2694:
2163:
1867:
2689:{\displaystyle \left\{\left\in K^{n}:{\begin{alignedat}{7}x_{1}&&\;=\;&&a_{11}t_{1}&&\;+\;&&a_{12}t_{2}&&\;+\cdots +\;&&a_{1m}t_{m}&\\x_{2}&&\;=\;&&a_{21}t_{1}&&\;+\;&&a_{22}t_{2}&&\;+\cdots +\;&&a_{2m}t_{m}&\\&&\vdots \;\;&&&&&&&&&&&\\x_{n}&&\;=\;&&a_{n1}t_{1}&&\;+\;&&a_{n2}t_{2}&&\;+\cdots +\;&&a_{nm}t_{m}&\\\end{alignedat}}{\text{ for some }}t_{1},\ldots ,t_{m}\in K\right\}.}
1412:
8448:
40:
35:
29:
24:
1862:{\displaystyle \left\{\left\in K^{n}:{\begin{alignedat}{6}a_{11}x_{1}&&\;+\;&&a_{12}x_{2}&&\;+\cdots +\;&&a_{1n}x_{n}&&\;=0&\\a_{21}x_{1}&&\;+\;&&a_{22}x_{2}&&\;+\cdots +\;&&a_{2n}x_{n}&&\;=0&\\&&&&&&&&&&\vdots \quad &\\a_{m1}x_{1}&&\;+\;&&a_{m2}x_{2}&&\;+\cdots +\;&&a_{mn}x_{n}&&\;=0&\end{alignedat}}\right\}.}
4500:
590:
8712:
3822:
7173:
6971:
3335:
3044:
7379:
6149:
This produces a basis for the column space that is a subset of the original column vectors. It works because the columns with pivots are a basis for the column space of the echelon form, and row reduction does not change the linear dependence relationships between the columns.
1373:
1229:
4040:
7168:{\displaystyle {\begin{alignedat}{1}\mathbf {c} _{3}&=-3\mathbf {c} _{1}+5\mathbf {c} _{2}\\\mathbf {c} _{5}&=2\mathbf {c} _{1}-\mathbf {c} _{2}+7\mathbf {c} _{4}\\\mathbf {c} _{6}&=4\mathbf {c} _{2}-9\mathbf {c} _{4}\end{alignedat}}}
4853:
2857:
3172:
3767:
2128:
2894:
4418:
2256:
5939:
into reduced row echelon form, then the resulting basis for the row space is uniquely determined. This provides an algorithm for checking whether two row spaces are equal and, by extension, whether two subspaces of
4965:
4243:
3157:
3617:
7194:
4969:
Here, the minimum only occurs if one subspace is contained in the other, while the maximum is the most general case. The dimension of the intersection and the sum are related by the following equation:
1970:
1277: = 0 of codimension 1. Subspaces of codimension 1 specified by two linear functionals are equal, if and only if one functional can be obtained from another with scalar multiplication (in the
5069:
6739:
1505:
61:(0, 0), marked with green circles, belongs to any of six 1-subspaces, while each of 24 remaining points belongs to exactly one; a property which holds for 1-subspaces over any field and in all
3917:
1287:
1143:
5532:
4139:. The number of elements in a basis is always equal to the geometric dimension of the subspace. Any spanning set for a subspace can be changed into a basis by removing redundant vectors (see
3472:
5212:
4775:
2043:
5829:
5694:
5577:
3629:-plane can be reached from the origin by first moving some distance in the direction of (1, 0, 0) and then moving some distance in the direction of (0, 0, 1).
8001:
6660:
be the columns of the reduced row echelon form. For each column without a pivot, write an equation expressing the column as a linear combination of the columns with pivots.
5648:
5457:
5370:
2717:
5136:
5107:
3330:{\displaystyle {\text{Span}}\{\mathbf {v} _{1},\ldots ,\mathbf {v} _{k}\}=\left\{t_{1}\mathbf {v} _{1}+\cdots +t_{k}\mathbf {v} _{k}:t_{1},\ldots ,t_{k}\in K\right\}.}
1028:
under sums and under scalar multiples. Equivalently, subspaces can be characterized by the property of being closed under linear combinations. That is, a nonempty set
6523:
6938:
5741:
6497:
1137:
to all possible scalar values. 1-subspaces specified by two vectors are equal if and only if one vector can be obtained from another with scalar multiplication:
5784:
5714:
5621:
5597:
5430:
5410:
5390:
5343:
5319:
5299:
5275:
3650:
4860:
3049:
The expression on the right is called a linear combination of the vectors (2, 5, −1) and (3, −4, 2). These two vectors are said to
3039:{\displaystyle {\begin{bmatrix}x\\y\\z\end{bmatrix}}\;=\;t_{1}\!{\begin{bmatrix}2\\5\\-1\end{bmatrix}}+t_{2}\!{\begin{bmatrix}3\\-4\\2\end{bmatrix}}.}
2058:
1894:
4262:
5109:. An equivalent restatement is that a direct sum is a subspace sum under the condition that every subspace contributes to the span of the sum.
4973:
4163:
4253:. In particular, every vector that satisfies the above equations can be written uniquely as a linear combination of the two basis vectors:
3089:
7374:{\displaystyle {\begin{alignedat}{1}x_{3}&=-3x_{1}+5x_{2}\\x_{5}&=2x_{1}-x_{2}+7x_{4}\\x_{6}&=4x_{2}-9x_{4}.\end{alignedat}}}
3497:
8261:
5143:
8306:
8639:
8697:
4035:{\displaystyle t_{1}\mathbf {v} _{1}+\cdots +t_{k}\mathbf {v} _{k}\;\neq \;u_{1}\mathbf {v} _{1}+\cdots +u_{k}\mathbf {v} _{k}}
1395:
1087:
903:
8111:
8101:
8082:
8034:
7986:
7912:
5866:
Row reduction does not change the span of the row vectors, i.e. the reduced matrix has the same row space as the original.
4857:
For example, the sum of two lines is the plane that contains them both. The dimension of the sum satisfies the inequality
3695:
1368:{\displaystyle \exists c\in K:\mathbf {F} '=c\mathbf {F} {\text{ (or }}\mathbf {F} ={\frac {1}{c}}\mathbf {F} '{\text{)}}}
1224:{\displaystyle \exists c\in K:\mathbf {v} '=c\mathbf {v} {\text{ (or }}\mathbf {v} ={\frac {1}{c}}\mathbf {v} '{\text{)}}}
1048:. The equivalent definition states that it is also equivalent to consider linear combinations of two elements at a time.
8009:
3644:
A system of linear parametric equations in a finite-dimensional space can also be written as a single matrix equation:
7691:
Basic facts about
Hilbert Space — class notes from Colorado State University on Partial Differential Equations (M645).
5462:
3404:
8156:
8064:
7964:
7939:
4434:
passing through the points (0, 0, 0, 0), (2, 1, 0, 0), and (0, 0, 5, 1).
3795:
The row space of a matrix is the subspace spanned by its row vectors. The row space is interesting because it is the
2006:
In a finite-dimensional space, a homogeneous system of linear equations can be written as a single matrix equation:
8687:
7956:
7904:
8649:
8585:
7416:
5138:
is the same as the sum of subspaces, but may be shortened because the dimension of the trivial subspace is zero.
2012:
6439:
and the remaining free variables are zero. The resulting collection of vectors is a basis for the null space of
5789:
8026:
7456:. In the case of vector spaces over the reals, linear subspaces, flats, and affine subspaces are also called
7411:
5653:
5537:
234:
As a corollary, all vector spaces are equipped with at least two (possibly different) linear subspaces: the
8746:
8427:
8299:
3861:
spanned by the three vectors (1, 0, 0), (0, 0, 1), and (2, 0, 3) is just the
8532:
8382:
8437:
8331:
7199:
6976:
4848:{\displaystyle U+W=\left\{\mathbf {u} +\mathbf {w} \colon \mathbf {u} \in U,\mathbf {w} \in W\right\}.}
8677:
8326:
8206:
5929:
4524:
8741:
8669:
8552:
7931:
7478:
5856:
5848:
5831:. As a result, this operation does not turn the lattice of subspaces into a Boolean algebra (nor a
3816:
2852:{\displaystyle x=2t_{1}+3t_{2},\;\;\;\;y=5t_{1}-4t_{2},\;\;\;\;{\text{and}}\;\;\;\;z=-t_{1}+2t_{2}}
1052:
62:
3484:-plane is spanned by the vectors (1, 0, 0) and (0, 0, 1). Every vector in the
2888:
In linear algebra, the system of parametric equations can be written as a single vector equation:
992:
Again, we know from calculus that the product of a continuous function and a number is continuous.
8736:
8715:
8644:
8422:
8292:
8265:
8237:
5759:
1270:
1006:
5626:
5435:
5348:
8479:
8412:
8402:
5755:
5115:
5086:
3812:
1239:
1114:. Geometrically (especially over the field of real numbers and its subfields), a subspace is a
939:
107:
8494:
8489:
8484:
8417:
8362:
6502:
5322:
5079:
4249:
Then the vectors (2, 1, 0, 0) and (0, 0, 5, 1) are a basis for
3796:
3639:
1254:
1126:
1025:
268:
6345:
If the final column of the reduced row echelon form contains a pivot, then the input vector
8504:
8469:
8456:
8347:
8132:
8126:
8056:
7482:
6729:
6526:
5726:
2154:
1111:
1091:
58:
6306:
Express the final column of the reduced echelon form as a linear combination of the first
8:
8682:
8562:
8537:
8387:
8198:
7926:
A First Course In Linear
Algebra: with Optional Introduction to Groups, Rings, and Fields
6476:
5278:
3808:
3777:
3762:{\displaystyle \mathbf {x} =A\mathbf {t} \;\;\;\;{\text{where}}\;\;\;\;A=\left{\text{.}}}
2052:
of the matrix. For example, the subspace described above is the null space of the matrix
1379:
1013:
955:
123:
8392:
8018:
7924:
7406:
6450:
5769:
5699:
5606:
5582:
5415:
5395:
5375:
5328:
5304:
5284:
5260:
4449:
1258:
1067:
8117:
5077:
when the only intersection between any pair of subspaces is the trivial subspace. The
3865:-plane, with each point on the plane described by infinitely many different values of
1024:
From the definition of vector spaces, it follows that subspaces are nonempty, and are
8590:
8547:
8474:
8367:
8172:
8152:
8136:
8107:
8078:
8060:
8049:
8030:
8005:
7982:
7960:
7935:
7908:
7426:
6419:
are free. Write equations for the dependent variables in terms of the free variables.
5234:
4737:
1382:. The following two subsections will present this latter description in details, and
1262:
1130:
1075:
235:
6136:. The corresponding columns of the original matrix are a basis for the column space.
8595:
8499:
8352:
6270:
6133:
5852:
5242:
1403:
1972:
is a one-dimensional subspace. More generally, that is to say that given a set of
72:(i.e. a 5 × 5 square) is pictured four times for a better visualization
8447:
8407:
8397:
7453:
7449:
7421:
7401:
6143:
5832:
5230:
1888:
1115:
548:
8659:
8580:
8315:
8257:
8229:
8044:
4507:, the intersection of two distinct two-dimensional subspaces is one-dimensional
1033:
83:
8233:
8176:
5249:, the greatest element, is an identity element of the intersection operation.
2123:{\displaystyle A={\begin{bmatrix}1&3&2\\2&-4&5\end{bmatrix}}.}
1001:
Keep the same field and vector space as before, but now consider the set Diff(
8730:
8692:
8615:
8575:
8542:
8522:
7896:
5844:
5751:
5238:
4453:
5762:, for example, orthogonal complements exist. However, these spaces may have
1090:, the subset of Euclidean space described by a system of homogeneous linear
8625:
8514:
8464:
8357:
7974:
5744:
3857:. However, there are exceptions to this rule. For example, the subspace of
2871:
1103:
158:
95:
48:
39:
34:
28:
23:
5869:
Row reduction does not affect the linear dependence of the column vectors.
5766:
that are orthogonal to themselves, and consequently there exist subspaces
4413:{\displaystyle (2t_{1},t_{1},5t_{2},t_{2})=t_{1}(2,1,0,0)+t_{2}(0,0,5,1).}
102:
of some larger vector space. A linear subspace is usually simply called a
8605:
8570:
8527:
8372:
7948:
6702:
matrix corresponding to this system is the desired matrix with nullspace
5763:
3772:
In this case, the subspace consists of all possible values of the vector
2883:
1235:
1095:
1071:
1009:. The same sort of argument as before shows that this is a subspace too.
989:
We know from calculus that the sum of continuous functions is continuous.
276:
239:
79:
5650:. Applying orthogonal complements twice returns the original subspace:
4499:
8634:
8377:
5600:
4459:
A subspace cannot lie in any subspace of lesser dimension. If dim
2049:
2001:
1278:
1099:
1063:
1037:
589:
47:
One-dimensional subspaces in the two-dimensional vector space over the
6456:
5914:
The nonzero rows of the echelon form are a basis for the row space of
4960:{\displaystyle \max(\dim U,\dim W)\leq \dim(U+W)\leq \dim(U)+\dim(W).}
8432:
8181:
6396:
Using the reduced row echelon form, determine which of the variables
5925:
4238:{\displaystyle x_{1}=2x_{2}\;\;\;\;{\text{and}}\;\;\;\;x_{3}=5x_{4}.}
1107:
166:
3821:
3776:. In linear algebra, this subspace is known as the column space (or
3625:
Geometrically, this corresponds to the fact that every point on the
3152:{\displaystyle t_{1}\mathbf {v} _{1}+\cdots +t_{k}\mathbf {v} _{k}.}
1070:
subspace is always closed. The same is true for subspaces of finite
8600:
7461:
5747:) orthocomplemented lattice (although not a distributive lattice).
5720:
3612:{\displaystyle (t_{1},0,t_{2})=t_{1}(1,0,0)+t_{2}(0,0,1){\text{.}}}
1981:
8284:
8277:
8249:
7486:
8140:
8610:
242:
alone and the entire vector space itself. These are called the
99:
106:
when the context serves to distinguish it from other types of
1074:(i.e., subspaces determined by a finite number of continuous
3364:. Geometrically, the span is the flat through the origin in
7391:
are a basis for the null space of the corresponding matrix.
5863:
The reduced matrix has the same null space as the original.
3488:-plane can be written as a linear combination of these two:
1965:{\displaystyle x+3y+2z=0\quad {\text{and}}\quad 2x-4y+5z=0}
1386:
four subsections further describe the idea of linear span.
6053:
If the echelon form has a row of zeroes, then the vectors
3162:
The set of all possible linear combinations is called the
8106:, Society for Industrial and Applied Mathematics (SIAM),
6339:
entries in the final column of the reduced echelon form.)
5859:. Row reduction has the following important properties:
5064:{\displaystyle \dim(U+W)=\dim(U)+\dim(W)-\dim(U\cap W).}
4438:
1976:
independent functions, the dimension of the subspace in
1086:
Descriptions of subspaces include the solution set to a
6310:
columns. The coefficients used are the desired numbers
4135:
is a set of linearly independent vectors whose span is
2137:
can be described as the null space of some matrix (see
4430:
is two-dimensional. Geometrically, it is the plane in
3002:
2952:
2903:
2180:
2073:
2048:
The set of solutions to this equation is known as the
1429:
906:
will yield a subspace. (The equation in example I was
7498:
This definition is often stated differently: vectors
7489:, but some integers may equal to zero in some fields.
7197:
6974:
6732:
6505:
6479:
5792:
5772:
5729:
5702:
5656:
5629:
5609:
5585:
5540:
5465:
5438:
5418:
5398:
5378:
5351:
5331:
5307:
5287:
5263:
5146:
5118:
5089:
4976:
4863:
4778:
4265:
4166:
3920:
3802:
3653:
3500:
3407:
3175:
3092:
2897:
2720:
2166:
2061:
2015:
1897:
1415:
1290:
1146:
8017:
7793:
1234:
This idea is generalized for higher dimensions with
898:
In general, any subset of the real coordinate space
6457:
Basis for the sum and intersection of two subspaces
5843:Most algorithms for dealing with subspaces involve
5754:, some but not all of these results still hold. In
5534:. Moreover, no vector is orthogonal to itself, so
1261:(usually implemented as linear equations). One non-
910: = 0, and the equation in example II was
8048:
7923:
7922:Beauregard, Raymond A.; Fraleigh, John B. (1973),
7921:
7616:
7481:that the given integer matrix has the appropriate
7373:
7167:
6932:
6517:
6491:
5823:
5778:
5735:
5708:
5688:
5642:
5615:
5591:
5571:
5526:
5451:
5424:
5404:
5384:
5364:
5337:
5313:
5293:
5269:
5206:
5130:
5101:
5063:
4959:
4847:
4412:
4237:
4124:for a vector in the span are uniquely determined.
4034:
3761:
3611:
3466:
3329:
3151:
3038:
2851:
2688:
2122:
2037:
1964:
1861:
1367:
1223:
6132:Determine which columns of the echelon form have
5245:of the sum operation, and the identical subspace
3833:are a basis for this two-dimensional subspace of
2996:
2946:
2233:
2232:
2178:
2177:
1482:
1481:
1427:
1426:
1378:It is generalized for higher codimensions with a
8728:
7889:Elementary Linear Algebra (Applications Version)
5527:{\displaystyle \dim(N)+\dim(N^{\perp })=\dim(V)}
5083:is the sum of independent subspaces, written as
4864:
4456:on the set of all subspaces (of any dimension).
3467:{\displaystyle x=t_{1},\;\;\;y=0,\;\;\;z=t_{2}.}
1012:Examples that extend these themes are common in
5207:{\displaystyle \dim(U\oplus W)=\dim(U)+\dim(W)}
6429:, choose a vector in the null space for which
5743:), makes the lattice of subspaces a (possibly
2144:
1389:
8300:
3633:
3360:components, then their span is a subspace of
1125:A natural description of a 1-subspace is the
5818:
5812:
5566:
5560:
3368:-dimensional space determined by the points
3217:
3181:
2153:described by a system of homogeneous linear
6532:
6153:
6093:
2038:{\displaystyle A\mathbf {x} =\mathbf {0} .}
8307:
8293:
8103:Matrix Analysis and Applied Linear Algebra
6916:
6915:
6914:
6913:
6912:
6905:
6904:
6903:
6902:
6901:
6894:
6893:
6892:
6891:
6890:
6883:
6882:
6881:
6880:
6879:
6872:
6871:
6870:
6869:
6868:
5824:{\displaystyle N\cap N^{\perp }\neq \{0\}}
4205:
4204:
4203:
4202:
4196:
4195:
4194:
4193:
3978:
3974:
3719:
3718:
3682:
3681:
3680:
3679:
3673:
3672:
3671:
3670:
3444:
3443:
3442:
3429:
3428:
3427:
2935:
2931:
2813:
2812:
2811:
2810:
2804:
2803:
2802:
2801:
2762:
2761:
2760:
2759:
2606:
2596:
2566:
2562:
2532:
2528:
2499:
2498:
2461:
2451:
2424:
2420:
2393:
2389:
2344:
2334:
2307:
2303:
2276:
2272:
1839:
1809:
1799:
1769:
1765:
1708:
1678:
1668:
1641:
1637:
1602:
1572:
1562:
1535:
1531:
1019:
809:, that is, a point in the plane such that
8196:
8171:
7688:
7676:
6924:
6671:linear equations involving the variables
6356:
5252:
3748:
1995:
649:, that is, points in the plane such that
8043:
8023:A (Terse) Introduction to Linear Algebra
7995:
7973:
7769:
7712:
7664:
7640:
7628:
6663:This results in a homogeneous system of
5873:
5229:make the set of all subspaces a bounded
4498:
3820:
588:
7981:(4th ed.). Orthogonal Publishing.
6335:. (These should be precisely the first
6142:See the article on column space for an
5689:{\displaystyle (N^{\perp })^{\perp }=N}
5459:satisfy the complementary relationship
5216:
1122:-space that passes through the origin.
8729:
8698:Comparison of linear algebra libraries
8256:
8228:
8124:
7947:
7817:
7805:
7757:
7736:
7724:
7700:
7652:
7592:
7467:
6077:are linearly dependent, and therefore
5947:
5572:{\displaystyle N\cap N^{\perp }=\{0\}}
5412:is a subspace, then the dimensions of
3399:can be parameterized by the equations
1396:homogeneous system of linear equations
1088:homogeneous system of linear equations
904:homogeneous system of linear equations
8288:
8149:Linear Algebra: A Modern Introduction
8146:
8099:
8095:(7th ed.), Pearson Prentice Hall
7895:
7886:
7865:
7853:
7841:
7829:
7781:
7604:
7573:. The two definitions are equivalent.
6628:Use elementary row operations to put
6449:See the article on null space for an
6389:Use elementary row operations to put
6299:Use elementary row operations to put
6125:Use elementary row operations to put
6046:Use elementary row operations to put
5907:Use elementary row operations to put
5851:to a matrix, until it reaches either
4439:Operations and relations on subspaces
2699:For example, the set of all vectors (
8100:Meyer, Carl D. (February 15, 2001),
8090:
7460:for emphasizing that there are also
2874:(such as real or rational numbers).
1871:For example, the set of all vectors
1383:
1098:of a collection of vectors, and the
543:again, but now let the vector space
8262:"The big picture of linear algebra"
8075:Linear Algebra and Its Applications
8072:
7891:(9th ed.), Wiley International
7448:is sometimes used for referring to
7181:It follows that the row vectors of
6716:If the reduced row echelon form of
4105:are linearly independent, then the
13:
8314:
7794:Katznelson & Katznelson (2008)
7742:
7387:In particular, the row vectors of
5847:. This is the process of applying
5730:
3803:Independence, basis, and dimension
3784:. It is precisely the subspace of
2877:
1291:
1147:
14:
8758:
8222:
8199:"Basic facts about Hilbert Space"
8073:Lay, David C. (August 22, 2005),
8021:; Katznelson, Yonatan R. (2008).
5222:
3788:spanned by the column vectors of
2862:is a two-dimensional subspace of
2711:) parameterized by the equations
8711:
8710:
8688:Basic Linear Algebra Subprograms
8446:
8234:"The four fundamental subspaces"
8128:Linear Algebra and Matrix Theory
8093:Linear Algebra With Applications
8077:(3rd ed.), Addison Wesley,
8051:Advanced Engineering Mathematics
7953:Finite-Dimensional Vector Spaces
7151:
7133:
7111:
7095:
7077:
7062:
7040:
7024:
7006:
6981:
6119:A basis for the column space of
4827:
4813:
4805:
4797:
4022:
3991:
3964:
3933:
3666:
3655:
3271:
3240:
3207:
3186:
3136:
3105:
2028:
2020:
1352:
1333:
1323:
1308:
1208:
1189:
1179:
1164:
316:, then they can be expressed as
287:whose last component is 0. Then
283:to be the set of all vectors in
38:
33:
27:
22:
8586:Seven-dimensional cross product
8268:from the original on 2021-12-11
8240:from the original on 2021-12-11
7859:
7847:
7835:
7823:
7811:
7799:
7787:
7775:
7763:
7751:
7748:Vector space related operators.
7730:
7718:
7706:
7694:
7682:
7617:Beauregard & Fraleigh (1973
7492:
7417:Quotient space (linear algebra)
4494:
4140:
3799:of the null space (see below).
2138:
1931:
1925:
1733:
1402:variables is a subspace in the
1081:
930:, but now let the vector space
16:In mathematics, vector subspace
8197:DuChateau, Paul (5 Sep 2002).
7715:p. 100, ch. 2, Definition 2.13
7670:
7667:p. 100, ch. 2, Definition 2.13
7658:
7646:
7634:
7622:
7610:
7598:
7586:
7438:
6632:into reduced row echelon form.
6383:A basis for the null space of
6303:into reduced row echelon form.
5719:This operation, understood as
5671:
5657:
5521:
5515:
5503:
5490:
5478:
5472:
5201:
5195:
5183:
5177:
5165:
5153:
5112:The dimension of a direct sum
5055:
5043:
5031:
5025:
5013:
5007:
4995:
4983:
4951:
4945:
4933:
4927:
4915:
4903:
4891:
4867:
4404:
4380:
4364:
4340:
4324:
4266:
3601:
3583:
3567:
3549:
3533:
3501:
1988:, the composite matrix of the
954:) be the subset consisting of
921:
1:
8151:(2nd ed.), Brooks/Cole,
8027:American Mathematical Society
7412:Multilinear subspace learning
6292:, with the last column being
5935:If we instead put the matrix
5900:A basis for the row space of
5838:
1980:will be the dimension of the
1257:description is provided with
1246:-spaces specified by sets of
996:
534:
113:
8428:Eigenvalues and eigenvectors
8002:Blaisdell Publishing Company
7579:
7514:are linearly independent if
6525:can be calculated using the
6393:in reduced row echelon form.
6015: + 1) ×
4716:belongs to both sets. Thus,
4546: is an element of both
4452:binary relation specifies a
4443:
1036:every linear combination of
254:
7:
7880:
7395:
6589:matrix whose null space is
6023:whose rows are the vectors
2145:Linear parametric equations
1891:) satisfying the equations
1390:Systems of linear equations
1250:vectors are not so simple.
975:We know from calculus that
926:Again take the field to be
249:
149:if under the operations of
82:, and more specifically in
10:
8763:
8131:(2nd ed.), New York:
8055:(3rd ed.), New York:
7875:
7485:in it. All fields include
5643:{\displaystyle N^{\perp }}
5452:{\displaystyle N^{\perp }}
5392:is finite-dimensional and
5372:, is again a subspace. If
5365:{\displaystyle N^{\perp }}
3849:parameters (or spanned by
3841:In general, a subspace of
3806:
3637:
3634:Column space and row space
3083:is any vector of the form
2881:
1999:
1062:need not be topologically
8706:
8668:
8624:
8561:
8513:
8455:
8444:
8340:
8322:
8207:Colorado State University
7901:Linear Algebra Done Right
7477:can be any field of such
5849:elementary row operations
5131:{\displaystyle U\oplus W}
5102:{\displaystyle U\oplus W}
4450:set-theoretical inclusion
4158:defined by the equations
558:to be the set of points (
122:is a vector space over a
8173:Weisstein, Eric Wolfgang
8125:Nering, Evar D. (1970),
8091:Leon, Steven J. (2006),
7996:Herstein, I. N. (1964),
7932:Houghton Mifflin Company
7432:
6946:then the column vectors
6533:Equations for a subspace
6154:Coordinates for a vector
6094:Basis for a column space
5857:reduced row echelon form
5760:symplectic vector spaces
5226:
4748:itself are subspaces of
4712:are vector spaces, then
4554:} is also a subspace of
3817:Dimension (vector space)
3053:the resulting subspace.
1394:The solution set to any
1053:topological vector space
1007:differentiable functions
173:is a linear subspace of
6518:{\displaystyle U\cap W}
6422:For each free variable
6030:, ... ,
5756:pseudo-Euclidean spaces
4732:For every vector space
4467:, a finite number, and
4080:, ... ,
4059:, ... ,
3898:, ... ,
3853:vectors) has dimension
3375:, ... ,
3347:, ... ,
3074:, ... ,
1020:Properties of subspaces
8413:Row and column vectors
8165:
7887:Anton, Howard (2005),
7375:
7169:
6934:
6519:
6493:
6357:Basis for a null space
6129:into row echelon form.
6050:into row echelon form.
5911:into row echelon form.
5825:
5780:
5737:
5710:
5690:
5644:
5617:
5593:
5573:
5528:
5453:
5426:
5406:
5386:
5366:
5339:
5315:
5295:
5271:
5253:Orthogonal complements
5208:
5132:
5103:
5073:A set of subspaces is
5065:
4961:
4849:
4755:
4508:
4414:
4239:
4036:
3838:
3813:Basis (linear algebra)
3763:
3613:
3468:
3331:
3153:
3040:
2853:
2690:
2124:
2039:
1996:Null space of a matrix
1966:
1863:
1369:
1225:
594:
593:Example II Illustrated
8418:Row and column spaces
8363:Scalar multiplication
8147:Poole, David (2006),
7376:
7185:satisfy the equations
7170:
6962:satisfy the equations
6935:
6933:{\displaystyle \left}
6520:
6499:and the intersection
6494:
6473:, a basis of the sum
5874:Basis for a row space
5826:
5781:
5750:In spaces with other
5738:
5736:{\displaystyle \neg }
5711:
5691:
5645:
5618:
5594:
5574:
5529:
5454:
5427:
5407:
5387:
5367:
5340:
5323:orthogonal complement
5316:
5296:
5272:
5209:
5133:
5104:
5066:
4962:
4850:
4768:are subspaces, their
4502:
4415:
4240:
4037:
3824:
3797:orthogonal complement
3764:
3640:Row and column spaces
3614:
3469:
3332:
3154:
3041:
2854:
2691:
2125:
2040:
1967:
1864:
1370:
1226:
1127:scalar multiplication
902:that is defined by a
592:
269:real coordinate space
246:of the vector space.
8553:Gram–Schmidt process
8505:Gaussian elimination
7949:Halmos, Paul Richard
7772:p. 148, ch. 2, §4.10
7195:
6972:
6730:
6527:Zassenhaus algorithm
6503:
6477:
6461:Given two subspaces
5790:
5770:
5727:
5700:
5654:
5627:
5607:
5583:
5538:
5463:
5436:
5416:
5396:
5376:
5349:
5329:
5305:
5285:
5261:
5217:Lattice of subspaces
5144:
5116:
5087:
4974:
4861:
4776:
4619:is a subspace, then
4603:is a subspace, then
4263:
4164:
3918:
3909:linearly independent
3891:In general, vectors
3651:
3498:
3405:
3173:
3090:
2895:
2718:
2640: for some
2164:
2155:parametric equations
2059:
2013:
1895:
1413:
1288:
1144:
1092:parametric equations
956:continuous functions
259:In the vector space
8747:Functional analysis
8683:Numerical stability
8563:Multilinear algebra
8538:Inner product space
8388:Linear independence
8019:Katznelson, Yitzhak
6492:{\displaystyle U+W}
6001:Determines whether
5948:Subspace membership
5924:See the article on
5696:for every subspace
5279:inner product space
4615:. Similarly, since
4154:be the subspace of
3809:Linear independence
3480:As a subspace, the
1380:system of equations
1238:, but criteria for
1014:functional analysis
962:) is a subspace of
805:) be an element of
8393:Linear combination
7897:Axler, Sheldon Jay
7796:pp. 10-11, § 1.2.5
7571:) ≠ (0, 0, ..., 0)
7407:Invariant subspace
7371:
7369:
7165:
7163:
6930:
6922:
6515:
6489:
6276:whose columns are
5821:
5776:
5733:
5706:
5686:
5640:
5613:
5589:
5569:
5524:
5449:
5422:
5402:
5382:
5362:
5335:
5311:
5291:
5267:
5204:
5128:
5099:
5061:
4957:
4845:
4666:be a scalar. Then
4519:of a vector space
4509:
4410:
4235:
4032:
3839:
3759:
3746:
3609:
3464:
3327:
3149:
3058:linear combination
3036:
3027:
2977:
2925:
2849:
2686:
2636:
2230:
2133:Every subspace of
2120:
2111:
2035:
1962:
1859:
1849:
1479:
1365:
1265:linear functional
1259:linear functionals
1221:
1076:linear functionals
1068:finite-dimensional
595:
238:consisting of the
211:, it follows that
165:. Equivalently, a
8724:
8723:
8591:Geometric algebra
8548:Kronecker product
8383:Linear projection
8368:Vector projection
8113:978-0-89871-454-8
8084:978-0-321-28713-7
8036:978-0-8218-4419-9
7998:Topics In Algebra
7988:978-1-944325-11-4
7914:978-3-319-11079-0
7427:Subspace topology
6564:} for a subspace
6185:} for a subspace
6005:is an element of
5979:} for a subspace
5779:{\displaystyle N}
5709:{\displaystyle N}
5616:{\displaystyle N}
5592:{\displaystyle V}
5425:{\displaystyle N}
5405:{\displaystyle N}
5385:{\displaystyle V}
5338:{\displaystyle N}
5314:{\displaystyle V}
5294:{\displaystyle N}
5270:{\displaystyle V}
4200:
4143:below for more).
3757:
3677:
3607:
3179:
2808:
2641:
2141:below for more).
2139:§ Algorithms
1929:
1363:
1348:
1330:
1219:
1204:
1186:
1040:many elements of
890:is an element of
780:is an element of
582:is a subspace of
539:Let the field be
526:is an element of
433:is an element of
291:is a subspace of
244:trivial subspaces
236:zero vector space
76:
75:
8754:
8714:
8713:
8596:Exterior algebra
8533:Hadamard product
8450:
8438:Linear equations
8309:
8302:
8295:
8286:
8285:
8281:
8275:
8273:
8253:
8247:
8245:
8218:
8216:
8214:
8203:
8193:
8191:
8189:
8161:
8143:
8121:
8120:on March 1, 2001
8116:, archived from
8096:
8087:
8069:
8054:
8040:
8014:
7992:
7970:
7955:(2nd ed.).
7944:
7929:
7918:
7903:(3rd ed.).
7892:
7869:
7863:
7857:
7851:
7845:
7839:
7833:
7827:
7821:
7815:
7809:
7803:
7797:
7791:
7785:
7779:
7773:
7767:
7761:
7755:
7749:
7746:
7740:
7734:
7728:
7722:
7716:
7710:
7704:
7698:
7692:
7689:DuChateau (2002)
7686:
7680:
7677:MathWorld (2021)
7674:
7668:
7662:
7656:
7650:
7644:
7638:
7632:
7626:
7620:
7614:
7608:
7602:
7596:
7590:
7574:
7572:
7547:
7496:
7490:
7471:
7465:
7458:linear manifolds
7454:affine subspaces
7442:
7380:
7378:
7377:
7372:
7370:
7363:
7362:
7347:
7346:
7327:
7326:
7313:
7312:
7297:
7296:
7284:
7283:
7264:
7263:
7250:
7249:
7234:
7233:
7211:
7210:
7174:
7172:
7171:
7166:
7164:
7160:
7159:
7154:
7142:
7141:
7136:
7120:
7119:
7114:
7104:
7103:
7098:
7086:
7085:
7080:
7071:
7070:
7065:
7049:
7048:
7043:
7033:
7032:
7027:
7015:
7014:
7009:
6990:
6989:
6984:
6961:
6939:
6937:
6936:
6931:
6929:
6925:
6923:
6910:
6899:
6888:
6877:
6866:
6850:
6844:
6838:
6832:
6826:
6813:
6804:
6798:
6792:
6786:
6773:
6767:
6761:
6752:
6746:
6701:
6659:
6624:
6596:Create a matrix
6524:
6522:
6521:
6516:
6498:
6496:
6495:
6490:
6472:
6468:
6464:
6438:
6418:
6349:does not lie in
6334:
6271:augmented matrix
6266:
6202:
6086:
6076:
5853:row echelon form
5830:
5828:
5827:
5822:
5808:
5807:
5785:
5783:
5782:
5777:
5742:
5740:
5739:
5734:
5715:
5713:
5712:
5707:
5695:
5693:
5692:
5687:
5679:
5678:
5669:
5668:
5649:
5647:
5646:
5641:
5639:
5638:
5622:
5620:
5619:
5614:
5598:
5596:
5595:
5590:
5578:
5576:
5575:
5570:
5556:
5555:
5533:
5531:
5530:
5525:
5502:
5501:
5458:
5456:
5455:
5450:
5448:
5447:
5431:
5429:
5428:
5423:
5411:
5409:
5408:
5403:
5391:
5389:
5388:
5383:
5371:
5369:
5368:
5363:
5361:
5360:
5344:
5342:
5341:
5336:
5320:
5318:
5317:
5312:
5300:
5298:
5297:
5292:
5276:
5274:
5273:
5268:
5243:identity element
5213:
5211:
5210:
5205:
5137:
5135:
5134:
5129:
5108:
5106:
5105:
5100:
5070:
5068:
5067:
5062:
4966:
4964:
4963:
4958:
4854:
4852:
4851:
4846:
4841:
4837:
4830:
4816:
4808:
4800:
4772:is the subspace
4693:belongs to both
4670:belongs to both
4511:Given subspaces
4475:, then dim
4419:
4417:
4416:
4411:
4379:
4378:
4339:
4338:
4323:
4322:
4310:
4309:
4294:
4293:
4281:
4280:
4244:
4242:
4241:
4236:
4231:
4230:
4215:
4214:
4201:
4198:
4192:
4191:
4176:
4175:
4123:
4104:
4041:
4039:
4038:
4033:
4031:
4030:
4025:
4019:
4018:
4000:
3999:
3994:
3988:
3987:
3973:
3972:
3967:
3961:
3960:
3942:
3941:
3936:
3930:
3929:
3887:
3780:) of the matrix
3768:
3766:
3765:
3760:
3758:
3755:
3753:
3749:
3747:
3744:
3738:
3727:
3716:
3708:
3702:
3678:
3675:
3669:
3658:
3618:
3616:
3615:
3610:
3608:
3605:
3582:
3581:
3548:
3547:
3532:
3531:
3513:
3512:
3473:
3471:
3470:
3465:
3460:
3459:
3423:
3422:
3336:
3334:
3333:
3328:
3323:
3319:
3312:
3311:
3293:
3292:
3280:
3279:
3274:
3268:
3267:
3249:
3248:
3243:
3237:
3236:
3216:
3215:
3210:
3195:
3194:
3189:
3180:
3177:
3158:
3156:
3155:
3150:
3145:
3144:
3139:
3133:
3132:
3114:
3113:
3108:
3102:
3101:
3045:
3043:
3042:
3037:
3032:
3031:
2995:
2994:
2982:
2981:
2945:
2944:
2930:
2929:
2858:
2856:
2855:
2850:
2848:
2847:
2832:
2831:
2809:
2806:
2797:
2796:
2781:
2780:
2755:
2754:
2739:
2738:
2695:
2693:
2692:
2687:
2682:
2678:
2671:
2670:
2652:
2651:
2642:
2639:
2637:
2634:
2632:
2631:
2622:
2621:
2608:
2594:
2592:
2591:
2582:
2581:
2568:
2560:
2558:
2557:
2548:
2547:
2534:
2526:
2524:
2523:
2511:
2510:
2509:
2508:
2507:
2506:
2505:
2504:
2503:
2502:
2501:
2493:
2492:
2489:
2487:
2486:
2477:
2476:
2463:
2449:
2447:
2446:
2437:
2436:
2426:
2418:
2416:
2415:
2406:
2405:
2395:
2387:
2385:
2384:
2372:
2370:
2369:
2360:
2359:
2346:
2332:
2330:
2329:
2320:
2319:
2309:
2301:
2299:
2298:
2289:
2288:
2278:
2270:
2268:
2267:
2251:
2250:
2238:
2234:
2231:
2227:
2226:
2206:
2205:
2192:
2191:
2129:
2127:
2126:
2121:
2116:
2115:
2044:
2042:
2041:
2036:
2031:
2023:
1971:
1969:
1968:
1963:
1930:
1927:
1889:rational numbers
1886:
1868:
1866:
1865:
1860:
1855:
1851:
1850:
1847:
1837:
1835:
1834:
1825:
1824:
1811:
1797:
1795:
1794:
1785:
1784:
1771:
1763:
1761:
1760:
1751:
1750:
1735:
1728:
1727:
1726:
1725:
1724:
1723:
1722:
1721:
1720:
1719:
1716:
1706:
1704:
1703:
1694:
1693:
1680:
1666:
1664:
1663:
1654:
1653:
1643:
1635:
1633:
1632:
1623:
1622:
1610:
1600:
1598:
1597:
1588:
1587:
1574:
1560:
1558:
1557:
1548:
1547:
1537:
1529:
1527:
1526:
1517:
1516:
1500:
1499:
1487:
1483:
1480:
1476:
1475:
1455:
1454:
1441:
1440:
1404:coordinate space
1374:
1372:
1371:
1366:
1364:
1361:
1359:
1355:
1349:
1341:
1336:
1331:
1328:
1326:
1315:
1311:
1230:
1228:
1227:
1222:
1220:
1217:
1215:
1211:
1205:
1197:
1192:
1187:
1184:
1182:
1171:
1167:
1044:also belongs to
985:
854:
715:
644:
623:
518:
476:
424:
357:
336:
226:
207:are elements of
206:
193:are elements of
192:
42:
37:
31:
26:
19:
18:
8762:
8761:
8757:
8756:
8755:
8753:
8752:
8751:
8742:Operator theory
8727:
8726:
8725:
8720:
8702:
8664:
8620:
8557:
8509:
8451:
8442:
8408:Change of basis
8398:Multilinear map
8336:
8318:
8313:
8271:
8269:
8258:Strang, Gilbert
8243:
8241:
8230:Strang, Gilbert
8225:
8212:
8210:
8201:
8187:
8185:
8168:
8159:
8114:
8085:
8067:
8045:Kreyszig, Erwin
8037:
8012:
7989:
7967:
7942:
7915:
7883:
7878:
7873:
7872:
7864:
7860:
7852:
7848:
7840:
7836:
7828:
7824:
7820:pp. 30-31, § 19
7816:
7812:
7808:pp. 28-29, § 18
7804:
7800:
7792:
7788:
7780:
7776:
7770:Hefferon (2020)
7768:
7764:
7756:
7752:
7747:
7743:
7735:
7731:
7723:
7719:
7713:Hefferon (2020)
7711:
7707:
7699:
7695:
7687:
7683:
7675:
7671:
7665:Hefferon (2020)
7663:
7659:
7651:
7647:
7639:
7635:
7627:
7623:
7615:
7611:
7603:
7599:
7595:pp. 16-17, § 10
7591:
7587:
7582:
7577:
7569:
7563:
7556:
7549:
7542:
7533:
7527:
7521:
7515:
7513:
7504:
7497:
7493:
7472:
7468:
7446:linear subspace
7443:
7439:
7435:
7422:Signal subspace
7402:Cyclic subspace
7398:
7368:
7367:
7358:
7354:
7342:
7338:
7328:
7322:
7318:
7315:
7314:
7308:
7304:
7292:
7288:
7279:
7275:
7265:
7259:
7255:
7252:
7251:
7245:
7241:
7229:
7225:
7212:
7206:
7202:
7198:
7196:
7193:
7192:
7162:
7161:
7155:
7150:
7149:
7137:
7132:
7131:
7121:
7115:
7110:
7109:
7106:
7105:
7099:
7094:
7093:
7081:
7076:
7075:
7066:
7061:
7060:
7050:
7044:
7039:
7038:
7035:
7034:
7028:
7023:
7022:
7010:
7005:
7004:
6991:
6985:
6980:
6979:
6975:
6973:
6970:
6969:
6960:
6953:
6947:
6921:
6920:
6909:
6898:
6887:
6876:
6865:
6859:
6858:
6849:
6843:
6837:
6831:
6825:
6819:
6818:
6812:
6803:
6797:
6791:
6785:
6779:
6778:
6772:
6766:
6760:
6751:
6745:
6738:
6737:
6733:
6731:
6728:
6727:
6688:
6686:
6677:
6658:
6649:
6642:
6636:
6623:
6614:
6607:
6601:
6600:whose rows are
6563:
6554:
6547:
6535:
6504:
6501:
6500:
6478:
6475:
6474:
6470:
6466:
6462:
6459:
6435:
6430:
6427:
6416:
6410:
6403:
6397:
6359:
6333:
6324:
6317:
6311:
6291:
6282:
6265:
6257:
6248:
6242:
6232:
6230:
6221:
6214:
6194:
6193:, and a vector
6184:
6175:
6168:
6156:
6096:
6078:
6070:
6061:
6054:
6038:
6029:
5987:, and a vector
5978:
5969:
5962:
5950:
5876:
5841:
5833:Heyting algebra
5803:
5799:
5791:
5788:
5787:
5771:
5768:
5767:
5728:
5725:
5724:
5701:
5698:
5697:
5674:
5670:
5664:
5660:
5655:
5652:
5651:
5634:
5630:
5628:
5625:
5624:
5608:
5605:
5604:
5584:
5581:
5580:
5551:
5547:
5539:
5536:
5535:
5497:
5493:
5464:
5461:
5460:
5443:
5439:
5437:
5434:
5433:
5417:
5414:
5413:
5397:
5394:
5393:
5377:
5374:
5373:
5356:
5352:
5350:
5347:
5346:
5330:
5327:
5326:
5306:
5303:
5302:
5301:is a subset of
5286:
5283:
5282:
5262:
5259:
5258:
5255:
5231:modular lattice
5221:The operations
5219:
5145:
5142:
5141:
5117:
5114:
5113:
5088:
5085:
5084:
4975:
4972:
4971:
4862:
4859:
4858:
4826:
4812:
4804:
4796:
4795:
4791:
4777:
4774:
4773:
4758:
4686:are subspaces,
4591:belong to both
4575:be elements of
4497:
4483:if and only if
4446:
4441:
4374:
4370:
4334:
4330:
4318:
4314:
4305:
4301:
4289:
4285:
4276:
4272:
4264:
4261:
4260:
4226:
4222:
4210:
4206:
4197:
4187:
4183:
4171:
4167:
4165:
4162:
4161:
4131:for a subspace
4121:
4115:
4109:
4103:
4094:
4088:
4085:
4079:
4072:
4064:
4058:
4051:
4026:
4021:
4020:
4014:
4010:
3995:
3990:
3989:
3983:
3979:
3968:
3963:
3962:
3956:
3952:
3937:
3932:
3931:
3925:
3921:
3919:
3916:
3915:
3906:
3897:
3886:
3879:
3872:
3866:
3819:
3807:Main articles:
3805:
3754:
3745:
3743:
3737:
3728:
3726:
3715:
3709:
3707:
3701:
3694:
3693:
3689:
3674:
3665:
3654:
3652:
3649:
3648:
3642:
3636:
3604:
3577:
3573:
3543:
3539:
3527:
3523:
3508:
3504:
3499:
3496:
3495:
3455:
3451:
3418:
3414:
3406:
3403:
3402:
3383:
3374:
3355:
3346:
3340:If the vectors
3307:
3303:
3288:
3284:
3275:
3270:
3269:
3263:
3259:
3244:
3239:
3238:
3232:
3228:
3227:
3223:
3211:
3206:
3205:
3190:
3185:
3184:
3176:
3174:
3171:
3170:
3140:
3135:
3134:
3128:
3124:
3109:
3104:
3103:
3097:
3093:
3091:
3088:
3087:
3082:
3073:
3066:
3026:
3025:
3019:
3018:
3009:
3008:
2998:
2997:
2990:
2986:
2976:
2975:
2966:
2965:
2959:
2958:
2948:
2947:
2940:
2936:
2924:
2923:
2917:
2916:
2910:
2909:
2899:
2898:
2896:
2893:
2892:
2886:
2880:
2878:Span of vectors
2843:
2839:
2827:
2823:
2805:
2792:
2788:
2776:
2772:
2750:
2746:
2734:
2730:
2719:
2716:
2715:
2666:
2662:
2647:
2643:
2638:
2635:
2633:
2627:
2623:
2614:
2610:
2607:
2593:
2587:
2583:
2574:
2570:
2567:
2559:
2553:
2549:
2540:
2536:
2533:
2525:
2519:
2515:
2512:
2500:
2490:
2488:
2482:
2478:
2469:
2465:
2462:
2448:
2442:
2438:
2432:
2428:
2425:
2417:
2411:
2407:
2401:
2397:
2394:
2386:
2380:
2376:
2373:
2371:
2365:
2361:
2352:
2348:
2345:
2331:
2325:
2321:
2315:
2311:
2308:
2300:
2294:
2290:
2284:
2280:
2277:
2269:
2263:
2259:
2255:
2246:
2242:
2229:
2228:
2222:
2218:
2215:
2214:
2208:
2207:
2201:
2197:
2194:
2193:
2187:
2183:
2179:
2176:
2172:
2171:
2167:
2165:
2162:
2161:
2157:is a subspace:
2147:
2110:
2109:
2104:
2096:
2090:
2089:
2084:
2079:
2069:
2068:
2060:
2057:
2056:
2027:
2019:
2014:
2011:
2010:
2004:
1998:
1926:
1896:
1893:
1892:
1872:
1848:
1846:
1836:
1830:
1826:
1817:
1813:
1810:
1796:
1790:
1786:
1777:
1773:
1770:
1762:
1756:
1752:
1743:
1739:
1736:
1734:
1717:
1715:
1705:
1699:
1695:
1686:
1682:
1679:
1665:
1659:
1655:
1649:
1645:
1642:
1634:
1628:
1624:
1618:
1614:
1611:
1609:
1599:
1593:
1589:
1580:
1576:
1573:
1559:
1553:
1549:
1543:
1539:
1536:
1528:
1522:
1518:
1512:
1508:
1504:
1495:
1491:
1478:
1477:
1471:
1467:
1464:
1463:
1457:
1456:
1450:
1446:
1443:
1442:
1436:
1432:
1428:
1425:
1421:
1420:
1416:
1414:
1411:
1410:
1392:
1360:
1351:
1350:
1340:
1332:
1329: (or
1327:
1322:
1307:
1306:
1289:
1286:
1285:
1216:
1207:
1206:
1196:
1188:
1185: (or
1183:
1178:
1163:
1162:
1145:
1142:
1141:
1084:
1022:
999:
976:
924:
882:
875:
868:
861:
852:
845:
832:
827:be a scalar in
822:
815:
804:
797:
771:
764:
757:
750:
743:
736:
729:
722:
713:
706:
699:
692:
678:
676:
669:
662:
655:
645:be elements of
642:
635:
625:
621:
614:
604:
549:Cartesian plane
537:
516:
509:
498:
491:
478:
474:
467:
457:
422:
415:
408:
401:
394:
387:
380:
373:
359:
355:
348:
338:
334:
327:
317:
271:over the field
257:
252:
225:
218:
212:
198:
191:
184:
178:
143:linear subspace
133:is a subset of
116:
92:vector subspace
88:linear subspace
71:
56:
32:
17:
12:
11:
5:
8760:
8750:
8749:
8744:
8739:
8737:Linear algebra
8722:
8721:
8719:
8718:
8707:
8704:
8703:
8701:
8700:
8695:
8690:
8685:
8680:
8678:Floating-point
8674:
8672:
8666:
8665:
8663:
8662:
8660:Tensor product
8657:
8652:
8647:
8645:Function space
8642:
8637:
8631:
8629:
8622:
8621:
8619:
8618:
8613:
8608:
8603:
8598:
8593:
8588:
8583:
8581:Triple product
8578:
8573:
8567:
8565:
8559:
8558:
8556:
8555:
8550:
8545:
8540:
8535:
8530:
8525:
8519:
8517:
8511:
8510:
8508:
8507:
8502:
8497:
8495:Transformation
8492:
8487:
8485:Multiplication
8482:
8477:
8472:
8467:
8461:
8459:
8453:
8452:
8445:
8443:
8441:
8440:
8435:
8430:
8425:
8420:
8415:
8410:
8405:
8400:
8395:
8390:
8385:
8380:
8375:
8370:
8365:
8360:
8355:
8350:
8344:
8342:
8341:Basic concepts
8338:
8337:
8335:
8334:
8329:
8323:
8320:
8319:
8316:Linear algebra
8312:
8311:
8304:
8297:
8289:
8283:
8282:
8260:(5 May 2020).
8254:
8232:(7 May 2009).
8224:
8223:External links
8221:
8220:
8219:
8194:
8167:
8164:
8163:
8162:
8157:
8144:
8122:
8112:
8097:
8088:
8083:
8070:
8065:
8041:
8035:
8015:
8011:978-1114541016
8010:
7993:
7987:
7979:Linear Algebra
7971:
7965:
7945:
7940:
7919:
7913:
7893:
7882:
7879:
7877:
7874:
7871:
7870:
7868:p. 195, § 6.51
7858:
7856:p. 194, § 6.47
7846:
7844:p. 195, § 6.50
7834:
7832:p. 193, § 6.46
7822:
7810:
7798:
7786:
7774:
7762:
7750:
7741:
7729:
7717:
7705:
7693:
7681:
7669:
7657:
7645:
7643:, p. 200)
7641:Kreyszig (1972
7633:
7631:, p. 132)
7629:Herstein (1964
7621:
7619:, p. 176)
7609:
7607:, p. 155)
7597:
7584:
7583:
7581:
7578:
7576:
7575:
7567:
7561:
7554:
7538:
7531:
7525:
7519:
7509:
7502:
7491:
7479:characteristic
7466:
7436:
7434:
7431:
7430:
7429:
7424:
7419:
7414:
7409:
7404:
7397:
7394:
7393:
7392:
7384:
7383:
7382:
7381:
7366:
7361:
7357:
7353:
7350:
7345:
7341:
7337:
7334:
7331:
7329:
7325:
7321:
7317:
7316:
7311:
7307:
7303:
7300:
7295:
7291:
7287:
7282:
7278:
7274:
7271:
7268:
7266:
7262:
7258:
7254:
7253:
7248:
7244:
7240:
7237:
7232:
7228:
7224:
7221:
7218:
7215:
7213:
7209:
7205:
7201:
7200:
7187:
7186:
7178:
7177:
7176:
7175:
7158:
7153:
7148:
7145:
7140:
7135:
7130:
7127:
7124:
7122:
7118:
7113:
7108:
7107:
7102:
7097:
7092:
7089:
7084:
7079:
7074:
7069:
7064:
7059:
7056:
7053:
7051:
7047:
7042:
7037:
7036:
7031:
7026:
7021:
7018:
7013:
7008:
7003:
7000:
6997:
6994:
6992:
6988:
6983:
6978:
6977:
6964:
6963:
6958:
6951:
6943:
6942:
6941:
6940:
6928:
6919:
6911:
6908:
6900:
6897:
6889:
6886:
6878:
6875:
6867:
6864:
6861:
6860:
6857:
6854:
6851:
6848:
6845:
6842:
6839:
6836:
6833:
6830:
6827:
6824:
6821:
6820:
6817:
6814:
6811:
6808:
6805:
6802:
6799:
6796:
6793:
6790:
6787:
6784:
6781:
6780:
6777:
6774:
6771:
6768:
6765:
6762:
6759:
6756:
6753:
6750:
6747:
6744:
6741:
6740:
6736:
6722:
6721:
6714:
6710:
6709:
6708:
6707:
6682:
6675:
6661:
6654:
6647:
6640:
6633:
6626:
6619:
6612:
6605:
6585:) ×
6572:
6559:
6552:
6545:
6534:
6531:
6514:
6511:
6508:
6488:
6485:
6482:
6458:
6455:
6447:
6446:
6445:
6444:
6433:
6425:
6420:
6414:
6408:
6401:
6394:
6378:
6358:
6355:
6343:
6342:
6341:
6340:
6329:
6322:
6315:
6304:
6297:
6287:
6280:
6261:
6253:
6246:
6240:
6226:
6219:
6212:
6203:
6180:
6173:
6166:
6155:
6152:
6140:
6139:
6138:
6137:
6130:
6114:
6095:
6092:
6091:
6090:
6089:
6088:
6066:
6059:
6051:
6044:
6034:
6027:
5996:
5974:
5967:
5960:
5949:
5946:
5922:
5921:
5920:
5919:
5912:
5895:
5875:
5872:
5871:
5870:
5867:
5864:
5840:
5837:
5820:
5817:
5814:
5811:
5806:
5802:
5798:
5795:
5775:
5752:bilinear forms
5732:
5705:
5685:
5682:
5677:
5673:
5667:
5663:
5659:
5637:
5633:
5612:
5588:
5568:
5565:
5562:
5559:
5554:
5550:
5546:
5543:
5523:
5520:
5517:
5514:
5511:
5508:
5505:
5500:
5496:
5492:
5489:
5486:
5483:
5480:
5477:
5474:
5471:
5468:
5446:
5442:
5421:
5401:
5381:
5359:
5355:
5334:
5310:
5290:
5266:
5254:
5251:
5218:
5215:
5203:
5200:
5197:
5194:
5191:
5188:
5185:
5182:
5179:
5176:
5173:
5170:
5167:
5164:
5161:
5158:
5155:
5152:
5149:
5127:
5124:
5121:
5098:
5095:
5092:
5060:
5057:
5054:
5051:
5048:
5045:
5042:
5039:
5036:
5033:
5030:
5027:
5024:
5021:
5018:
5015:
5012:
5009:
5006:
5003:
5000:
4997:
4994:
4991:
4988:
4985:
4982:
4979:
4956:
4953:
4950:
4947:
4944:
4941:
4938:
4935:
4932:
4929:
4926:
4923:
4920:
4917:
4914:
4911:
4908:
4905:
4902:
4899:
4896:
4893:
4890:
4887:
4884:
4881:
4878:
4875:
4872:
4869:
4866:
4844:
4840:
4836:
4833:
4829:
4825:
4822:
4819:
4815:
4811:
4807:
4803:
4799:
4794:
4790:
4787:
4784:
4781:
4757:
4754:
4730:
4729:
4702:
4648:
4496:
4493:
4445:
4442:
4440:
4437:
4436:
4435:
4423:
4422:
4421:
4420:
4409:
4406:
4403:
4400:
4397:
4394:
4391:
4388:
4385:
4382:
4377:
4373:
4369:
4366:
4363:
4360:
4357:
4354:
4351:
4348:
4345:
4342:
4337:
4333:
4329:
4326:
4321:
4317:
4313:
4308:
4304:
4300:
4297:
4292:
4288:
4284:
4279:
4275:
4271:
4268:
4255:
4254:
4247:
4246:
4245:
4234:
4229:
4225:
4221:
4218:
4213:
4209:
4190:
4186:
4182:
4179:
4174:
4170:
4148:
4119:
4113:
4099:
4092:
4083:
4077:
4070:
4062:
4056:
4049:
4043:
4042:
4029:
4024:
4017:
4013:
4009:
4006:
4003:
3998:
3993:
3986:
3982:
3977:
3971:
3966:
3959:
3955:
3951:
3948:
3945:
3940:
3935:
3928:
3924:
3902:
3895:
3884:
3877:
3870:
3845:determined by
3804:
3801:
3770:
3769:
3752:
3742:
3739:
3736:
3733:
3730:
3729:
3725:
3722:
3717:
3714:
3711:
3710:
3706:
3703:
3700:
3697:
3696:
3692:
3688:
3685:
3668:
3664:
3661:
3657:
3638:Main article:
3635:
3632:
3631:
3630:
3622:
3621:
3620:
3619:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3580:
3576:
3572:
3569:
3566:
3563:
3560:
3557:
3554:
3551:
3546:
3542:
3538:
3535:
3530:
3526:
3522:
3519:
3516:
3511:
3507:
3503:
3490:
3489:
3477:
3476:
3475:
3474:
3463:
3458:
3454:
3450:
3447:
3441:
3438:
3435:
3432:
3426:
3421:
3417:
3413:
3410:
3389:
3379:
3372:
3351:
3344:
3338:
3337:
3326:
3322:
3318:
3315:
3310:
3306:
3302:
3299:
3296:
3291:
3287:
3283:
3278:
3273:
3266:
3262:
3258:
3255:
3252:
3247:
3242:
3235:
3231:
3226:
3222:
3219:
3214:
3209:
3204:
3201:
3198:
3193:
3188:
3183:
3160:
3159:
3148:
3143:
3138:
3131:
3127:
3123:
3120:
3117:
3112:
3107:
3100:
3096:
3078:
3071:
3064:
3056:In general, a
3047:
3046:
3035:
3030:
3024:
3021:
3020:
3017:
3014:
3011:
3010:
3007:
3004:
3003:
3001:
2993:
2989:
2985:
2980:
2974:
2971:
2968:
2967:
2964:
2961:
2960:
2957:
2954:
2953:
2951:
2943:
2939:
2934:
2928:
2922:
2919:
2918:
2915:
2912:
2911:
2908:
2905:
2904:
2902:
2882:Main article:
2879:
2876:
2860:
2859:
2846:
2842:
2838:
2835:
2830:
2826:
2822:
2819:
2816:
2800:
2795:
2791:
2787:
2784:
2779:
2775:
2771:
2768:
2765:
2758:
2753:
2749:
2745:
2742:
2737:
2733:
2729:
2726:
2723:
2697:
2696:
2685:
2681:
2677:
2674:
2669:
2665:
2661:
2658:
2655:
2650:
2646:
2630:
2626:
2620:
2617:
2613:
2609:
2605:
2602:
2599:
2595:
2590:
2586:
2580:
2577:
2573:
2569:
2565:
2561:
2556:
2552:
2546:
2543:
2539:
2535:
2531:
2527:
2522:
2518:
2514:
2513:
2497:
2494:
2491:
2485:
2481:
2475:
2472:
2468:
2464:
2460:
2457:
2454:
2450:
2445:
2441:
2435:
2431:
2427:
2423:
2419:
2414:
2410:
2404:
2400:
2396:
2392:
2388:
2383:
2379:
2375:
2374:
2368:
2364:
2358:
2355:
2351:
2347:
2343:
2340:
2337:
2333:
2328:
2324:
2318:
2314:
2310:
2306:
2302:
2297:
2293:
2287:
2283:
2279:
2275:
2271:
2266:
2262:
2258:
2257:
2254:
2249:
2245:
2241:
2237:
2225:
2221:
2217:
2216:
2213:
2210:
2209:
2204:
2200:
2196:
2195:
2190:
2186:
2182:
2181:
2175:
2170:
2149:The subset of
2146:
2143:
2131:
2130:
2119:
2114:
2108:
2105:
2103:
2100:
2097:
2095:
2092:
2091:
2088:
2085:
2083:
2080:
2078:
2075:
2074:
2072:
2067:
2064:
2046:
2045:
2034:
2030:
2026:
2022:
2018:
2000:Main article:
1997:
1994:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1887:(over real or
1858:
1854:
1845:
1842:
1838:
1833:
1829:
1823:
1820:
1816:
1812:
1808:
1805:
1802:
1798:
1793:
1789:
1783:
1780:
1776:
1772:
1768:
1764:
1759:
1755:
1749:
1746:
1742:
1738:
1737:
1732:
1729:
1718:
1714:
1711:
1707:
1702:
1698:
1692:
1689:
1685:
1681:
1677:
1674:
1671:
1667:
1662:
1658:
1652:
1648:
1644:
1640:
1636:
1631:
1627:
1621:
1617:
1613:
1612:
1608:
1605:
1601:
1596:
1592:
1586:
1583:
1579:
1575:
1571:
1568:
1565:
1561:
1556:
1552:
1546:
1542:
1538:
1534:
1530:
1525:
1521:
1515:
1511:
1507:
1506:
1503:
1498:
1494:
1490:
1486:
1474:
1470:
1466:
1465:
1462:
1459:
1458:
1453:
1449:
1445:
1444:
1439:
1435:
1431:
1430:
1424:
1419:
1391:
1388:
1376:
1375:
1358:
1354:
1347:
1344:
1339:
1335:
1325:
1321:
1318:
1314:
1310:
1305:
1302:
1299:
1296:
1293:
1269:specifies its
1232:
1231:
1214:
1210:
1203:
1200:
1195:
1191:
1181:
1177:
1174:
1170:
1166:
1161:
1158:
1155:
1152:
1149:
1083:
1080:
1034:if and only if
1032:is a subspace
1021:
1018:
998:
995:
994:
993:
990:
987:
923:
920:
896:
895:
880:
873:
866:
859:
850:
843:
820:
813:
802:
795:
785:
769:
762:
755:
748:
741:
734:
727:
720:
711:
704:
697:
690:
674:
667:
660:
653:
640:
633:
619:
612:
536:
533:
532:
531:
514:
507:
496:
489:
472:
465:
438:
420:
413:
406:
399:
392:
385:
378:
371:
353:
346:
332:
325:
256:
253:
251:
248:
223:
216:
189:
182:
115:
112:
84:linear algebra
74:
73:
69:
54:
44:
43:
15:
9:
6:
4:
3:
2:
8759:
8748:
8745:
8743:
8740:
8738:
8735:
8734:
8732:
8717:
8709:
8708:
8705:
8699:
8696:
8694:
8693:Sparse matrix
8691:
8689:
8686:
8684:
8681:
8679:
8676:
8675:
8673:
8671:
8667:
8661:
8658:
8656:
8653:
8651:
8648:
8646:
8643:
8641:
8638:
8636:
8633:
8632:
8630:
8628:constructions
8627:
8623:
8617:
8616:Outermorphism
8614:
8612:
8609:
8607:
8604:
8602:
8599:
8597:
8594:
8592:
8589:
8587:
8584:
8582:
8579:
8577:
8576:Cross product
8574:
8572:
8569:
8568:
8566:
8564:
8560:
8554:
8551:
8549:
8546:
8544:
8543:Outer product
8541:
8539:
8536:
8534:
8531:
8529:
8526:
8524:
8523:Orthogonality
8521:
8520:
8518:
8516:
8512:
8506:
8503:
8501:
8500:Cramer's rule
8498:
8496:
8493:
8491:
8488:
8486:
8483:
8481:
8478:
8476:
8473:
8471:
8470:Decomposition
8468:
8466:
8463:
8462:
8460:
8458:
8454:
8449:
8439:
8436:
8434:
8431:
8429:
8426:
8424:
8421:
8419:
8416:
8414:
8411:
8409:
8406:
8404:
8401:
8399:
8396:
8394:
8391:
8389:
8386:
8384:
8381:
8379:
8376:
8374:
8371:
8369:
8366:
8364:
8361:
8359:
8356:
8354:
8351:
8349:
8346:
8345:
8343:
8339:
8333:
8330:
8328:
8325:
8324:
8321:
8317:
8310:
8305:
8303:
8298:
8296:
8291:
8290:
8287:
8279:
8267:
8263:
8259:
8255:
8251:
8239:
8235:
8231:
8227:
8226:
8209:
8208:
8200:
8195:
8184:
8183:
8178:
8174:
8170:
8169:
8160:
8158:0-534-99845-3
8154:
8150:
8145:
8142:
8138:
8134:
8130:
8129:
8123:
8119:
8115:
8109:
8105:
8104:
8098:
8094:
8089:
8086:
8080:
8076:
8071:
8068:
8066:0-471-50728-8
8062:
8058:
8053:
8052:
8046:
8042:
8038:
8032:
8028:
8024:
8020:
8016:
8013:
8007:
8003:
7999:
7994:
7990:
7984:
7980:
7976:
7975:Hefferon, Jim
7972:
7968:
7966:0-387-90093-4
7962:
7958:
7954:
7950:
7946:
7943:
7941:0-395-14017-X
7937:
7933:
7928:
7927:
7920:
7916:
7910:
7906:
7902:
7898:
7894:
7890:
7885:
7884:
7867:
7862:
7855:
7850:
7843:
7838:
7831:
7826:
7819:
7818:Halmos (1974)
7814:
7807:
7806:Halmos (1974)
7802:
7795:
7790:
7783:
7778:
7771:
7766:
7760:, p. 22)
7759:
7754:
7745:
7739:, p. 21)
7738:
7733:
7727:, p. 20)
7726:
7721:
7714:
7709:
7703:, p. 21)
7702:
7697:
7690:
7685:
7678:
7673:
7666:
7661:
7655:, p. 20)
7654:
7649:
7642:
7637:
7630:
7625:
7618:
7613:
7606:
7601:
7594:
7593:Halmos (1974)
7589:
7585:
7570:
7560:
7553:
7546:
7541:
7537:
7534:
7524:
7518:
7512:
7508:
7501:
7495:
7488:
7484:
7480:
7476:
7470:
7463:
7459:
7455:
7451:
7447:
7441:
7437:
7428:
7425:
7423:
7420:
7418:
7415:
7413:
7410:
7408:
7405:
7403:
7400:
7399:
7390:
7386:
7385:
7364:
7359:
7355:
7351:
7348:
7343:
7339:
7335:
7332:
7330:
7323:
7319:
7309:
7305:
7301:
7298:
7293:
7289:
7285:
7280:
7276:
7272:
7269:
7267:
7260:
7256:
7246:
7242:
7238:
7235:
7230:
7226:
7222:
7219:
7216:
7214:
7207:
7203:
7191:
7190:
7189:
7188:
7184:
7180:
7179:
7156:
7146:
7143:
7138:
7128:
7125:
7123:
7116:
7100:
7090:
7087:
7082:
7072:
7067:
7057:
7054:
7052:
7045:
7029:
7019:
7016:
7011:
7001:
6998:
6995:
6993:
6986:
6968:
6967:
6966:
6965:
6957:
6950:
6945:
6944:
6926:
6917:
6906:
6895:
6884:
6873:
6862:
6855:
6852:
6846:
6840:
6834:
6828:
6822:
6815:
6809:
6806:
6800:
6794:
6788:
6782:
6775:
6769:
6763:
6757:
6754:
6748:
6742:
6734:
6726:
6725:
6724:
6723:
6719:
6715:
6712:
6711:
6705:
6700:
6696:
6692:
6685:
6681:
6674:
6670:
6666:
6662:
6657:
6653:
6646:
6639:
6634:
6631:
6627:
6622:
6618:
6611:
6604:
6599:
6595:
6594:
6592:
6588:
6584:
6581: −
6580:
6576:
6573:
6571:
6567:
6562:
6558:
6551:
6544:
6540:
6537:
6536:
6530:
6528:
6512:
6509:
6506:
6486:
6483:
6480:
6454:
6452:
6442:
6436:
6428:
6421:
6417:
6407:
6400:
6395:
6392:
6388:
6387:
6386:
6382:
6379:
6376:
6372:
6369: ×
6368:
6364:
6361:
6360:
6354:
6352:
6348:
6338:
6332:
6328:
6321:
6314:
6309:
6305:
6302:
6298:
6295:
6290:
6286:
6279:
6275:
6272:
6268:
6267:
6264:
6260:
6256:
6252:
6245:
6239:
6235:
6229:
6225:
6218:
6211:
6207:
6204:
6201:
6197:
6192:
6188:
6183:
6179:
6172:
6165:
6161:
6158:
6157:
6151:
6147:
6145:
6135:
6131:
6128:
6124:
6123:
6122:
6118:
6115:
6113:
6109:
6106: ×
6105:
6101:
6098:
6097:
6085:
6081:
6074:
6069:
6065:
6058:
6052:
6049:
6045:
6042:
6037:
6033:
6026:
6022:
6018:
6014:
6010:
6009:
6008:
6004:
6000:
5997:
5994:
5990:
5986:
5982:
5977:
5973:
5966:
5959:
5955:
5952:
5951:
5945:
5943:
5938:
5933:
5931:
5927:
5917:
5913:
5910:
5906:
5905:
5903:
5899:
5896:
5893:
5889:
5886: ×
5885:
5881:
5878:
5877:
5868:
5865:
5862:
5861:
5860:
5858:
5854:
5850:
5846:
5845:row reduction
5836:
5834:
5815:
5809:
5804:
5800:
5796:
5793:
5773:
5765:
5761:
5757:
5753:
5748:
5746:
5722:
5717:
5703:
5683:
5680:
5675:
5665:
5661:
5635:
5631:
5610:
5602:
5586:
5563:
5557:
5552:
5548:
5544:
5541:
5518:
5512:
5509:
5506:
5498:
5494:
5487:
5484:
5481:
5475:
5469:
5466:
5444:
5440:
5419:
5399:
5379:
5357:
5353:
5332:
5324:
5308:
5288:
5280:
5264:
5250:
5248:
5244:
5240:
5239:least element
5236:
5232:
5228:
5224:
5214:
5198:
5192:
5189:
5186:
5180:
5174:
5171:
5168:
5162:
5159:
5156:
5150:
5147:
5139:
5125:
5122:
5119:
5110:
5096:
5093:
5090:
5082:
5081:
5076:
5071:
5058:
5052:
5049:
5046:
5040:
5037:
5034:
5028:
5022:
5019:
5016:
5010:
5004:
5001:
4998:
4992:
4989:
4986:
4980:
4977:
4967:
4954:
4948:
4942:
4939:
4936:
4930:
4924:
4921:
4918:
4912:
4909:
4906:
4900:
4897:
4894:
4888:
4885:
4882:
4879:
4876:
4873:
4870:
4855:
4842:
4838:
4834:
4831:
4823:
4820:
4817:
4809:
4801:
4792:
4788:
4785:
4782:
4779:
4771:
4767:
4763:
4753:
4751:
4747:
4743:
4741:
4735:
4727:
4724: ∩
4723:
4719:
4715:
4711:
4707:
4703:
4700:
4696:
4692:
4689:
4685:
4681:
4677:
4673:
4669:
4665:
4661:
4658: ∩
4657:
4653:
4649:
4646:
4643: ∩
4642:
4638:
4635: +
4634:
4630:
4626:
4623: +
4622:
4618:
4614:
4610:
4607: +
4606:
4602:
4598:
4594:
4590:
4586:
4582:
4579: ∩
4578:
4574:
4570:
4566:
4565:
4564:
4563:
4559:
4557:
4553:
4549:
4545:
4541:
4538: ∈
4537:
4533:
4530: ∩
4529:
4526:
4523:, then their
4522:
4518:
4514:
4506:
4501:
4492:
4490:
4487: =
4486:
4482:
4479: =
4478:
4474:
4471: ⊂
4470:
4466:
4463: =
4462:
4457:
4455:
4454:partial order
4451:
4433:
4429:
4426:The subspace
4425:
4424:
4407:
4401:
4398:
4395:
4392:
4389:
4386:
4383:
4375:
4371:
4367:
4361:
4358:
4355:
4352:
4349:
4346:
4343:
4335:
4331:
4327:
4319:
4315:
4311:
4306:
4302:
4298:
4295:
4290:
4286:
4282:
4277:
4273:
4269:
4259:
4258:
4257:
4256:
4252:
4248:
4232:
4227:
4223:
4219:
4216:
4211:
4207:
4188:
4184:
4180:
4177:
4172:
4168:
4160:
4159:
4157:
4153:
4149:
4146:
4145:
4144:
4142:
4138:
4134:
4130:
4125:
4122:
4112:
4108:
4102:
4098:
4091:
4086:
4076:
4069:
4065:
4055:
4048:
4027:
4015:
4011:
4007:
4004:
4001:
3996:
3984:
3980:
3975:
3969:
3957:
3953:
3949:
3946:
3943:
3938:
3926:
3922:
3914:
3913:
3912:
3910:
3905:
3901:
3894:
3889:
3883:
3876:
3869:
3864:
3860:
3856:
3852:
3848:
3844:
3836:
3832:
3828:
3823:
3818:
3814:
3810:
3800:
3798:
3793:
3791:
3787:
3783:
3779:
3775:
3750:
3740:
3734:
3731:
3723:
3720:
3712:
3704:
3698:
3690:
3686:
3683:
3662:
3659:
3647:
3646:
3645:
3641:
3628:
3624:
3623:
3598:
3595:
3592:
3589:
3586:
3578:
3574:
3570:
3564:
3561:
3558:
3555:
3552:
3544:
3540:
3536:
3528:
3524:
3520:
3517:
3514:
3509:
3505:
3494:
3493:
3492:
3491:
3487:
3483:
3479:
3478:
3461:
3456:
3452:
3448:
3445:
3439:
3436:
3433:
3430:
3424:
3419:
3415:
3411:
3408:
3401:
3400:
3398:
3394:
3390:
3387:
3386:
3385:
3382:
3378:
3371:
3367:
3363:
3359:
3354:
3350:
3343:
3324:
3320:
3316:
3313:
3308:
3304:
3300:
3297:
3294:
3289:
3285:
3281:
3276:
3264:
3260:
3256:
3253:
3250:
3245:
3233:
3229:
3224:
3220:
3212:
3202:
3199:
3196:
3191:
3169:
3168:
3167:
3165:
3146:
3141:
3129:
3125:
3121:
3118:
3115:
3110:
3098:
3094:
3086:
3085:
3084:
3081:
3077:
3070:
3063:
3059:
3054:
3052:
3033:
3028:
3022:
3015:
3012:
3005:
2999:
2991:
2987:
2983:
2978:
2972:
2969:
2962:
2955:
2949:
2941:
2937:
2932:
2926:
2920:
2913:
2906:
2900:
2891:
2890:
2889:
2885:
2875:
2873:
2869:
2865:
2844:
2840:
2836:
2833:
2828:
2824:
2820:
2817:
2814:
2798:
2793:
2789:
2785:
2782:
2777:
2773:
2769:
2766:
2763:
2756:
2751:
2747:
2743:
2740:
2735:
2731:
2727:
2724:
2721:
2714:
2713:
2712:
2710:
2706:
2702:
2683:
2679:
2675:
2672:
2667:
2663:
2659:
2656:
2653:
2648:
2644:
2628:
2624:
2618:
2615:
2611:
2603:
2600:
2597:
2588:
2584:
2578:
2575:
2571:
2563:
2554:
2550:
2544:
2541:
2537:
2529:
2520:
2516:
2495:
2483:
2479:
2473:
2470:
2466:
2458:
2455:
2452:
2443:
2439:
2433:
2429:
2421:
2412:
2408:
2402:
2398:
2390:
2381:
2377:
2366:
2362:
2356:
2353:
2349:
2341:
2338:
2335:
2326:
2322:
2316:
2312:
2304:
2295:
2291:
2285:
2281:
2273:
2264:
2260:
2252:
2247:
2243:
2239:
2235:
2223:
2219:
2211:
2202:
2198:
2188:
2184:
2173:
2168:
2160:
2159:
2158:
2156:
2152:
2142:
2140:
2136:
2117:
2112:
2106:
2101:
2098:
2093:
2086:
2081:
2076:
2070:
2065:
2062:
2055:
2054:
2053:
2051:
2032:
2024:
2016:
2009:
2008:
2007:
2003:
1993:
1991:
1987:
1983:
1979:
1975:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1890:
1884:
1880:
1876:
1869:
1856:
1852:
1843:
1840:
1831:
1827:
1821:
1818:
1814:
1806:
1803:
1800:
1791:
1787:
1781:
1778:
1774:
1766:
1757:
1753:
1747:
1744:
1740:
1730:
1712:
1709:
1700:
1696:
1690:
1687:
1683:
1675:
1672:
1669:
1660:
1656:
1650:
1646:
1638:
1629:
1625:
1619:
1615:
1606:
1603:
1594:
1590:
1584:
1581:
1577:
1569:
1566:
1563:
1554:
1550:
1544:
1540:
1532:
1523:
1519:
1513:
1509:
1501:
1496:
1492:
1488:
1484:
1472:
1468:
1460:
1451:
1447:
1437:
1433:
1422:
1417:
1408:
1405:
1401:
1397:
1387:
1385:
1384:the remaining
1381:
1356:
1345:
1342:
1337:
1319:
1316:
1312:
1303:
1300:
1297:
1294:
1284:
1283:
1282:
1280:
1276:
1272:
1268:
1264:
1260:
1256:
1251:
1249:
1245:
1241:
1237:
1212:
1201:
1198:
1193:
1175:
1172:
1168:
1159:
1156:
1153:
1150:
1140:
1139:
1138:
1136:
1132:
1128:
1123:
1121:
1117:
1113:
1109:
1105:
1101:
1097:
1093:
1089:
1079:
1077:
1073:
1069:
1065:
1061:
1058:, a subspace
1057:
1054:
1049:
1047:
1043:
1039:
1035:
1031:
1027:
1017:
1015:
1010:
1008:
1004:
991:
988:
984:
980:
974:
973:
972:
971:
967:
965:
961:
957:
953:
949:
945:
941:
937:
933:
929:
919:
917:
914: =
913:
909:
905:
901:
893:
889:
886:
879:
872:
865:
858:
849:
842:
838:
835:
830:
826:
819:
812:
808:
801:
794:
790:
786:
783:
779:
775:
768:
761:
754:
747:
740:
733:
726:
719:
710:
703:
696:
689:
685:
681:
673:
666:
659:
652:
648:
639:
632:
628:
618:
611:
607:
602:
601:
600:
599:
591:
587:
585:
581:
577:
573:
569:
565:
561:
557:
553:
550:
546:
542:
529:
525:
522:
513:
506:
502:
495:
488:
484:
481:
471:
464:
460:
455:
451:
448:and a scalar
447:
443:
439:
436:
432:
428:
419:
412:
405:
398:
391:
384:
377:
370:
366:
362:
352:
345:
341:
331:
324:
320:
315:
311:
307:
303:
302:
301:
300:
296:
294:
290:
286:
282:
278:
274:
270:
266:
262:
247:
245:
241:
237:
232:
230:
222:
215:
210:
205:
201:
196:
188:
181:
177:if, whenever
176:
172:
168:
164:
160:
156:
152:
148:
144:
140:
136:
132:
128:
125:
121:
111:
109:
105:
101:
97:
93:
89:
85:
81:
68:
64:
60:
53:
50:
46:
45:
41:
36:
30:
25:
21:
20:
8654:
8626:Vector space
8358:Vector space
8276:– via
8270:. Retrieved
8248:– via
8242:. Retrieved
8211:. Retrieved
8205:
8186:. Retrieved
8180:
8148:
8127:
8118:the original
8102:
8092:
8074:
8050:
8022:
7997:
7978:
7952:
7925:
7900:
7888:
7866:Axler (2015)
7861:
7854:Axler (2015)
7849:
7842:Axler (2015)
7837:
7830:Axler (2015)
7825:
7813:
7801:
7789:
7784:p. 21 § 1.40
7782:Axler (2015)
7777:
7765:
7758:Nering (1970
7753:
7744:
7737:Nering (1970
7732:
7725:Nering (1970
7720:
7708:
7701:Nering (1970
7696:
7684:
7672:
7660:
7653:Nering (1970
7648:
7636:
7624:
7612:
7600:
7588:
7565:
7558:
7551:
7544:
7539:
7535:
7529:
7522:
7516:
7510:
7506:
7499:
7494:
7474:
7469:
7457:
7445:
7440:
7388:
7182:
6955:
6948:
6717:
6703:
6698:
6694:
6690:
6683:
6679:
6672:
6668:
6664:
6655:
6651:
6644:
6637:
6629:
6620:
6616:
6609:
6602:
6597:
6590:
6586:
6582:
6578:
6574:
6569:
6565:
6560:
6556:
6549:
6542:
6538:
6460:
6448:
6440:
6431:
6423:
6412:
6405:
6398:
6390:
6384:
6380:
6374:
6370:
6366:
6362:
6350:
6346:
6344:
6336:
6330:
6326:
6319:
6312:
6307:
6300:
6293:
6288:
6284:
6277:
6273:
6262:
6258:
6254:
6250:
6243:
6237:
6233:
6227:
6223:
6216:
6209:
6205:
6199:
6195:
6190:
6186:
6181:
6177:
6170:
6163:
6159:
6148:
6141:
6126:
6120:
6116:
6111:
6107:
6103:
6099:
6083:
6079:
6072:
6067:
6063:
6056:
6047:
6040:
6035:
6031:
6024:
6020:
6016:
6012:
6006:
6002:
5998:
5992:
5988:
5984:
5980:
5975:
5971:
5964:
5957:
5953:
5941:
5936:
5934:
5923:
5915:
5908:
5901:
5897:
5891:
5887:
5883:
5879:
5842:
5764:null vectors
5749:
5718:
5256:
5246:
5235:{0} subspace
5233:, where the
5223:intersection
5220:
5140:
5111:
5078:
5074:
5072:
4968:
4856:
4769:
4765:
4761:
4759:
4749:
4745:
4739:
4733:
4731:
4725:
4721:
4717:
4713:
4709:
4705:
4698:
4694:
4690:
4687:
4683:
4679:
4675:
4671:
4667:
4663:
4659:
4655:
4651:
4644:
4640:
4636:
4632:
4628:
4624:
4620:
4616:
4612:
4608:
4604:
4600:
4596:
4592:
4588:
4584:
4580:
4576:
4572:
4568:
4561:
4560:
4555:
4551:
4547:
4543:
4539:
4535:
4531:
4527:
4525:intersection
4520:
4516:
4512:
4510:
4504:
4495:Intersection
4488:
4484:
4480:
4476:
4472:
4468:
4464:
4460:
4458:
4447:
4431:
4427:
4250:
4155:
4151:
4141:§ Algorithms
4136:
4132:
4128:
4126:
4117:
4110:
4106:
4100:
4096:
4089:
4081:
4074:
4067:
4060:
4053:
4046:
4044:
3908:
3903:
3899:
3892:
3890:
3881:
3874:
3867:
3862:
3858:
3854:
3850:
3846:
3842:
3840:
3834:
3830:
3826:
3825:The vectors
3794:
3789:
3785:
3781:
3773:
3771:
3643:
3626:
3485:
3481:
3396:
3392:
3380:
3376:
3369:
3365:
3361:
3357:
3352:
3348:
3341:
3339:
3163:
3161:
3079:
3075:
3068:
3061:
3057:
3055:
3050:
3048:
2887:
2872:number field
2867:
2863:
2861:
2708:
2704:
2700:
2698:
2150:
2148:
2134:
2132:
2047:
2005:
1989:
1985:
1977:
1973:
1882:
1878:
1874:
1870:
1406:
1399:
1393:
1377:
1274:
1266:
1252:
1247:
1243:
1233:
1134:
1124:
1119:
1104:column space
1085:
1082:Descriptions
1059:
1055:
1050:
1045:
1041:
1029:
1023:
1011:
1002:
1000:
982:
978:
969:
968:
963:
959:
951:
947:
943:
935:
931:
927:
925:
915:
911:
907:
899:
897:
891:
887:
884:
877:
870:
863:
856:
847:
840:
836:
833:
828:
824:
817:
810:
806:
799:
792:
788:
781:
777:
773:
766:
759:
752:
745:
738:
731:
724:
717:
708:
701:
694:
687:
683:
679:
671:
664:
657:
650:
646:
637:
630:
626:
616:
609:
605:
597:
596:
583:
579:
575:
571:
567:
563:
559:
555:
551:
544:
540:
538:
527:
523:
520:
511:
504:
500:
493:
486:
482:
479:
477:again, then
469:
462:
458:
453:
449:
445:
441:
434:
430:
426:
417:
410:
403:
396:
389:
382:
375:
368:
364:
360:
350:
343:
339:
329:
322:
318:
313:
309:
305:
298:
297:
292:
288:
284:
280:
277:real numbers
272:
264:
260:
258:
243:
233:
228:
220:
213:
208:
203:
199:
194:
186:
179:
174:
170:
162:
159:vector space
154:
150:
146:
142:
138:
134:
130:
126:
119:
117:
103:
96:vector space
91:
87:
77:
66:
51:
49:finite field
8606:Multivector
8571:Determinant
8528:Dot product
8373:Linear span
8000:, Waltham:
7605:Anton (2005
7473:Generally,
5995:components.
5944:are equal.
5321:, then the
5075:independent
4720:belongs to
4639:belongs to
4627:belongs to
4611:belongs to
4107:coordinates
3907:are called
3060:of vectors
2884:Linear span
1992:functions.
1236:linear span
1129:of one non-
1072:codimension
934:be the set
922:Example III
240:zero vector
80:mathematics
8731:Categories
8640:Direct sum
8475:Invertible
8378:Linear map
8177:"Subspace"
7930:, Boston:
6269:Create an
6231:such that
6011:Create a (
5839:Algorithms
5786:such that
5601:direct sum
5345:, denoted
5080:direct sum
4662:, and let
4654:belong to
4599:. Because
4534: := {
4066:) ≠ (
3395:-plane in
2050:null space
2002:Null space
1279:dual space
1100:null space
997:Example IV
823:, and let
570:such that
535:Example II
395:, 0+0) = (
114:Definition
98:that is a
63:dimensions
8670:Numerical
8433:Transpose
8182:MathWorld
7951:(1974) .
7679:Subspace.
7580:Citations
7462:manifolds
7444:The term
7349:−
7286:−
7220:−
7144:−
7073:−
6999:−
6853:−
6807:−
6755:−
6541:A basis {
6510:∩
6162:A basis {
5956:A basis {
5926:row space
5810:≠
5805:⊥
5797:∩
5731:¬
5676:⊥
5666:⊥
5636:⊥
5553:⊥
5545:∩
5513:
5499:⊥
5488:
5470:
5445:⊥
5358:⊥
5193:
5175:
5160:⊕
5151:
5123:⊕
5094:⊕
5050:∩
5041:
5035:−
5023:
5005:
4981:
4943:
4925:
4919:≤
4901:
4895:≤
4886:
4874:
4832:∈
4818:∈
4810::
4697:and
4550:and
4444:Inclusion
4005:⋯
3976:≠
3947:⋯
3732:−
3721:−
3314:∈
3298:…
3254:⋯
3200:…
3119:⋯
3013:−
2970:−
2821:−
2783:−
2673:∈
2657:…
2601:⋯
2496:⋮
2456:⋯
2339:⋯
2240:∈
2212:⋮
2099:−
1939:−
1804:⋯
1731:⋮
1673:⋯
1567:⋯
1489:∈
1461:⋮
1298:∈
1292:∃
1273:subspace
1154:∈
1148:∃
1108:row space
1005:) of all
958:. Then C(
940:functions
255:Example I
108:subspaces
8716:Category
8655:Subspace
8650:Quotient
8601:Bivector
8515:Bilinear
8457:Matrices
8332:Glossary
8266:Archived
8238:Archived
8141:76091646
8047:(1972),
7977:(2020).
7957:Springer
7905:Springer
7899:(2015).
7881:Textbook
7528:+ ··· +
7487:integers
7396:See also
6249:+ ··· +
6208:Numbers
5745:infinite
5721:negation
5241:, is an
4678:. Since
4631:. Thus,
4542: :
1982:null set
1357:′
1313:′
1240:equality
1213:′
1169:′
1066:, but a
1038:finitely
950:. Let C(
855:; since
716:; since
519:. Thus,
425:. Thus,
279:), take
250:Examples
167:nonempty
104:subspace
8327:Outline
8278:YouTube
8250:YouTube
7876:Sources
7564:, ...,
7505:, ...,
6954:, ...,
6713:Example
6650:, ...,
6615:, ...,
6555:, ...,
6451:example
6411:, ...,
6373:matrix
6325:, ...,
6222:, ...,
6176:, ...,
6144:example
6110:matrix
6062:, ...,
6019:matrix
5970:, ...,
5930:example
5928:for an
5890:matrix
5599:is the
4583:. Then
4147:Example
4116:, ...,
4095:, ...,
4073:,
4052:,
3388:Example
3067:,
2707:,
2703:,
1133:vector
938:of all
869:, then
831:. Then
744:, then
677:. Then
578:. Then
554:. Take
547:be the
358:. Then
169:subset
137:, then
129:and if
8611:Tensor
8423:Kernel
8353:Vector
8348:Scalar
8272:17 Feb
8244:17 Feb
8213:17 Feb
8188:16 Feb
8155:
8139:
8110:
8081:
8063:
8033:
8008:
7985:
7963:
7938:
7911:
6687:. The
6575:Output
6381:Output
6206:Output
6134:pivots
6117:Output
5999:Output
5898:Output
5277:is an
5237:, the
4736:, the
4704:Since
4562:Proof:
4087:). If
3815:, and
1271:kernel
1118:in an
1112:matrix
1106:, and
1094:, the
1064:closed
1026:closed
977:0 ∈ C(
970:Proof:
598:Proof:
503:0) = (
440:Given
437:, too.
304:Given
299:Proof:
227:is in
100:subset
65:. All
59:origin
57:. The
8480:Minor
8465:Block
8403:Basis
8202:(PDF)
8133:Wiley
8057:Wiley
7450:flats
7433:Notes
6678:,...,
6539:Input
6363:Input
6283:,...,
6160:Input
6100:Input
5991:with
5954:Input
5880:Input
4738:set {
4129:basis
4045:for (
3778:image
3676:where
3356:have
2870:is a
2866:, if
1398:with
1110:of a
1051:In a
942:from
883:, so
772:, so
566:) of
456:, if
267:(the
161:over
157:is a
141:is a
124:field
94:is a
8635:Dual
8490:Rank
8274:2021
8246:2021
8215:2021
8190:2021
8153:ISBN
8137:LCCN
8108:ISBN
8079:ISBN
8061:ISBN
8031:ISBN
8006:ISBN
7983:ISBN
7961:ISBN
7936:ISBN
7909:ISBN
7548:for
7483:rank
7452:and
6697:) ×
6635:Let
6577:An (
6465:and
6039:and
5758:and
5623:and
5579:and
5432:and
5281:and
5225:and
4764:and
4744:and
4708:and
4682:and
4674:and
4650:Let
4595:and
4587:and
4571:and
4567:Let
4515:and
4448:The
4150:Let
3829:and
3391:The
3178:Span
3164:span
3051:span
1263:zero
1255:dual
1131:zero
1116:flat
1096:span
981:) ⊂
787:Let
730:and
663:and
624:and
603:Let
530:too.
475:, 0)
423:, 0)
356:, 0)
337:and
335:, 0)
308:and
197:and
86:, a
8166:Web
6568:of
6469:of
6437:= 1
6365:An
6189:of
6102:An
5983:of
5882:An
5855:or
5835:).
5603:of
5510:dim
5485:dim
5467:dim
5325:of
5257:If
5227:sum
5190:dim
5172:dim
5148:dim
5038:dim
5020:dim
5002:dim
4978:dim
4940:dim
4922:dim
4898:dim
4883:dim
4871:dim
4865:max
4770:sum
4760:If
4756:Sum
4503:In
4199:and
3911:if
2807:and
1984:of
1928:and
1281:):
1242:of
1078:).
946:to
918:.)
839:= (
791:= (
686:= (
629:= (
608:= (
517:,0)
485:= (
461:= (
452:in
444:in
367:= (
342:= (
321:= (
312:in
275:of
145:of
118:If
90:or
78:In
8733::
8264:.
8236:.
8204:.
8179:.
8175:.
8135:,
8059:,
8029:.
8025:.
8004:,
7959:.
7934:,
7907:.
7557:,
7543:≠
6720:is
6693:−
6667:−
6643:,
6608:,
6593:.
6548:,
6529:.
6453:.
6404:,
6353:.
6318:,
6236:=
6215:,
6198:∈
6169:,
6146:.
6082:∈
6075:}
6071:,
5963:,
5932:.
5904:.
5716:.
4752:.
4558:.
4491:.
4127:A
3888:.
3880:,
3873:,
3863:xz
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3792:.
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3384:.
3166::
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1514:11
1409::
1253:A
1102:,
1016:.
966:.
878:cp
876:=
871:cp
862:=
848:cp
846:,
841:cp
816:=
798:,
776:+
765:+
758:=
751:+
737:=
723:=
700:,
682:+
670:=
656:=
636:,
615:,
586:.
574:=
562:,
512:cu
510:,
505:cu
499:,
494:cu
492:,
487:cu
468:,
429:+
409:,
381:,
363:+
349:,
328:,
295:.
263:=
231:.
221:βw
219:+
214:αw
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185:,
153:,
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7217:=
7208:3
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7183:A
7157:4
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7112:c
7101:4
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7088:+
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6927:]
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6443:.
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6399:x
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6377:.
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6316:1
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6301:A
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6112:A
6108:n
6104:m
6087:.
6084:S
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6057:b
6055:{
6048:A
6043:.
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5918:.
5916:A
5909:A
5902:A
5894:.
5892:A
5888:n
5884:m
5819:}
5816:0
5813:{
5801:N
5794:N
5774:N
5723:(
5704:N
5684:N
5681:=
5672:)
5662:N
5658:(
5632:N
5611:N
5587:V
5567:}
5564:0
5561:{
5558:=
5549:N
5542:N
5522:)
5519:V
5516:(
5507:=
5504:)
5495:N
5491:(
5482:+
5479:)
5476:N
5473:(
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5400:N
5380:V
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5199:W
5196:(
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5178:(
5169:=
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5163:W
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5154:(
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5097:W
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5059:.
5056:)
5053:W
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5044:(
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5008:(
4999:=
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4993:W
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4984:(
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4952:)
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4889:W
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4839:}
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4821:U
4814:u
4806:w
4802:+
4798:u
4793:{
4789:=
4786:W
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4766:W
4762:U
4750:V
4746:V
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4740:0
4734:V
4728:.
4726:W
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4718:0
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4706:U
4701:.
4699:W
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4691:v
4688:c
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4660:W
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4647:.
4645:W
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4637:w
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4505:R
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4477:W
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4428:S
4408:.
4405:)
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4399:,
4396:5
4393:,
4390:0
4387:,
4384:0
4381:(
4376:2
4372:t
4368:+
4365:)
4362:0
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4350:1
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4332:t
4328:=
4325:)
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4267:(
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4233:.
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4217:=
4212:3
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4178:=
4173:1
4169:x
4156:R
4152:S
4137:S
4133:S
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4101:k
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3855:k
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3837:.
3835:R
3831:v
3827:u
3790:A
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3774:x
3756:.
3751:]
3741:2
3735:1
3724:4
3713:5
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3691:[
3687:=
3684:A
3667:t
3663:A
3660:=
3656:x
3606:.
3602:)
3599:1
3596:,
3593:0
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3502:(
3462:.
3457:2
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3409:x
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2815:z
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2764:y
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2684:.
2680:}
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2459:+
2453:+
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2265:1
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2169:{
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2118:.
2113:]
2107:5
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2082:3
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2066:=
2063:A
2033:.
2029:0
2025:=
2021:x
2017:A
1990:n
1986:A
1978:K
1974:n
1960:0
1957:=
1954:z
1951:5
1948:+
1945:y
1942:4
1936:x
1933:2
1923:0
1920:=
1917:z
1914:2
1911:+
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1873:(
1857:.
1853:}
1844:0
1841:=
1832:n
1828:x
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1819:m
1815:a
1807:+
1801:+
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1779:m
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1767:+
1758:1
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1748:1
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1741:a
1713:0
1710:=
1701:n
1697:x
1691:n
1688:2
1684:a
1676:+
1670:+
1661:2
1657:x
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1639:+
1630:1
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1616:a
1607:0
1604:=
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1309:F
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