:
4390:
4385:
4381:
4377:
4374:
4371:
4367:
4362:
4358:
4354:
4350:
4345:
4341:
4337:
4333:
4327:
4323:
4317:
4313:
4309:
4306:
4301:
4297:
4291:
4287:
4282:
4278:
4258:
4252:
4249:
4246:
4243:
4240:
4220:
4214:
4211:
4208:
4188:
4184:
4179:
4175:
4171:
4167:
4162:
4158:
4154:
4151:
4148:
4144:
4139:
4135:
4131:
4127:
4122:
4118:
4114:
4111:
4091:
4086:
4082:
4076:
4072:
4068:
4065:
4062:
4057:
4053:
4047:
4043:
4039:
4036:
4016:
4013:
4008:
4004:
4000:
3997:
3994:
3989:
3985:
3962:
3958:
3954:
3951:
3948:
3943:
3939:
3918:
3915:
3912:
3892:
3886:
3883:
3880:
3877:
3874:
3854:
3848:
3845:
3842:
3833:
3820:
3817:
3797:
3794:
3791:
3788:
3768:
3748:
3742:
3739:
3736:
3716:
3696:
3676:
3656:
3636:
3616:
3613:
3593:
3590:
3587:
3564:
3558:
3555:
3549:
3543:
3523:
3516:
3500:
3480:
3474:
3471:
3468:
3460:
3447:
3444:
3424:
3397:
3394:
3391:
3388:
3380:
3364:
3358:
3355:
3352:
3332:
3312:
3304:
3294:
3260:
3257:
3254:
3243:
3237:
3214:
3211:
3208:
3205:
3199:
3193:
3190:
3187:
3181:
3174:The function
3170:
3165:
3139:
3136:
3122:
3119:
3108:
3097:
3074:
3071:
3068:
3065:
3059:
3053:
3050:
3047:
3041:
3034:The function
3030:
3025:
3009:
2989:
2966:
2963:
2960:
2957:
2951:
2945:
2942:
2939:
2933:
2911:
2896:
2886:
2883:
2876:The function
2872:
2867:
2866:
2862:
2843:
2837:
2834:
2831:
2825:
2822:
2816:
2793:
2787:
2784:
2778:
2772:
2769:
2763:
2760:
2757:
2751:
2731:
2711:
2703:
2699:
2695:
2692:
2688:
2683:
2679:
2662:
2656:
2653:
2650:
2618:
2598:
2590:
2587:
2539:
2535:
2531:
2517:
2514:
2511:
2505:
2499:
2494:
2489:
2485:
2481:
2478:
2475:
2472:
2466:
2460:
2455:
2450:
2446:
2442:
2439:
2436:
2433:
2429:
2422:
2416:
2413:
2410:
2404:
2398:
2395:
2391:
2385:
2380:
2376:
2338:
2334:
2330:
2316:
2310:
2307:
2299:
2293:
2290:
2284:
2281:
2275:
2272:
2264:
2258:
2255:
2249:
2246:
2242:
2235:
2229:
2226:
2223:
2217:
2211:
2208:
2204:
2197:
2194:
2190:
2180:
2176:
2172:
2168:
2164:
2160:
2157:
2142:
2122:
2119:
2113:
2107:
2099:
2098:normed spaces
2096:between real
2095:
2079:
2073:
2070:
2067:
2059:
2056:
2052:
2034:
2024:
2016:
1994:
1984:
1972:
1971:column vector
1969:by sending a
1954:
1925:
1898:
1890:
1875:
1872:
1869:
1849:
1841:
1838:
1822:
1819:
1816:
1810:
1802:
1785:
1781:
1774:
1766:
1763:
1759:
1756:
1723:The zero map
1722:
1720:is a scalar).
1698:
1683:
1679:
1676:
1660:
1657:
1651:
1648:
1632:
1629:
1621:
1620:
1614:
1600:
1580:
1575:
1562:
1560:
1544:
1524:
1518:
1511:A linear map
1509:
1496:
1491:
1481:
1467:
1464:
1461:
1450:
1444:
1441:
1433:
1420:
1400:
1397:
1372:
1369:
1366:
1346:
1341:
1331:
1323:
1310:
1288:
1259:
1232:
1212:
1203:
1201:
1185:
1177:
1164:
1159:
1155:
1151:
1148:
1145:
1137:
1124:
1119:
1115:
1111:
1103:
1091:
1087:
1083:
1080:
1077:
1072:
1060:
1056:
1049:
1029:
1026:
1023:
1018:
1014:
1010:
1007:
1004:
999:
995:
974:
971:
966:
956:
953:
950:
945:
930:
925:
923:
893:
890:
887:
876:
870:
862:
859:
834:
831:
817:
814:
803:
792:
784:
781:
780:
779:
765:
762:
759:
739:
736:
728:
715:
699:
693:
690:
687:
667:
660:
644:
624:
610:
608:
604:
600:
596:
591:
589:
585:
569:
549:
541:
525:
505:
497:
493:
489:
473:
453:
445:
429:
409:
389:
369:
360:
359:it does not.
358:
354:
350:
349:
344:
340:
324:
304:
301:
298:
290:
274:
254:
246:
238:
222:
219:
216:
208:
200:
195:
193:
189:
185:
181:
177:
173:
172:vector spaces
170:between two
157:
151:
144:
140:
136:
132:
128:
124:
120:
116:
105:
102:
94:
91:December 2021
84:
80:
74:
73:
67:
62:
53:
52:
47:
40:
33:
19:
16580:Hermann Weyl
16384:Vector space
16369:Pseudotensor
16348:
16334:Fiber bundle
16287:abstractions
16182:Mixed tensor
16167:Tensor field
15974:
15834:Vector space
15585:
15566:Vector space
15467:
15441:
15411:
15384:
15350:
15315:
15291:
15262:
15235:
15205:
15182:
15156:
15133:
15109:
15086:
15076:Bibliography
14939:another map
14909:
14694:
14692:1.18 Theorem
14691:
14683:
14668:
14658:
14616:
14604:
14599:p. 52, § 3.3
14597:Axler (2015)
14592:
14559:
14552:
14522:
14460:
14089:
14083:
13753:
13711:
13700:. Retrieved
13696:
13687:
13679:
13667:
13407:
13387:
13154:
13121:. A mapping
13118:
13102:
13071:
12889:balanced set
12759:, p. 14
12752:
12536:
12520:
12507:
12435:Additive map
12425:techniques.
12416:
12401:
12398:Applications
12389:
12382:
12378:
12373:
12344:
12342:
12316:
12311:
12307:
12304:
12300:
12298:
12134:
12129:
12121:
12117:
12113:
12109:
12106:endomorphism
12103:
12069:
12065:
12049:
12045:
12028:
12006:
12002:
11998:
11995:
11972:
11967:
11953:identity map
11947:
11941:
11937:
11933:
11917:
11913:
11907:
11903:
11897:
11893:
11889:
11883:
11879:
11875:
11854:
11849:
11826:
11820:
11815:
11801:identity map
11795:
11789:
11785:
11781:
11765:
11761:
11755:
11751:
11745:
11741:
11737:
11731:
11727:
11723:
11703:
11693:
11688:
11679:as a map of
11665:monomorphism
11663:
11657:
11652:
11649:Monomorphism
11641:
11637:
11633:
11618:
11615:
11593:
11589:
11583:
11578:
11574:
11572:
11472:
11470:
11458:
11454:
11447:
11387:
11383:
11379:
11376:endomorphism
11315:
11308:
11302:
11298:
11291:
11231:
11227:
11223:
11221:
11216:
11212:
11208:
11204:
11156:
11152:
11148:
11144:
11140:
11136:
11132:
11128:
11124:
11120:
11116:
11112:
11108:
11104:
11100:
11096:
11092:
11088:
11084:
11080:
11076:
11072:
11070:
11065:
11061:
11057:
11055:
11038:
11034:
11030:
11026:
11018:
11014:
11010:
11008:
10927:
10921:
10916:
10912:
10908:
10907:This is the
10906:
10816:
10781:
10389:
10177:
9936:
9677:
9643:in the ring
9469:automorphism
9442:
9407:identity map
9323:endomorphism
9288:
9285:Automorphism
9281:Endomorphism
9262:
9056:
8892:
8662:
8655:
8642:
8638:
8527:
8515:
8446:
8288:
8284:
8283:skew of the
8033:
7963:
7962:through the
7893:
7892:through the
7792:
7791:by an angle
7709:
7703:
7327:
7123:relative to
7061:relative to
7036:
6582:is a vector
6385:
6094:
6092:
5883:
5628:
5397:
5187:Indeed, the
5116:
4609:
3834:
3411:
3292:
2681:
2677:
1762:identity map
1757:) is linear.
1593:over a ring
1563:
1510:
1204:
987:and scalars
926:
921:
919:
713:
616:
592:
494:through the
361:
352:
346:
240:
236:
202:
196:
138:
134:
130:
126:
122:
112:
97:
88:
69:
16520:Élie Cartan
16468:Spin tensor
16442:Weyl tensor
16400:Mathematics
16364:Multivector
16155:definitions
16053:Engineering
15992:Mathematics
15814:Multivector
15779:Determinant
15736:Dot product
15581:Linear span
15232:Lang, Serge
15002:then so is
14933:is said to
14653:p. 90, § 50
13627:instead of
11974:isomorphism
11964:Isomorphism
11828:epimorphism
11812:Epimorphism
11217:obstruction
11083:, given by
11050:constraints
11031:homogeneous
10390:The number
9465:isomorphism
8594:composition
7706:dimensional
7103:Matrix for
7041:Matrix for
3493:defined on
3303:linear span
2331:A definite
2181:). Indeed,
1889:real matrix
861:Homogeneity
355:, while in
115:mathematics
83:introducing
16595:Categories
16349:Linear map
16217:Operations
15848:Direct sum
15683:Invertible
15586:Linear map
14834:is closed.
14688:Rudin 1991
14463:column of
13962:such that
13758:Rudin 1976
13702:2021-02-17
13672:Rudin 1991
13662:is linear.
13392:Rudin 1976
13382:is linear.
13107:Rudin 1991
13072:null space
12885:convex set
12757:Rudin 1991
12529:Rudin 1976
12357:continuous
12329:Continuity
12059:idempotent
11991:bimorphism
11945:such that
11822:surjective
11793:such that
11677:one-to-one
11143:, 0) <
11021:to solve,
9826:isomorphic
9752:isomorphic
8649:, forms a
8443:projection
7888:reflection
6522:. To get
6165:. Now let
5510:is a real
3709:to all of
2859:, but the
2368:. Indeed,
2335:over some
2100:such that
783:Additivity
714:linear map
603:equivalent
353:linear map
123:linear map
66:references
16488:EM tensor
16324:Dimension
16275:Transpose
15878:Numerical
15641:Transpose
15486:849801114
15403:175294365
15373:840278135
15224:754555941
15132:(1974) .
14879:Λ
14850:Λ
14819:Λ
14790:Λ
14763:∈
14750:for some
14735:≠
14727:Λ
14719:. Assume
14703:Λ
14675:EMS Press
14609:Tu (2011)
14429:…
14300:…
14257:⋮
14252:⋱
14247:⋮
14242:⋮
14217:…
14156:…
14075:rows and
14050:≤
14044:≤
13989:∑
13899:∈
13848:…
13788:…
13583:∈
13370:Λ
13339:Λ
13314:Λ
13294:β
13274:α
13251:∈
13213:Λ
13210:β
13199:Λ
13196:α
13182:β
13171:α
13165:Λ
13138:→
13129:Λ
13082:Λ
13051:Λ
13023:Λ
13014:∈
12980:−
12976:Λ
12939:−
12935:Λ
12899:Λ
12856:Λ
12830:⊂
12804:⊂
12778:→
12769:Λ
12724:∈
12691:λ
12677:λ
12647:∈
12355:, may be
12228:−
12038:nilpotent
11983:bijection
11659:injective
11543:
11534:
11528:−
11513:
11504:
11486:
11416:↦
11350:ℵ
11331:ℵ
11260:↦
11182:↦
11033:equation
11027:solutions
10992:→
10980:
10974:→
10968:→
10962:→
10950:
10944:→
10881:
10832:
10799:→
10627:ν
10598:
10539:
10530:
10498:ρ
10469:
10410:
10401:
10364:
10343:
10334:
10313:
10304:
10287:dimension
10244:
10188:
10153:∈
10114:∈
10086:
10044:∈
10016:
9954:→
9880:×
9842:
9769:×
9729:
9654:
9618:
9582:
9550:
9425:→
9356:
9306:→
9231:
9167:
9022:α
8999:α
8973:α
8933:α
8910:→
8743:pointwise
8726:→
8687:→
8647:morphisms
8618:→
8606:∘
8577:→
8545:→
8445:onto the
8347:θ
8344:
8329:θ
8326:
8320:−
8116:θ
8110:
8104:−
8099:θ
8093:
8083:θ
8077:
8069:θ
8063:
7990:−
7937:−
7863:θ
7860:
7852:θ
7849:
7839:θ
7836:
7830:−
7825:θ
7822:
7754:−
7607:−
7571:−
7300:−
6986:⋯
6906:…
6806:⋮
6795:⋯
6775:⋯
6615:⋮
6447:×
6341:⋯
6191:…
6129:…
6040:⋯
5972:⋯
5901:→
5842:⋯
5754:…
5718:∈
5655:…
5598:→
5521:×
5278:≤
5231:→
5088:−
5061:−
5002:∈
4920:→
4825:
4737:→
4699:−
4696:→
4592:→
4586:
4559:→
4497:…
4448:…
4375:⋯
4310:⋯
4256:→
4250:
4218:→
4152:⋯
4066:⋯
4014:∈
3998:…
3952:…
3890:→
3884:
3852:→
3792:
3746:→
3591:∈
3478:→
3392:⊆
3362:→
3258:λ
3244:λ
2902:→
2660:→
2486:∫
2447:∫
2377:∫
2077:→
2025:∈
1985:∈
1873:×
1814:↦
1778:↦
1736:↦
1696:↦
1682:homothety
1655:↦
1641:→
1522:→
1398:∈
1149:⋯
1081:⋯
1024:∈
1008:…
972:∈
954:…
763:∈
737:∈
697:→
599:morphisms
488:dimension
199:bijection
155:→
16354:Manifold
16339:Geodesic
16097:Notation
15924:Category
15863:Subspace
15858:Quotient
15809:Bivector
15723:Bilinear
15665:Matrices
15540:Glossary
15466:(2013).
15446:Springer
15440:(2011).
15430:24909067
15383:(1996).
15339:21163277
15313:(1991).
15289:(1976).
15259:(1973).
15234:(1987),
15138:Springer
15091:Springer
15085:(2015).
14959:if when
14913:One map
13540:for all
13230:for all
12711:for all
12626:for all
12429:See also
12347:between
12310:, being
12277:′
12265:′
12250:′
12212:′
12179:′
12154:′
12073:, where
11911:implies
11759:implies
11702:dim(ker
11631:and let
11596:→ 0. In
11577:) − dim(
11297:= 0 and
11095:) = (0,
10922:quotient
10778:Cokernel
10772:Cokernel
10211:subspace
9973:and the
9828:to the
9511:forms a
9252:, is an
8651:category
7721:rotation
7714:matrices
7681:′
7640:′
7543:′
7501:′
7460:′
7437:′
7410:′
7355:′
7274:′
7187:′
7160:′
7135:′
5388:Matrices
5213:seminorm
3835:The map
3669:then a (
3580:for all
3379:function
2861:variance
2744:we have
2534:integral
2337:interval
2333:integral
2179:codomain
2094:isometry
2057:, below.
1617:Examples
584:matrices
357:analysis
16451:Physics
16285:Related
16048:Physics
15966:Tensors
15535:Outline
13362:, when
12887:, or a
12365:bounded
12323:tensors
12319:variant
12053:, then
11951:is the
11799:is the
11215:is the
11029:to the
10928:target.
10926:of the
10917:domain,
10814:is the
10565:nullity
9869:of all
9758:of all
9754:to the
9718:, then
8658:inverse
8141:scaling
7704:In two-
6462:matrix
3515:extends
3410:Then a
3301:to the
1891:, then
186:over a
184:modules
143:mapping
141:) is a
79:improve
16379:Vector
16374:Spinor
16359:Matrix
16153:Tensor
15819:Tensor
15631:Kernel
15561:Vector
15556:Scalar
15484:
15474:
15452:
15428:
15418:
15401:
15391:
15371:
15361:
15337:
15327:
15299:
15275:
15246:
15222:
15212:
15193:
15167:
15144:
15118:
15097:
14936:extend
14090:matrix
13497:
13155:linear
12196:hence
12018:, the
11825:or an
10819:kernel
9971:kernel
9931:, and
9515:, the
9321:is an
9256:under
8449:axis:
7966:axis:
7896:axis:
7708:space
6846:where
6798:
6772:
5488:matrix
2175:domain
2092:is an
496:origin
190:; see
68:, but
16299:Basis
15984:Scope
15688:Minor
15673:Block
15611:Basis
14895:of 0.
14553:range
12564:, if
12560:into
12552:into
12515:into
12499:Notes
11847:coker
11715:monic
11706:) = 0
11662:or a
11540:coker
11473:index
11467:Index
11446:with
11290:with
11135:) + (
11111:) = (
10977:coker
10924:space
10829:coker
10648:. If
10209:is a
9979:range
9975:image
9641:units
9513:group
8635:class
7594:, or
5622:(see
5444:basis
3513:that
3377:is a
2926:with
2700:of a
2135:then
1862:is a
1755:field
1675:graph
1537:with
659:field
492:plane
337:is a
15843:Dual
15698:Rank
15482:OCLC
15472:ISBN
15450:ISBN
15426:OCLC
15416:ISBN
15399:OCLC
15389:ISBN
15369:OCLC
15359:ISBN
15335:OCLC
15325:ISBN
15297:ISBN
15273:ISBN
15244:ISBN
15220:OCLC
15210:ISBN
15191:ISBN
15165:ISBN
15142:ISBN
15116:ISBN
15095:ISBN
15022:and
14695:Let
13884:and
13820:and
13410:if:
13286:and
13113:and
12819:and
12544:and
12540:Let
12388:cos(
12377:sin(
12337:and
12098:and
11887:and
11867:epic
11852:= {0
11840:onto
11735:and
11691:= {0
11687:ker
11681:sets
11623:and
11619:Let
11475:as:
11314:for
11017:) =
10909:dual
10668:and
10595:null
10466:rank
10436:rank
10233:and
9403:ring
9283:and
8745:sum
8702:and
8656:The
8560:and
5629:Let
5438:are
5418:and
4822:span
4714:and
4645:and
4583:span
4476:and
4247:span
3916:>
3881:span
3789:span
3325:and
2809:and
2724:and
2696:The
2639:and
2611:and
2536:(or
2177:and
1760:The
1385:and
1359:Let
1274:and
1225:and
637:and
617:Let
540:line
538:, a
289:real
287:are
267:and
188:ring
178:and
121:, a
15355:GTM
14555:of
14525:of
13658:if
13157:if
13074:of
12922:If
12879:If
12527:."
12523:on
12063:If
12043:If
11996:If
11955:on
11927:is
11865:is
11838:is
11803:on
11775:is
11713:is
11675:is
11531:dim
11510:ker
11501:dim
11483:ind
11461:+ 1
11320:sum
11305:+ 1
10947:ker
10913:sub
10619:or
10567:of
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4728:,
4725:0
4722:(
4702:1
4693:)
4690:0
4687:,
4684:1
4681:(
4660:R
4656:=
4653:Y
4631:2
4626:R
4621:=
4618:X
4595:Y
4589:S
4562:Y
4556:S
4553::
4550:f
4530:S
4508:n
4504:s
4500:,
4494:,
4489:1
4485:s
4464:,
4459:n
4455:c
4451:,
4445:,
4440:1
4436:c
4432:,
4429:n
4408:)
4403:n
4399:s
4395:(
4391:f
4386:n
4382:c
4378:+
4372:+
4368:)
4363:1
4359:s
4355:(
4351:f
4346:1
4342:c
4338:=
4334:)
4328:n
4324:s
4318:n
4314:c
4307:+
4302:1
4298:s
4292:1
4288:c
4283:(
4279:F
4259:Y
4253:S
4244::
4241:F
4221:Y
4215:S
4212::
4209:f
4189:.
4185:)
4180:n
4176:s
4172:(
4168:f
4163:n
4159:c
4155:+
4149:+
4145:)
4140:1
4136:s
4132:(
4128:f
4123:1
4119:c
4115:=
4112:0
4092:,
4087:n
4083:s
4077:n
4073:c
4069:+
4063:+
4058:1
4054:s
4048:1
4044:c
4040:=
4037:0
4017:S
4009:n
4005:s
4001:,
3995:,
3990:1
3986:s
3963:n
3959:c
3955:,
3949:,
3944:1
3940:c
3919:0
3913:n
3893:Y
3887:S
3878::
3875:F
3855:Y
3849:S
3846::
3843:f
3821:.
3818:X
3798:,
3795:S
3769:f
3749:Y
3743:S
3740::
3737:f
3717:X
3697:f
3677:Y
3657:X
3637:S
3617:.
3614:f
3594:S
3588:s
3568:)
3565:s
3562:(
3559:f
3556:=
3553:)
3550:s
3547:(
3544:F
3524:f
3501:X
3481:Y
3475:X
3472::
3469:F
3448:,
3445:X
3425:f
3398:.
3395:X
3389:S
3365:Y
3359:S
3356::
3353:f
3333:Y
3313:X
3272:)
3268:a
3264:(
3261:f
3255:=
3252:)
3248:a
3241:(
3238:f
3218:)
3215:y
3212:,
3209:x
3206:2
3203:(
3200:=
3197:)
3194:y
3191:,
3188:x
3185:(
3182:f
3151:)
3147:b
3143:(
3140:f
3137:+
3134:)
3130:a
3126:(
3123:f
3120:=
3117:)
3113:b
3109:+
3105:a
3101:(
3098:f
3078:)
3075:y
3072:,
3069:x
3066:2
3063:(
3060:=
3057:)
3054:y
3051:,
3048:x
3045:(
3042:f
3022:.
3010:2
2990:x
2970:)
2967:y
2964:,
2961:x
2958:2
2955:(
2952:=
2949:)
2946:y
2943:,
2940:x
2937:(
2934:f
2912:2
2907:R
2897:2
2892:R
2887::
2884:f
2847:]
2844:X
2841:[
2838:E
2835:a
2832:=
2829:]
2826:X
2823:a
2820:[
2817:E
2797:]
2794:Y
2791:[
2788:E
2785:+
2782:]
2779:X
2776:[
2773:E
2770:=
2767:]
2764:Y
2761:+
2758:X
2755:[
2752:E
2732:Y
2712:X
2693:.
2682:m
2678:n
2663:W
2657:V
2654::
2651:f
2641:n
2637:m
2633:F
2619:W
2599:V
2571:R
2549:R
2518:.
2515:x
2512:d
2509:)
2506:x
2503:(
2500:g
2495:v
2490:u
2482:b
2479:+
2476:x
2473:d
2470:)
2467:x
2464:(
2461:f
2456:v
2451:u
2443:a
2440:=
2437:x
2434:d
2430:)
2426:)
2423:x
2420:(
2417:g
2414:b
2411:+
2408:)
2405:x
2402:(
2399:f
2396:a
2392:(
2386:v
2381:u
2355:R
2344:I
2340:I
2317:.
2311:x
2308:d
2303:)
2300:x
2297:(
2294:g
2291:d
2285:b
2282:+
2276:x
2273:d
2268:)
2265:x
2262:(
2259:f
2256:d
2250:a
2247:=
2243:)
2239:)
2236:x
2233:(
2230:g
2227:b
2224:+
2221:)
2218:x
2215:(
2212:f
2209:a
2205:(
2198:x
2195:d
2191:d
2143:f
2123:0
2120:=
2117:)
2114:0
2111:(
2108:f
2080:W
2074:V
2071::
2068:f
2035:m
2030:R
2021:x
2017:A
1995:n
1990:R
1981:x
1955:m
1950:R
1926:n
1921:R
1899:A
1876:n
1870:m
1850:A
1823:1
1820:+
1817:x
1811:x
1786:2
1782:x
1775:x
1740:0
1732:x
1718:c
1703:v
1699:c
1692:v
1661:x
1658:c
1652:x
1649::
1645:R
1637:R
1633::
1630:f
1601:R
1581:M
1576:R
1545:K
1525:K
1519:V
1497:.
1492:W
1487:0
1482:=
1479:)
1475:v
1471:(
1468:f
1465:0
1462:=
1459:)
1455:v
1451:0
1448:(
1445:f
1442:=
1439:)
1434:V
1429:0
1424:(
1421:f
1401:V
1394:v
1373:0
1370:=
1367:c
1347:.
1342:W
1337:0
1332:=
1329:)
1324:V
1319:0
1314:(
1311:f
1289:W
1284:0
1260:V
1255:0
1233:W
1213:V
1186:.
1183:)
1178:n
1173:u
1168:(
1165:f
1160:n
1156:c
1152:+
1146:+
1143:)
1138:1
1133:u
1128:(
1125:f
1120:1
1116:c
1112:=
1109:)
1104:n
1099:u
1092:n
1088:c
1084:+
1078:+
1073:1
1068:u
1061:1
1057:c
1053:(
1050:f
1030:,
1027:K
1019:n
1015:c
1011:,
1005:,
1000:1
996:c
975:V
967:n
962:u
957:,
951:,
946:1
941:u
905:)
901:u
897:(
894:f
891:c
888:=
885:)
881:u
877:c
874:(
871:f
846:)
842:v
838:(
835:f
832:+
829:)
825:u
821:(
818:f
815:=
812:)
808:v
804:+
800:u
796:(
793:f
766:K
760:c
740:V
733:v
729:,
725:u
700:W
694:V
691::
688:f
668:K
645:W
625:V
570:W
550:W
526:W
506:V
474:W
454:V
430:W
410:V
390:W
370:V
325:V
305:W
302:=
299:V
275:W
255:V
223:W
220:=
217:V
158:W
152:V
104:)
98:(
93:)
89:(
75:.
48:.
41:.
34:.
20:)
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