2705:
2174:
1878:
2700:{\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\}.}
1423:
8459:
51:
46:
40:
35:
1873:{\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\}.}
4511:
601:
8723:
3833:
7184:
6982:
3346:
3055:
7390:
6160:
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.
1384:
1240:
4051:
7179:{\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}}}
4864:
2868:
3183:
3778:
2139:
2905:
4429:
2267:
5950:
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
4976:
4254:
3168:
3628:
7205:
4980:
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:
1981:
1288: = 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
5080:
6750:
1516:
72:(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
3928:
1298:
1154:
5543:
4150:. 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
3483:
5223:
4786:
2054:
5840:
5705:
5588:
3640:-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).
8012:
6671:
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.
5659:
5468:
5381:
2728:
5147:
5118:
3341:{\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\}.}
1039:
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
6534:
6949:
5752:
6508:
1148:
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:
5795:
5725:
5632:
5608:
5441:
5421:
5401:
5354:
5330:
5310:
5286:
3661:
4871:
3060:
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
3050:{\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}}.}
2069:
1905:
4273:
5120:. 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.
4984:
4174:
4264:. In particular, every vector that satisfies the above equations can be written uniquely as a linear combination of the two basis vectors:
3100:
7385:{\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}}}
3508:
8272:
5154:
8317:
8650:
8708:
4046:{\displaystyle t_{1}\mathbf {v} _{1}+\cdots +t_{k}\mathbf {v} _{k}\;\neq \;u_{1}\mathbf {v} _{1}+\cdots +u_{k}\mathbf {v} _{k}}
1406:
1098:
914:
8122:
8112:
8093:
8045:
7997:
7923:
5877:
Row reduction does not change the span of the row vectors, i.e. the reduced matrix has the same row space as the original.
4868:
For example, the sum of two lines is the plane that contains them both. The dimension of the sum satisfies the inequality
3706:
1379:{\displaystyle \exists c\in K:\mathbf {F} '=c\mathbf {F} {\text{ (or }}\mathbf {F} ={\frac {1}{c}}\mathbf {F} '{\text{)}}}
1235:{\displaystyle \exists c\in K:\mathbf {v} '=c\mathbf {v} {\text{ (or }}\mathbf {v} ={\frac {1}{c}}\mathbf {v} '{\text{)}}}
1059:. The equivalent definition states that it is also equivalent to consider linear combinations of two elements at a time.
8020:
3655:
A system of linear parametric equations in a finite-dimensional space can also be written as a single matrix equation:
7702:
Basic facts about
Hilbert Space — class notes from Colorado State University on Partial Differential Equations (M645).
5473:
3415:
8167:
8075:
7975:
7950:
4445:
passing through the points (0, 0, 0, 0), (2, 1, 0, 0), and (0, 0, 5, 1).
3806:
The row space of a matrix is the subspace spanned by its row vectors. The row space is interesting because it is the
2017:
In a finite-dimensional space, a homogeneous system of linear equations can be written as a single matrix equation:
8698:
7967:
7915:
8660:
8596:
7427:
5149:
is the same as the sum of subspaces, but may be shortened because the dimension of the trivial subspace is zero.
2023:
17:
6450:
and the remaining free variables are zero. The resulting collection of vectors is a basis for the null space of
5800:
8037:
7467:. In the case of vector spaces over the reals, linear subspaces, flats, and affine subspaces are also called
7422:
5664:
5548:
245:
As a corollary, all vector spaces are equipped with at least two (possibly different) linear subspaces: the
8757:
8438:
8310:
3872:
spanned by the three vectors (1, 0, 0), (0, 0, 1), and (2, 0, 3) is just the
8543:
8393:
8448:
8342:
7210:
6987:
4859:{\displaystyle U+W=\left\{\mathbf {u} +\mathbf {w} \colon \mathbf {u} \in U,\mathbf {w} \in W\right\}.}
8688:
8337:
8217:
5940:
4535:
8752:
8680:
8563:
7942:
7489:
5867:
5859:
5842:. As a result, this operation does not turn the lattice of subspaces into a Boolean algebra (nor a
3827:
2863:{\displaystyle x=2t_{1}+3t_{2},\;\;\;\;y=5t_{1}-4t_{2},\;\;\;\;{\text{and}}\;\;\;\;z=-t_{1}+2t_{2}}
1063:
73:
3495:-plane is spanned by the vectors (1, 0, 0) and (0, 0, 1). Every vector in the
2899:
In linear algebra, the system of parametric equations can be written as a single vector equation:
1003:
Again, we know from calculus that the product of a continuous function and a number is continuous.
8747:
8726:
8655:
8433:
8303:
8276:
8248:
5770:
1281:
1017:
5637:
5446:
5359:
8490:
8423:
8413:
5766:
5126:
5097:
3823:
1250:
1125:. Geometrically (especially over the field of real numbers and its subfields), a subspace is a
950:
118:
8505:
8500:
8495:
8428:
8373:
6513:
5333:
5090:
4260:
Then the vectors (2, 1, 0, 0) and (0, 0, 5, 1) are a basis for
3807:
3650:
1265:
1137:
1036:
279:
6356:
If the final column of the reduced row echelon form contains a pivot, then the input vector
8515:
8480:
8467:
8358:
8143:
8137:
8067:
7493:
6740:
6537:
5737:
2165:
1122:
1102:
69:
6317:
Express the final column of the reduced echelon form as a linear combination of the first
8:
8693:
8573:
8548:
8398:
8209:
7937:
A First Course In Linear
Algebra: with Optional Introduction to Groups, Rings, and Fields
6487:
5289:
3819:
3788:
3773:{\displaystyle \mathbf {x} =A\mathbf {t} \;\;\;\;{\text{where}}\;\;\;\;A=\left{\text{.}}}
2063:
of the matrix. For example, the subspace described above is the null space of the matrix
1390:
1024:
966:
134:
8403:
8029:
7935:
7417:
6461:
5780:
5710:
5617:
5593:
5426:
5406:
5386:
5339:
5315:
5295:
5271:
4460:
1269:
1078:
8128:
5088:
when the only intersection between any pair of subspaces is the trivial subspace. The
3876:-plane, with each point on the plane described by infinitely many different values of
1035:
From the definition of vector spaces, it follows that subspaces are nonempty, and are
8601:
8558:
8485:
8378:
8183:
8163:
8147:
8118:
8089:
8071:
8060:
8041:
8016:
7993:
7971:
7946:
7919:
7437:
6430:
are free. Write equations for the dependent variables in terms of the free variables.
5245:
4748:
1393:. The following two subsections will present this latter description in details, and
1273:
1141:
1086:
246:
6147:. The corresponding columns of the original matrix are a basis for the column space.
8606:
8510:
8363:
6281:
6144:
5863:
5253:
1414:
1983:
is a one-dimensional subspace. More generally, that is to say that given a set of
83:(i.e. a 5 × 5 square) is pictured four times for a better visualization
8458:
8418:
8408:
7464:
7460:
7432:
7412:
6154:
5843:
5241:
1899:
1126:
559:
8670:
8591:
8326:
8268:
8240:
8055:
4518:, the intersection of two distinct two-dimensional subspaces is one-dimensional
1044:
94:
8244:
8187:
5260:, the greatest element, is an identity element of the intersection operation.
2134:{\displaystyle A={\begin{bmatrix}1&3&2\\2&-4&5\end{bmatrix}}.}
1012:
Keep the same field and vector space as before, but now consider the set Diff(
8741:
8703:
8626:
8586:
8553:
8533:
7907:
5855:
5762:
5249:
4464:
5773:, for example, orthogonal complements exist. However, these spaces may have
1101:, the subset of Euclidean space described by a system of homogeneous linear
8636:
8525:
8475:
8368:
7985:
5755:
3868:. However, there are exceptions to this rule. For example, the subspace of
2882:
1114:
169:
106:
59:
50:
45:
39:
34:
5880:
Row reduction does not affect the linear dependence of the column vectors.
5777:
that are orthogonal to themselves, and consequently there exist subspaces
4424:{\displaystyle (2t_{1},t_{1},5t_{2},t_{2})=t_{1}(2,1,0,0)+t_{2}(0,0,5,1).}
113:
of some larger vector space. A linear subspace is usually simply called a
8616:
8581:
8538:
8383:
7959:
6713:
matrix corresponding to this system is the desired matrix with nullspace
5774:
3783:
In this case, the subspace consists of all possible values of the vector
2894:
1246:
1106:
1082:
1020:. The same sort of argument as before shows that this is a subspace too.
1000:
We know from calculus that the sum of continuous functions is continuous.
287:
250:
90:
5661:. Applying orthogonal complements twice returns the original subspace:
4510:
8645:
8388:
5611:
4470:
A subspace cannot lie in any subspace of lesser dimension. If dim
2060:
2012:
1289:
1110:
1074:
1048:
600:
58:
One-dimensional subspaces in the two-dimensional vector space over the
6467:
5925:
The nonzero rows of the echelon form are a basis for the row space of
4971:{\displaystyle \max(\dim U,\dim W)\leq \dim(U+W)\leq \dim(U)+\dim(W).}
8443:
8192:
6407:
Using the reduced row echelon form, determine which of the variables
5936:
4249:{\displaystyle x_{1}=2x_{2}\;\;\;\;{\text{and}}\;\;\;\;x_{3}=5x_{4}.}
1118:
177:
3832:
3787:. In linear algebra, this subspace is known as the column space (or
3636:
Geometrically, this corresponds to the fact that every point on the
3163:{\displaystyle t_{1}\mathbf {v} _{1}+\cdots +t_{k}\mathbf {v} _{k}.}
1081:
subspace is always closed. The same is true for subspaces of finite
8611:
7472:
5758:) orthocomplemented lattice (although not a distributive lattice).
5731:
3623:{\displaystyle (t_{1},0,t_{2})=t_{1}(1,0,0)+t_{2}(0,0,1){\text{.}}}
1992:
8295:
8288:
8260:
7497:
8151:
8621:
253:
alone and the entire vector space itself. These are called the
110:
117:
when the context serves to distinguish it from other types of
1085:(i.e., subspaces determined by a finite number of continuous
3375:. Geometrically, the span is the flat through the origin in
7402:
are a basis for the null space of the corresponding matrix.
5874:
The reduced matrix has the same null space as the original.
3499:-plane can be written as a linear combination of these two:
1976:{\displaystyle x+3y+2z=0\quad {\text{and}}\quad 2x-4y+5z=0}
1397:
four subsections further describe the idea of linear span.
6064:
If the echelon form has a row of zeroes, then the vectors
3173:
The set of all possible linear combinations is called the
8117:, Society for Industrial and Applied Mathematics (SIAM),
6350:
entries in the final column of the reduced echelon form.)
5870:. Row reduction has the following important properties:
5075:{\displaystyle \dim(U+W)=\dim(U)+\dim(W)-\dim(U\cap W).}
4449:
1987:
independent functions, the dimension of the subspace in
1097:
Descriptions of subspaces include the solution set to a
6321:
columns. The coefficients used are the desired numbers
4146:
is a set of linearly independent vectors whose span is
2148:
can be described as the null space of some matrix (see
4441:
is two-dimensional. Geometrically, it is the plane in
3013:
2963:
2914:
2191:
2084:
2059:
The set of solutions to this equation is known as the
1440:
917:
will yield a subspace. (The equation in example I was
7509:
This definition is often stated differently: vectors
7500:, but some integers may equal to zero in some fields.
7208:
6985:
6743:
6516:
6490:
5803:
5783:
5740:
5713:
5667:
5640:
5620:
5596:
5551:
5476:
5449:
5429:
5409:
5389:
5362:
5342:
5318:
5298:
5274:
5157:
5129:
5100:
4987:
4874:
4789:
4276:
4177:
3931:
3813:
3664:
3511:
3418:
3186:
3103:
2908:
2731:
2177:
2072:
2026:
1908:
1426:
1301:
1157:
8028:
7804:
1245:
This idea is generalized for higher dimensions with
909:
In general, any subset of the real coordinate space
6468:
Basis for the sum and intersection of two subspaces
5854:Most algorithms for dealing with subspaces involve
5765:, some but not all of these results still hold. In
5545:. Moreover, no vector is orthogonal to itself, so
1272:(usually implemented as linear equations). One non-
921: = 0, and the equation in example II was
8059:
7934:
7933:Beauregard, Raymond A.; Fraleigh, John B. (1973),
7932:
7627:
7492:that the given integer matrix has the appropriate
7384:
7178:
6943:
6528:
6502:
5834:
5789:
5746:
5719:
5699:
5653:
5626:
5602:
5582:
5537:
5462:
5435:
5415:
5395:
5375:
5348:
5324:
5304:
5280:
5217:
5141:
5112:
5074:
4970:
4858:
4423:
4248:
4135:for a vector in the span are uniquely determined.
4045:
3772:
3622:
3477:
3340:
3162:
3049:
2862:
2699:
2133:
2048:
1975:
1872:
1378:
1234:
6143:Determine which columns of the echelon form have
5256:of the sum operation, and the identical subspace
3844:are a basis for this two-dimensional subspace of
3007:
2957:
2244:
2243:
2189:
2188:
1493:
1492:
1438:
1437:
1389:It is generalized for higher codimensions with a
8739:
7900:Elementary Linear Algebra (Applications Version)
5538:{\displaystyle \dim(N)+\dim(N^{\perp })=\dim(V)}
5094:is the sum of independent subspaces, written as
4875:
4467:on the set of all subspaces (of any dimension).
3478:{\displaystyle x=t_{1},\;\;\;y=0,\;\;\;z=t_{2}.}
1023:Examples that extend these themes are common in
5218:{\displaystyle \dim(U\oplus W)=\dim(U)+\dim(W)}
6440:, choose a vector in the null space for which
5754:), makes the lattice of subspaces a (possibly
2155:
1400:
8311:
3644:
3371:components, then their span is a subspace of
1136:A natural description of a 1-subspace is the
5829:
5823:
5577:
5571:
3379:-dimensional space determined by the points
3228:
3192:
2164:described by a system of homogeneous linear
6543:
6164:
6104:
2049:{\displaystyle A\mathbf {x} =\mathbf {0} .}
8318:
8304:
8114:Matrix Analysis and Applied Linear Algebra
6927:
6926:
6925:
6924:
6923:
6916:
6915:
6914:
6913:
6912:
6905:
6904:
6903:
6902:
6901:
6894:
6893:
6892:
6891:
6890:
6883:
6882:
6881:
6880:
6879:
5835:{\displaystyle N\cap N^{\perp }\neq \{0\}}
4216:
4215:
4214:
4213:
4207:
4206:
4205:
4204:
3989:
3985:
3730:
3729:
3693:
3692:
3691:
3690:
3684:
3683:
3682:
3681:
3455:
3454:
3453:
3440:
3439:
3438:
2946:
2942:
2824:
2823:
2822:
2821:
2815:
2814:
2813:
2812:
2773:
2772:
2771:
2770:
2617:
2607:
2577:
2573:
2543:
2539:
2510:
2509:
2472:
2462:
2435:
2431:
2404:
2400:
2355:
2345:
2318:
2314:
2287:
2283:
1850:
1820:
1810:
1780:
1776:
1719:
1689:
1679:
1652:
1648:
1613:
1583:
1573:
1546:
1542:
1030:
820:, that is, a point in the plane such that
8207:
8182:
7699:
7687:
6935:
6682:linear equations involving the variables
6367:
5263:
3759:
2006:
660:, that is, points in the plane such that
8054:
8034:A (Terse) Introduction to Linear Algebra
8006:
7984:
7780:
7723:
7675:
7651:
7639:
6674:This results in a homogeneous system of
5884:
5240:make the set of all subspaces a bounded
4509:
3831:
599:
7992:(4th ed.). Orthogonal Publishing.
6346:. (These should be precisely the first
6153:See the article on column space for an
5700:{\displaystyle (N^{\perp })^{\perp }=N}
5470:satisfy the complementary relationship
5227:
1133:-space that passes through the origin.
14:
8740:
8709:Comparison of linear algebra libraries
8267:
8239:
8135:
7958:
7828:
7816:
7768:
7747:
7735:
7711:
7663:
7603:
7478:
6088:are linearly dependent, and therefore
5958:
5583:{\displaystyle N\cap N^{\perp }=\{0\}}
5423:is a subspace, then the dimensions of
3410:can be parameterized by the equations
1407:homogeneous system of linear equations
1099:homogeneous system of linear equations
915:homogeneous system of linear equations
8299:
8160:Linear Algebra: A Modern Introduction
8157:
8110:
8106:(7th ed.), Pearson Prentice Hall
7906:
7897:
7876:
7864:
7852:
7840:
7792:
7615:
7584:. The two definitions are equivalent.
6639:Use elementary row operations to put
6460:See the article on null space for an
6400:Use elementary row operations to put
6310:Use elementary row operations to put
6136:Use elementary row operations to put
6057:Use elementary row operations to put
5918:Use elementary row operations to put
5862:to a matrix, until it reaches either
4450:Operations and relations on subspaces
2710:For example, the set of all vectors (
8111:Meyer, Carl D. (February 15, 2001),
8101:
7471:for emphasizing that there are also
2885:(such as real or rational numbers).
1882:For example, the set of all vectors
1394:
1109:of a collection of vectors, and the
554:again, but now let the vector space
8273:"The big picture of linear algebra"
8086:Linear Algebra and Its Applications
8083:
7902:(9th ed.), Wiley International
7459:is sometimes used for referring to
7192:It follows that the row vectors of
6727:If the reduced row echelon form of
4116:are linearly independent, then the
24:
8325:
7805:Katznelson & Katznelson (2008)
7753:
7398:In particular, the row vectors of
5858:. This is the process of applying
5741:
3814:Independence, basis, and dimension
3795:. It is precisely the subspace of
2888:
1302:
1158:
25:
8769:
8233:
8210:"Basic facts about Hilbert Space"
8084:Lay, David C. (August 22, 2005),
8032:; Katznelson, Yonatan R. (2008).
5233:
3799:spanned by the column vectors of
2873:is a two-dimensional subspace of
2722:) parameterized by the equations
8722:
8721:
8699:Basic Linear Algebra Subprograms
8457:
8245:"The four fundamental subspaces"
8139:Linear Algebra and Matrix Theory
8104:Linear Algebra With Applications
8088:(3rd ed.), Addison Wesley,
8062:Advanced Engineering Mathematics
7964:Finite-Dimensional Vector Spaces
7162:
7144:
7122:
7106:
7088:
7073:
7051:
7035:
7017:
6992:
6130:A basis for the column space of
4838:
4824:
4816:
4808:
4033:
4002:
3975:
3944:
3677:
3666:
3282:
3251:
3218:
3197:
3147:
3116:
2039:
2031:
1363:
1344:
1334:
1319:
1219:
1200:
1190:
1175:
327:, then they can be expressed as
298:whose last component is 0. Then
294:to be the set of all vectors in
49:
44:
38:
33:
8597:Seven-dimensional cross product
8279:from the original on 2021-12-11
8251:from the original on 2021-12-11
7870:
7858:
7846:
7834:
7822:
7810:
7798:
7786:
7774:
7762:
7759:Vector space related operators.
7741:
7729:
7717:
7705:
7693:
7628:Beauregard & Fraleigh (1973
7503:
7428:Quotient space (linear algebra)
4505:
4151:
3810:of the null space (see below).
2149:
1942:
1936:
1744:
1413:variables is a subspace in the
1092:
941:, but now let the vector space
27:In mathematics, vector subspace
8208:DuChateau, Paul (5 Sep 2002).
7726:p. 100, ch. 2, Definition 2.13
7681:
7678:p. 100, ch. 2, Definition 2.13
7669:
7657:
7645:
7633:
7621:
7609:
7597:
7449:
6643:into reduced row echelon form.
6394:A basis for the null space of
6314:into reduced row echelon form.
5730:This operation, understood as
5682:
5668:
5532:
5526:
5514:
5501:
5489:
5483:
5212:
5206:
5194:
5188:
5176:
5164:
5123:The dimension of a direct sum
5066:
5054:
5042:
5036:
5024:
5018:
5006:
4994:
4962:
4956:
4944:
4938:
4926:
4914:
4902:
4878:
4415:
4391:
4375:
4351:
4335:
4277:
3612:
3594:
3578:
3560:
3544:
3512:
1999:, the composite matrix of the
965:) be the subset consisting of
932:
13:
1:
8162:(2nd ed.), Brooks/Cole,
8038:American Mathematical Society
7423:Multilinear subspace learning
6303:, with the last column being
5946:If we instead put the matrix
5911:A basis for the row space of
5849:
1991:will be the dimension of the
1268:description is provided with
1257:-spaces specified by sets of
1007:
545:
124:
8439:Eigenvalues and eigenvectors
8013:Blaisdell Publishing Company
7590:
7525:are linearly independent if
6536:can be calculated using the
6404:in reduced row echelon form.
6026: + 1) ×
4727:belongs to both sets. Thus,
4557: is an element of both
4463:binary relation specifies a
4454:
1047:every linear combination of
265:
7:
7891:
7406:
6600:matrix whose null space is
6034:whose rows are the vectors
2156:Linear parametric equations
1902:) satisfying the equations
1401:Systems of linear equations
1261:vectors are not so simple.
986:We know from calculus that
937:Again take the field to be
260:
160:if under the operations of
93:, and more specifically in
10:
8774:
8142:(2nd ed.), New York:
8066:(3rd ed.), New York:
7886:
7496:in it. All fields include
5654:{\displaystyle N^{\perp }}
5463:{\displaystyle N^{\perp }}
5403:is finite-dimensional and
5383:, is again a subspace. If
5376:{\displaystyle N^{\perp }}
3860:parameters (or spanned by
3852:In general, a subspace of
3817:
3648:
3645:Column space and row space
3094:is any vector of the form
2892:
2010:
1073:need not be topologically
8717:
8679:
8635:
8572:
8524:
8466:
8455:
8351:
8333:
8218:Colorado State University
7912:Linear Algebra Done Right
7488:can be any field of such
5860:elementary row operations
5142:{\displaystyle U\oplus W}
5113:{\displaystyle U\oplus W}
4461:set-theoretical inclusion
4169:defined by the equations
569:to be the set of points (
133:is a vector space over a
8184:Weisstein, Eric Wolfgang
8136:Nering, Evar D. (1970),
8102:Leon, Steven J. (2006),
8007:Herstein, I. N. (1964),
7943:Houghton Mifflin Company
7443:
6957:then the column vectors
6544:Equations for a subspace
6165:Coordinates for a vector
6105:Basis for a column space
5868:reduced row echelon form
5771:symplectic vector spaces
5237:
4759:itself are subspaces of
4723:are vector spaces, then
4565:} is also a subspace of
3828:Dimension (vector space)
3064:the resulting subspace.
1405:The solution set to any
1064:topological vector space
1018:differentiable functions
184:is a linear subspace of
6529:{\displaystyle U\cap W}
6433:For each free variable
6041:, ... ,
5767:pseudo-Euclidean spaces
4743:For every vector space
4478:, a finite number, and
4091:, ... ,
4070:, ... ,
3909:, ... ,
3864:vectors) has dimension
3386:, ... ,
3358:, ... ,
3085:, ... ,
1031:Properties of subspaces
8424:Row and column vectors
8176:
7898:Anton, Howard (2005),
7386:
7180:
6945:
6530:
6504:
6368:Basis for a null space
6140:into row echelon form.
6061:into row echelon form.
5922:into row echelon form.
5836:
5791:
5748:
5721:
5701:
5655:
5628:
5604:
5584:
5539:
5464:
5437:
5417:
5397:
5377:
5350:
5326:
5306:
5282:
5264:Orthogonal complements
5219:
5143:
5114:
5084:A set of subspaces is
5076:
4972:
4860:
4766:
4519:
4425:
4250:
4047:
3849:
3824:Basis (linear algebra)
3774:
3624:
3479:
3342:
3164:
3051:
2864:
2701:
2135:
2050:
2007:Null space of a matrix
1977:
1874:
1380:
1236:
605:
604:Example II Illustrated
8429:Row and column spaces
8374:Scalar multiplication
8158:Poole, David (2006),
7387:
7196:satisfy the equations
7181:
6973:satisfy the equations
6946:
6944:{\displaystyle \left}
6531:
6510:and the intersection
6505:
6484:, a basis of the sum
5885:Basis for a row space
5837:
5792:
5761:In spaces with other
5749:
5747:{\displaystyle \neg }
5722:
5702:
5656:
5629:
5605:
5585:
5540:
5465:
5438:
5418:
5398:
5378:
5351:
5334:orthogonal complement
5327:
5307:
5283:
5220:
5144:
5115:
5077:
4973:
4861:
4779:are subspaces, their
4513:
4426:
4251:
4048:
3835:
3808:orthogonal complement
3775:
3651:Row and column spaces
3625:
3480:
3343:
3165:
3052:
2865:
2702:
2136:
2051:
1978:
1875:
1381:
1237:
1138:scalar multiplication
913:that is defined by a
603:
280:real coordinate space
257:of the vector space.
8564:Gram–Schmidt process
8516:Gaussian elimination
7960:Halmos, Paul Richard
7783:p. 148, ch. 2, §4.10
7206:
6983:
6741:
6538:Zassenhaus algorithm
6514:
6488:
6472:Given two subspaces
5801:
5781:
5738:
5711:
5665:
5638:
5618:
5594:
5549:
5474:
5447:
5427:
5407:
5387:
5360:
5340:
5316:
5296:
5272:
5228:Lattice of subspaces
5155:
5127:
5098:
4985:
4872:
4787:
4630:is a subspace, then
4614:is a subspace, then
4274:
4175:
3929:
3920:linearly independent
3902:In general, vectors
3662:
3509:
3416:
3184:
3101:
2906:
2729:
2651: for some
2175:
2166:parametric equations
2070:
2024:
1906:
1424:
1299:
1155:
1103:parametric equations
967:continuous functions
270:In the vector space
8758:Functional analysis
8694:Numerical stability
8574:Multilinear algebra
8549:Inner product space
8399:Linear independence
8030:Katznelson, Yitzhak
6503:{\displaystyle U+W}
6012:Determines whether
5959:Subspace membership
5935:See the article on
5707:for every subspace
5290:inner product space
4626:. Similarly, since
4165:be the subspace of
3820:Linear independence
3491:As a subspace, the
1391:system of equations
1249:, but criteria for
1025:functional analysis
973:) is a subspace of
816:) be an element of
8404:Linear combination
7908:Axler, Sheldon Jay
7807:pp. 10-11, § 1.2.5
7582:) ≠ (0, 0, ..., 0)
7418:Invariant subspace
7382:
7380:
7176:
7174:
6941:
6933:
6526:
6500:
6287:whose columns are
5832:
5787:
5744:
5717:
5697:
5651:
5624:
5600:
5580:
5535:
5460:
5433:
5413:
5393:
5373:
5346:
5322:
5302:
5278:
5215:
5139:
5110:
5072:
4968:
4856:
4677:be a scalar. Then
4530:of a vector space
4520:
4421:
4246:
4043:
3850:
3770:
3757:
3620:
3475:
3338:
3160:
3069:linear combination
3047:
3038:
2988:
2936:
2860:
2697:
2647:
2241:
2144:Every subspace of
2131:
2122:
2046:
1973:
1870:
1860:
1490:
1376:
1276:linear functional
1270:linear functionals
1232:
1087:linear functionals
1079:finite-dimensional
606:
249:consisting of the
222:, it follows that
176:. Equivalently, a
8735:
8734:
8602:Geometric algebra
8559:Kronecker product
8394:Linear projection
8379:Vector projection
8124:978-0-89871-454-8
8095:978-0-321-28713-7
8047:978-0-8218-4419-9
8009:Topics In Algebra
7999:978-1-944325-11-4
7925:978-3-319-11079-0
7438:Subspace topology
6575:} for a subspace
6196:} for a subspace
6016:is an element of
5990:} for a subspace
5790:{\displaystyle N}
5720:{\displaystyle N}
5627:{\displaystyle N}
5603:{\displaystyle V}
5436:{\displaystyle N}
5416:{\displaystyle N}
5396:{\displaystyle V}
5349:{\displaystyle N}
5325:{\displaystyle V}
5305:{\displaystyle N}
5281:{\displaystyle V}
4211:
4154:below for more).
3768:
3688:
3618:
3190:
2819:
2652:
2152:below for more).
2150:§ Algorithms
1940:
1374:
1359:
1341:
1230:
1215:
1197:
1051:many elements of
901:is an element of
791:is an element of
593:is a subspace of
550:Let the field be
537:is an element of
444:is an element of
302:is a subspace of
255:trivial subspaces
247:zero vector space
87:
86:
16:(Redirected from
8765:
8725:
8724:
8607:Exterior algebra
8544:Hadamard product
8461:
8449:Linear equations
8320:
8313:
8306:
8297:
8296:
8292:
8286:
8284:
8264:
8258:
8256:
8229:
8227:
8225:
8214:
8204:
8202:
8200:
8172:
8154:
8132:
8131:on March 1, 2001
8127:, archived from
8107:
8098:
8080:
8065:
8051:
8025:
8003:
7981:
7966:(2nd ed.).
7955:
7940:
7929:
7914:(3rd ed.).
7903:
7880:
7874:
7868:
7862:
7856:
7850:
7844:
7838:
7832:
7826:
7820:
7814:
7808:
7802:
7796:
7790:
7784:
7778:
7772:
7766:
7760:
7757:
7751:
7745:
7739:
7733:
7727:
7721:
7715:
7709:
7703:
7700:DuChateau (2002)
7697:
7691:
7688:MathWorld (2021)
7685:
7679:
7673:
7667:
7661:
7655:
7649:
7643:
7637:
7631:
7625:
7619:
7613:
7607:
7601:
7585:
7583:
7558:
7507:
7501:
7482:
7476:
7469:linear manifolds
7465:affine subspaces
7453:
7391:
7389:
7388:
7383:
7381:
7374:
7373:
7358:
7357:
7338:
7337:
7324:
7323:
7308:
7307:
7295:
7294:
7275:
7274:
7261:
7260:
7245:
7244:
7222:
7221:
7185:
7183:
7182:
7177:
7175:
7171:
7170:
7165:
7153:
7152:
7147:
7131:
7130:
7125:
7115:
7114:
7109:
7097:
7096:
7091:
7082:
7081:
7076:
7060:
7059:
7054:
7044:
7043:
7038:
7026:
7025:
7020:
7001:
7000:
6995:
6972:
6950:
6948:
6947:
6942:
6940:
6936:
6934:
6921:
6910:
6899:
6888:
6877:
6861:
6855:
6849:
6843:
6837:
6824:
6815:
6809:
6803:
6797:
6784:
6778:
6772:
6763:
6757:
6712:
6670:
6635:
6607:Create a matrix
6535:
6533:
6532:
6527:
6509:
6507:
6506:
6501:
6483:
6479:
6475:
6449:
6429:
6360:does not lie in
6345:
6282:augmented matrix
6277:
6213:
6097:
6087:
5864:row echelon form
5841:
5839:
5838:
5833:
5819:
5818:
5796:
5794:
5793:
5788:
5753:
5751:
5750:
5745:
5726:
5724:
5723:
5718:
5706:
5704:
5703:
5698:
5690:
5689:
5680:
5679:
5660:
5658:
5657:
5652:
5650:
5649:
5633:
5631:
5630:
5625:
5609:
5607:
5606:
5601:
5589:
5587:
5586:
5581:
5567:
5566:
5544:
5542:
5541:
5536:
5513:
5512:
5469:
5467:
5466:
5461:
5459:
5458:
5442:
5440:
5439:
5434:
5422:
5420:
5419:
5414:
5402:
5400:
5399:
5394:
5382:
5380:
5379:
5374:
5372:
5371:
5355:
5353:
5352:
5347:
5331:
5329:
5328:
5323:
5311:
5309:
5308:
5303:
5287:
5285:
5284:
5279:
5254:identity element
5224:
5222:
5221:
5216:
5148:
5146:
5145:
5140:
5119:
5117:
5116:
5111:
5081:
5079:
5078:
5073:
4977:
4975:
4974:
4969:
4865:
4863:
4862:
4857:
4852:
4848:
4841:
4827:
4819:
4811:
4783:is the subspace
4704:belongs to both
4681:belongs to both
4522:Given subspaces
4486:, then dim
4430:
4428:
4427:
4422:
4390:
4389:
4350:
4349:
4334:
4333:
4321:
4320:
4305:
4304:
4292:
4291:
4255:
4253:
4252:
4247:
4242:
4241:
4226:
4225:
4212:
4209:
4203:
4202:
4187:
4186:
4134:
4115:
4052:
4050:
4049:
4044:
4042:
4041:
4036:
4030:
4029:
4011:
4010:
4005:
3999:
3998:
3984:
3983:
3978:
3972:
3971:
3953:
3952:
3947:
3941:
3940:
3898:
3791:) of the matrix
3779:
3777:
3776:
3771:
3769:
3766:
3764:
3760:
3758:
3755:
3749:
3738:
3727:
3719:
3713:
3689:
3686:
3680:
3669:
3629:
3627:
3626:
3621:
3619:
3616:
3593:
3592:
3559:
3558:
3543:
3542:
3524:
3523:
3484:
3482:
3481:
3476:
3471:
3470:
3434:
3433:
3347:
3345:
3344:
3339:
3334:
3330:
3323:
3322:
3304:
3303:
3291:
3290:
3285:
3279:
3278:
3260:
3259:
3254:
3248:
3247:
3227:
3226:
3221:
3206:
3205:
3200:
3191:
3188:
3169:
3167:
3166:
3161:
3156:
3155:
3150:
3144:
3143:
3125:
3124:
3119:
3113:
3112:
3056:
3054:
3053:
3048:
3043:
3042:
3006:
3005:
2993:
2992:
2956:
2955:
2941:
2940:
2869:
2867:
2866:
2861:
2859:
2858:
2843:
2842:
2820:
2817:
2808:
2807:
2792:
2791:
2766:
2765:
2750:
2749:
2706:
2704:
2703:
2698:
2693:
2689:
2682:
2681:
2663:
2662:
2653:
2650:
2648:
2645:
2643:
2642:
2633:
2632:
2619:
2605:
2603:
2602:
2593:
2592:
2579:
2571:
2569:
2568:
2559:
2558:
2545:
2537:
2535:
2534:
2522:
2521:
2520:
2519:
2518:
2517:
2516:
2515:
2514:
2513:
2512:
2504:
2503:
2500:
2498:
2497:
2488:
2487:
2474:
2460:
2458:
2457:
2448:
2447:
2437:
2429:
2427:
2426:
2417:
2416:
2406:
2398:
2396:
2395:
2383:
2381:
2380:
2371:
2370:
2357:
2343:
2341:
2340:
2331:
2330:
2320:
2312:
2310:
2309:
2300:
2299:
2289:
2281:
2279:
2278:
2262:
2261:
2249:
2245:
2242:
2238:
2237:
2217:
2216:
2203:
2202:
2140:
2138:
2137:
2132:
2127:
2126:
2055:
2053:
2052:
2047:
2042:
2034:
1982:
1980:
1979:
1974:
1941:
1938:
1900:rational numbers
1897:
1879:
1877:
1876:
1871:
1866:
1862:
1861:
1858:
1848:
1846:
1845:
1836:
1835:
1822:
1808:
1806:
1805:
1796:
1795:
1782:
1774:
1772:
1771:
1762:
1761:
1746:
1739:
1738:
1737:
1736:
1735:
1734:
1733:
1732:
1731:
1730:
1727:
1717:
1715:
1714:
1705:
1704:
1691:
1677:
1675:
1674:
1665:
1664:
1654:
1646:
1644:
1643:
1634:
1633:
1621:
1611:
1609:
1608:
1599:
1598:
1585:
1571:
1569:
1568:
1559:
1558:
1548:
1540:
1538:
1537:
1528:
1527:
1511:
1510:
1498:
1494:
1491:
1487:
1486:
1466:
1465:
1452:
1451:
1415:coordinate space
1385:
1383:
1382:
1377:
1375:
1372:
1370:
1366:
1360:
1352:
1347:
1342:
1339:
1337:
1326:
1322:
1241:
1239:
1238:
1233:
1231:
1228:
1226:
1222:
1216:
1208:
1203:
1198:
1195:
1193:
1182:
1178:
1055:also belongs to
996:
865:
726:
655:
634:
529:
487:
435:
368:
347:
237:
218:are elements of
217:
204:are elements of
203:
53:
48:
42:
37:
30:
29:
21:
8773:
8772:
8768:
8767:
8766:
8764:
8763:
8762:
8753:Operator theory
8738:
8737:
8736:
8731:
8713:
8675:
8631:
8568:
8520:
8462:
8453:
8419:Change of basis
8409:Multilinear map
8347:
8329:
8324:
8282:
8280:
8269:Strang, Gilbert
8254:
8252:
8241:Strang, Gilbert
8236:
8223:
8221:
8212:
8198:
8196:
8179:
8170:
8125:
8096:
8078:
8056:Kreyszig, Erwin
8048:
8023:
8000:
7978:
7953:
7926:
7894:
7889:
7884:
7883:
7875:
7871:
7863:
7859:
7851:
7847:
7839:
7835:
7831:pp. 30-31, § 19
7827:
7823:
7819:pp. 28-29, § 18
7815:
7811:
7803:
7799:
7791:
7787:
7781:Hefferon (2020)
7779:
7775:
7767:
7763:
7758:
7754:
7746:
7742:
7734:
7730:
7724:Hefferon (2020)
7722:
7718:
7710:
7706:
7698:
7694:
7686:
7682:
7676:Hefferon (2020)
7674:
7670:
7662:
7658:
7650:
7646:
7638:
7634:
7626:
7622:
7614:
7610:
7606:pp. 16-17, § 10
7602:
7598:
7593:
7588:
7580:
7574:
7567:
7560:
7553:
7544:
7538:
7532:
7526:
7524:
7515:
7508:
7504:
7483:
7479:
7457:linear subspace
7454:
7450:
7446:
7433:Signal subspace
7413:Cyclic subspace
7409:
7379:
7378:
7369:
7365:
7353:
7349:
7339:
7333:
7329:
7326:
7325:
7319:
7315:
7303:
7299:
7290:
7286:
7276:
7270:
7266:
7263:
7262:
7256:
7252:
7240:
7236:
7223:
7217:
7213:
7209:
7207:
7204:
7203:
7173:
7172:
7166:
7161:
7160:
7148:
7143:
7142:
7132:
7126:
7121:
7120:
7117:
7116:
7110:
7105:
7104:
7092:
7087:
7086:
7077:
7072:
7071:
7061:
7055:
7050:
7049:
7046:
7045:
7039:
7034:
7033:
7021:
7016:
7015:
7002:
6996:
6991:
6990:
6986:
6984:
6981:
6980:
6971:
6964:
6958:
6932:
6931:
6920:
6909:
6898:
6887:
6876:
6870:
6869:
6860:
6854:
6848:
6842:
6836:
6830:
6829:
6823:
6814:
6808:
6802:
6796:
6790:
6789:
6783:
6777:
6771:
6762:
6756:
6749:
6748:
6744:
6742:
6739:
6738:
6699:
6697:
6688:
6669:
6660:
6653:
6647:
6634:
6625:
6618:
6612:
6611:whose rows are
6574:
6565:
6558:
6546:
6515:
6512:
6511:
6489:
6486:
6485:
6481:
6477:
6473:
6470:
6446:
6441:
6438:
6427:
6421:
6414:
6408:
6370:
6344:
6335:
6328:
6322:
6302:
6293:
6276:
6268:
6259:
6253:
6243:
6241:
6232:
6225:
6205:
6204:, and a vector
6195:
6186:
6179:
6167:
6107:
6089:
6081:
6072:
6065:
6049:
6040:
5998:, and a vector
5989:
5980:
5973:
5961:
5887:
5852:
5844:Heyting algebra
5814:
5810:
5802:
5799:
5798:
5782:
5779:
5778:
5739:
5736:
5735:
5712:
5709:
5708:
5685:
5681:
5675:
5671:
5666:
5663:
5662:
5645:
5641:
5639:
5636:
5635:
5619:
5616:
5615:
5595:
5592:
5591:
5562:
5558:
5550:
5547:
5546:
5508:
5504:
5475:
5472:
5471:
5454:
5450:
5448:
5445:
5444:
5428:
5425:
5424:
5408:
5405:
5404:
5388:
5385:
5384:
5367:
5363:
5361:
5358:
5357:
5341:
5338:
5337:
5317:
5314:
5313:
5312:is a subset of
5297:
5294:
5293:
5273:
5270:
5269:
5266:
5242:modular lattice
5232:The operations
5230:
5156:
5153:
5152:
5128:
5125:
5124:
5099:
5096:
5095:
4986:
4983:
4982:
4873:
4870:
4869:
4837:
4823:
4815:
4807:
4806:
4802:
4788:
4785:
4784:
4769:
4697:are subspaces,
4602:belong to both
4586:be elements of
4508:
4494:if and only if
4457:
4452:
4385:
4381:
4345:
4341:
4329:
4325:
4316:
4312:
4300:
4296:
4287:
4283:
4275:
4272:
4271:
4237:
4233:
4221:
4217:
4208:
4198:
4194:
4182:
4178:
4176:
4173:
4172:
4142:for a subspace
4132:
4126:
4120:
4114:
4105:
4099:
4096:
4090:
4083:
4075:
4069:
4062:
4037:
4032:
4031:
4025:
4021:
4006:
4001:
4000:
3994:
3990:
3979:
3974:
3973:
3967:
3963:
3948:
3943:
3942:
3936:
3932:
3930:
3927:
3926:
3917:
3908:
3897:
3890:
3883:
3877:
3830:
3818:Main articles:
3816:
3765:
3756:
3754:
3748:
3739:
3737:
3726:
3720:
3718:
3712:
3705:
3704:
3700:
3685:
3676:
3665:
3663:
3660:
3659:
3653:
3647:
3615:
3588:
3584:
3554:
3550:
3538:
3534:
3519:
3515:
3510:
3507:
3506:
3466:
3462:
3429:
3425:
3417:
3414:
3413:
3394:
3385:
3366:
3357:
3351:If the vectors
3318:
3314:
3299:
3295:
3286:
3281:
3280:
3274:
3270:
3255:
3250:
3249:
3243:
3239:
3238:
3234:
3222:
3217:
3216:
3201:
3196:
3195:
3187:
3185:
3182:
3181:
3151:
3146:
3145:
3139:
3135:
3120:
3115:
3114:
3108:
3104:
3102:
3099:
3098:
3093:
3084:
3077:
3037:
3036:
3030:
3029:
3020:
3019:
3009:
3008:
3001:
2997:
2987:
2986:
2977:
2976:
2970:
2969:
2959:
2958:
2951:
2947:
2935:
2934:
2928:
2927:
2921:
2920:
2910:
2909:
2907:
2904:
2903:
2897:
2891:
2889:Span of vectors
2854:
2850:
2838:
2834:
2816:
2803:
2799:
2787:
2783:
2761:
2757:
2745:
2741:
2730:
2727:
2726:
2677:
2673:
2658:
2654:
2649:
2646:
2644:
2638:
2634:
2625:
2621:
2618:
2604:
2598:
2594:
2585:
2581:
2578:
2570:
2564:
2560:
2551:
2547:
2544:
2536:
2530:
2526:
2523:
2511:
2501:
2499:
2493:
2489:
2480:
2476:
2473:
2459:
2453:
2449:
2443:
2439:
2436:
2428:
2422:
2418:
2412:
2408:
2405:
2397:
2391:
2387:
2384:
2382:
2376:
2372:
2363:
2359:
2356:
2342:
2336:
2332:
2326:
2322:
2319:
2311:
2305:
2301:
2295:
2291:
2288:
2280:
2274:
2270:
2266:
2257:
2253:
2240:
2239:
2233:
2229:
2226:
2225:
2219:
2218:
2212:
2208:
2205:
2204:
2198:
2194:
2190:
2187:
2183:
2182:
2178:
2176:
2173:
2172:
2168:is a subspace:
2158:
2121:
2120:
2115:
2107:
2101:
2100:
2095:
2090:
2080:
2079:
2071:
2068:
2067:
2038:
2030:
2025:
2022:
2021:
2015:
2009:
1937:
1907:
1904:
1903:
1883:
1859:
1857:
1847:
1841:
1837:
1828:
1824:
1821:
1807:
1801:
1797:
1788:
1784:
1781:
1773:
1767:
1763:
1754:
1750:
1747:
1745:
1728:
1726:
1716:
1710:
1706:
1697:
1693:
1690:
1676:
1670:
1666:
1660:
1656:
1653:
1645:
1639:
1635:
1629:
1625:
1622:
1620:
1610:
1604:
1600:
1591:
1587:
1584:
1570:
1564:
1560:
1554:
1550:
1547:
1539:
1533:
1529:
1523:
1519:
1515:
1506:
1502:
1489:
1488:
1482:
1478:
1475:
1474:
1468:
1467:
1461:
1457:
1454:
1453:
1447:
1443:
1439:
1436:
1432:
1431:
1427:
1425:
1422:
1421:
1403:
1371:
1362:
1361:
1351:
1343:
1340: (or
1338:
1333:
1318:
1317:
1300:
1297:
1296:
1227:
1218:
1217:
1207:
1199:
1196: (or
1194:
1189:
1174:
1173:
1156:
1153:
1152:
1095:
1033:
1010:
987:
935:
893:
886:
879:
872:
863:
856:
843:
838:be a scalar in
833:
826:
815:
808:
782:
775:
768:
761:
754:
747:
740:
733:
724:
717:
710:
703:
689:
687:
680:
673:
666:
656:be elements of
653:
646:
636:
632:
625:
615:
560:Cartesian plane
548:
527:
520:
509:
502:
489:
485:
478:
468:
433:
426:
419:
412:
405:
398:
391:
384:
370:
366:
359:
349:
345:
338:
328:
282:over the field
268:
263:
236:
229:
223:
209:
202:
195:
189:
154:linear subspace
144:is a subset of
127:
103:vector subspace
99:linear subspace
82:
67:
43:
28:
23:
22:
18:Vector subspace
15:
12:
11:
5:
8771:
8761:
8760:
8755:
8750:
8748:Linear algebra
8733:
8732:
8730:
8729:
8718:
8715:
8714:
8712:
8711:
8706:
8701:
8696:
8691:
8689:Floating-point
8685:
8683:
8677:
8676:
8674:
8673:
8671:Tensor product
8668:
8663:
8658:
8656:Function space
8653:
8648:
8642:
8640:
8633:
8632:
8630:
8629:
8624:
8619:
8614:
8609:
8604:
8599:
8594:
8592:Triple product
8589:
8584:
8578:
8576:
8570:
8569:
8567:
8566:
8561:
8556:
8551:
8546:
8541:
8536:
8530:
8528:
8522:
8521:
8519:
8518:
8513:
8508:
8506:Transformation
8503:
8498:
8496:Multiplication
8493:
8488:
8483:
8478:
8472:
8470:
8464:
8463:
8456:
8454:
8452:
8451:
8446:
8441:
8436:
8431:
8426:
8421:
8416:
8411:
8406:
8401:
8396:
8391:
8386:
8381:
8376:
8371:
8366:
8361:
8355:
8353:
8352:Basic concepts
8349:
8348:
8346:
8345:
8340:
8334:
8331:
8330:
8327:Linear algebra
8323:
8322:
8315:
8308:
8300:
8294:
8293:
8271:(5 May 2020).
8265:
8243:(7 May 2009).
8235:
8234:External links
8232:
8231:
8230:
8205:
8178:
8175:
8174:
8173:
8168:
8155:
8133:
8123:
8108:
8099:
8094:
8081:
8076:
8052:
8046:
8026:
8022:978-1114541016
8021:
8004:
7998:
7990:Linear Algebra
7982:
7976:
7956:
7951:
7930:
7924:
7904:
7893:
7890:
7888:
7885:
7882:
7881:
7879:p. 195, § 6.51
7869:
7867:p. 194, § 6.47
7857:
7855:p. 195, § 6.50
7845:
7843:p. 193, § 6.46
7833:
7821:
7809:
7797:
7785:
7773:
7761:
7752:
7740:
7728:
7716:
7704:
7692:
7680:
7668:
7656:
7654:, p. 200)
7652:Kreyszig (1972
7644:
7642:, p. 132)
7640:Herstein (1964
7632:
7630:, p. 176)
7620:
7618:, p. 155)
7608:
7595:
7594:
7592:
7589:
7587:
7586:
7578:
7572:
7565:
7549:
7542:
7536:
7530:
7520:
7513:
7502:
7490:characteristic
7477:
7447:
7445:
7442:
7441:
7440:
7435:
7430:
7425:
7420:
7415:
7408:
7405:
7404:
7403:
7395:
7394:
7393:
7392:
7377:
7372:
7368:
7364:
7361:
7356:
7352:
7348:
7345:
7342:
7340:
7336:
7332:
7328:
7327:
7322:
7318:
7314:
7311:
7306:
7302:
7298:
7293:
7289:
7285:
7282:
7279:
7277:
7273:
7269:
7265:
7264:
7259:
7255:
7251:
7248:
7243:
7239:
7235:
7232:
7229:
7226:
7224:
7220:
7216:
7212:
7211:
7198:
7197:
7189:
7188:
7187:
7186:
7169:
7164:
7159:
7156:
7151:
7146:
7141:
7138:
7135:
7133:
7129:
7124:
7119:
7118:
7113:
7108:
7103:
7100:
7095:
7090:
7085:
7080:
7075:
7070:
7067:
7064:
7062:
7058:
7053:
7048:
7047:
7042:
7037:
7032:
7029:
7024:
7019:
7014:
7011:
7008:
7005:
7003:
6999:
6994:
6989:
6988:
6975:
6974:
6969:
6962:
6954:
6953:
6952:
6951:
6939:
6930:
6922:
6919:
6911:
6908:
6900:
6897:
6889:
6886:
6878:
6875:
6872:
6871:
6868:
6865:
6862:
6859:
6856:
6853:
6850:
6847:
6844:
6841:
6838:
6835:
6832:
6831:
6828:
6825:
6822:
6819:
6816:
6813:
6810:
6807:
6804:
6801:
6798:
6795:
6792:
6791:
6788:
6785:
6782:
6779:
6776:
6773:
6770:
6767:
6764:
6761:
6758:
6755:
6752:
6751:
6747:
6733:
6732:
6725:
6721:
6720:
6719:
6718:
6693:
6686:
6672:
6665:
6658:
6651:
6644:
6637:
6630:
6623:
6616:
6596:) ×
6583:
6570:
6563:
6556:
6545:
6542:
6525:
6522:
6519:
6499:
6496:
6493:
6469:
6466:
6458:
6457:
6456:
6455:
6444:
6436:
6431:
6425:
6419:
6412:
6405:
6389:
6369:
6366:
6354:
6353:
6352:
6351:
6340:
6333:
6326:
6315:
6308:
6298:
6291:
6272:
6264:
6257:
6251:
6237:
6230:
6223:
6214:
6191:
6184:
6177:
6166:
6163:
6151:
6150:
6149:
6148:
6141:
6125:
6106:
6103:
6102:
6101:
6100:
6099:
6077:
6070:
6062:
6055:
6045:
6038:
6007:
5985:
5978:
5971:
5960:
5957:
5933:
5932:
5931:
5930:
5923:
5906:
5886:
5883:
5882:
5881:
5878:
5875:
5851:
5848:
5831:
5828:
5825:
5822:
5817:
5813:
5809:
5806:
5786:
5763:bilinear forms
5743:
5716:
5696:
5693:
5688:
5684:
5678:
5674:
5670:
5648:
5644:
5623:
5599:
5579:
5576:
5573:
5570:
5565:
5561:
5557:
5554:
5534:
5531:
5528:
5525:
5522:
5519:
5516:
5511:
5507:
5503:
5500:
5497:
5494:
5491:
5488:
5485:
5482:
5479:
5457:
5453:
5432:
5412:
5392:
5370:
5366:
5345:
5321:
5301:
5277:
5265:
5262:
5229:
5226:
5214:
5211:
5208:
5205:
5202:
5199:
5196:
5193:
5190:
5187:
5184:
5181:
5178:
5175:
5172:
5169:
5166:
5163:
5160:
5138:
5135:
5132:
5109:
5106:
5103:
5071:
5068:
5065:
5062:
5059:
5056:
5053:
5050:
5047:
5044:
5041:
5038:
5035:
5032:
5029:
5026:
5023:
5020:
5017:
5014:
5011:
5008:
5005:
5002:
4999:
4996:
4993:
4990:
4967:
4964:
4961:
4958:
4955:
4952:
4949:
4946:
4943:
4940:
4937:
4934:
4931:
4928:
4925:
4922:
4919:
4916:
4913:
4910:
4907:
4904:
4901:
4898:
4895:
4892:
4889:
4886:
4883:
4880:
4877:
4855:
4851:
4847:
4844:
4840:
4836:
4833:
4830:
4826:
4822:
4818:
4814:
4810:
4805:
4801:
4798:
4795:
4792:
4768:
4765:
4741:
4740:
4713:
4659:
4507:
4504:
4456:
4453:
4451:
4448:
4447:
4446:
4434:
4433:
4432:
4431:
4420:
4417:
4414:
4411:
4408:
4405:
4402:
4399:
4396:
4393:
4388:
4384:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4359:
4356:
4353:
4348:
4344:
4340:
4337:
4332:
4328:
4324:
4319:
4315:
4311:
4308:
4303:
4299:
4295:
4290:
4286:
4282:
4279:
4266:
4265:
4258:
4257:
4256:
4245:
4240:
4236:
4232:
4229:
4224:
4220:
4201:
4197:
4193:
4190:
4185:
4181:
4159:
4130:
4124:
4110:
4103:
4094:
4088:
4081:
4073:
4067:
4060:
4054:
4053:
4040:
4035:
4028:
4024:
4020:
4017:
4014:
4009:
4004:
3997:
3993:
3988:
3982:
3977:
3970:
3966:
3962:
3959:
3956:
3951:
3946:
3939:
3935:
3913:
3906:
3895:
3888:
3881:
3856:determined by
3815:
3812:
3781:
3780:
3763:
3753:
3750:
3747:
3744:
3741:
3740:
3736:
3733:
3728:
3725:
3722:
3721:
3717:
3714:
3711:
3708:
3707:
3703:
3699:
3696:
3679:
3675:
3672:
3668:
3649:Main article:
3646:
3643:
3642:
3641:
3633:
3632:
3631:
3630:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3591:
3587:
3583:
3580:
3577:
3574:
3571:
3568:
3565:
3562:
3557:
3553:
3549:
3546:
3541:
3537:
3533:
3530:
3527:
3522:
3518:
3514:
3501:
3500:
3488:
3487:
3486:
3485:
3474:
3469:
3465:
3461:
3458:
3452:
3449:
3446:
3443:
3437:
3432:
3428:
3424:
3421:
3400:
3390:
3383:
3362:
3355:
3349:
3348:
3337:
3333:
3329:
3326:
3321:
3317:
3313:
3310:
3307:
3302:
3298:
3294:
3289:
3284:
3277:
3273:
3269:
3266:
3263:
3258:
3253:
3246:
3242:
3237:
3233:
3230:
3225:
3220:
3215:
3212:
3209:
3204:
3199:
3194:
3171:
3170:
3159:
3154:
3149:
3142:
3138:
3134:
3131:
3128:
3123:
3118:
3111:
3107:
3089:
3082:
3075:
3067:In general, a
3058:
3057:
3046:
3041:
3035:
3032:
3031:
3028:
3025:
3022:
3021:
3018:
3015:
3014:
3012:
3004:
3000:
2996:
2991:
2985:
2982:
2979:
2978:
2975:
2972:
2971:
2968:
2965:
2964:
2962:
2954:
2950:
2945:
2939:
2933:
2930:
2929:
2926:
2923:
2922:
2919:
2916:
2915:
2913:
2893:Main article:
2890:
2887:
2871:
2870:
2857:
2853:
2849:
2846:
2841:
2837:
2833:
2830:
2827:
2811:
2806:
2802:
2798:
2795:
2790:
2786:
2782:
2779:
2776:
2769:
2764:
2760:
2756:
2753:
2748:
2744:
2740:
2737:
2734:
2708:
2707:
2696:
2692:
2688:
2685:
2680:
2676:
2672:
2669:
2666:
2661:
2657:
2641:
2637:
2631:
2628:
2624:
2620:
2616:
2613:
2610:
2606:
2601:
2597:
2591:
2588:
2584:
2580:
2576:
2572:
2567:
2563:
2557:
2554:
2550:
2546:
2542:
2538:
2533:
2529:
2525:
2524:
2508:
2505:
2502:
2496:
2492:
2486:
2483:
2479:
2475:
2471:
2468:
2465:
2461:
2456:
2452:
2446:
2442:
2438:
2434:
2430:
2425:
2421:
2415:
2411:
2407:
2403:
2399:
2394:
2390:
2386:
2385:
2379:
2375:
2369:
2366:
2362:
2358:
2354:
2351:
2348:
2344:
2339:
2335:
2329:
2325:
2321:
2317:
2313:
2308:
2304:
2298:
2294:
2290:
2286:
2282:
2277:
2273:
2269:
2268:
2265:
2260:
2256:
2252:
2248:
2236:
2232:
2228:
2227:
2224:
2221:
2220:
2215:
2211:
2207:
2206:
2201:
2197:
2193:
2192:
2186:
2181:
2160:The subset of
2157:
2154:
2142:
2141:
2130:
2125:
2119:
2116:
2114:
2111:
2108:
2106:
2103:
2102:
2099:
2096:
2094:
2091:
2089:
2086:
2085:
2083:
2078:
2075:
2057:
2056:
2045:
2041:
2037:
2033:
2029:
2011:Main article:
2008:
2005:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1898:(over real or
1869:
1865:
1856:
1853:
1849:
1844:
1840:
1834:
1831:
1827:
1823:
1819:
1816:
1813:
1809:
1804:
1800:
1794:
1791:
1787:
1783:
1779:
1775:
1770:
1766:
1760:
1757:
1753:
1749:
1748:
1743:
1740:
1729:
1725:
1722:
1718:
1713:
1709:
1703:
1700:
1696:
1692:
1688:
1685:
1682:
1678:
1673:
1669:
1663:
1659:
1655:
1651:
1647:
1642:
1638:
1632:
1628:
1624:
1623:
1619:
1616:
1612:
1607:
1603:
1597:
1594:
1590:
1586:
1582:
1579:
1576:
1572:
1567:
1563:
1557:
1553:
1549:
1545:
1541:
1536:
1532:
1526:
1522:
1518:
1517:
1514:
1509:
1505:
1501:
1497:
1485:
1481:
1477:
1476:
1473:
1470:
1469:
1464:
1460:
1456:
1455:
1450:
1446:
1442:
1441:
1435:
1430:
1402:
1399:
1387:
1386:
1369:
1365:
1358:
1355:
1350:
1346:
1336:
1332:
1329:
1325:
1321:
1316:
1313:
1310:
1307:
1304:
1280:specifies its
1243:
1242:
1225:
1221:
1214:
1211:
1206:
1202:
1192:
1188:
1185:
1181:
1177:
1172:
1169:
1166:
1163:
1160:
1094:
1091:
1045:if and only if
1043:is a subspace
1032:
1029:
1009:
1006:
1005:
1004:
1001:
998:
934:
931:
907:
906:
891:
884:
877:
870:
861:
854:
831:
824:
813:
806:
796:
780:
773:
766:
759:
752:
745:
738:
731:
722:
715:
708:
701:
685:
678:
671:
664:
651:
644:
630:
623:
547:
544:
543:
542:
525:
518:
507:
500:
483:
476:
449:
431:
424:
417:
410:
403:
396:
389:
382:
364:
357:
343:
336:
267:
264:
262:
259:
234:
227:
200:
193:
126:
123:
95:linear algebra
85:
84:
80:
65:
55:
54:
26:
9:
6:
4:
3:
2:
8770:
8759:
8756:
8754:
8751:
8749:
8746:
8745:
8743:
8728:
8720:
8719:
8716:
8710:
8707:
8705:
8704:Sparse matrix
8702:
8700:
8697:
8695:
8692:
8690:
8687:
8686:
8684:
8682:
8678:
8672:
8669:
8667:
8664:
8662:
8659:
8657:
8654:
8652:
8649:
8647:
8644:
8643:
8641:
8639:constructions
8638:
8634:
8628:
8627:Outermorphism
8625:
8623:
8620:
8618:
8615:
8613:
8610:
8608:
8605:
8603:
8600:
8598:
8595:
8593:
8590:
8588:
8587:Cross product
8585:
8583:
8580:
8579:
8577:
8575:
8571:
8565:
8562:
8560:
8557:
8555:
8554:Outer product
8552:
8550:
8547:
8545:
8542:
8540:
8537:
8535:
8534:Orthogonality
8532:
8531:
8529:
8527:
8523:
8517:
8514:
8512:
8511:Cramer's rule
8509:
8507:
8504:
8502:
8499:
8497:
8494:
8492:
8489:
8487:
8484:
8482:
8481:Decomposition
8479:
8477:
8474:
8473:
8471:
8469:
8465:
8460:
8450:
8447:
8445:
8442:
8440:
8437:
8435:
8432:
8430:
8427:
8425:
8422:
8420:
8417:
8415:
8412:
8410:
8407:
8405:
8402:
8400:
8397:
8395:
8392:
8390:
8387:
8385:
8382:
8380:
8377:
8375:
8372:
8370:
8367:
8365:
8362:
8360:
8357:
8356:
8354:
8350:
8344:
8341:
8339:
8336:
8335:
8332:
8328:
8321:
8316:
8314:
8309:
8307:
8302:
8301:
8298:
8290:
8278:
8274:
8270:
8266:
8262:
8250:
8246:
8242:
8238:
8237:
8220:
8219:
8211:
8206:
8195:
8194:
8189:
8185:
8181:
8180:
8171:
8169:0-534-99845-3
8165:
8161:
8156:
8153:
8149:
8145:
8141:
8140:
8134:
8130:
8126:
8120:
8116:
8115:
8109:
8105:
8100:
8097:
8091:
8087:
8082:
8079:
8077:0-471-50728-8
8073:
8069:
8064:
8063:
8057:
8053:
8049:
8043:
8039:
8035:
8031:
8027:
8024:
8018:
8014:
8010:
8005:
8001:
7995:
7991:
7987:
7986:Hefferon, Jim
7983:
7979:
7977:0-387-90093-4
7973:
7969:
7965:
7961:
7957:
7954:
7952:0-395-14017-X
7948:
7944:
7939:
7938:
7931:
7927:
7921:
7917:
7913:
7909:
7905:
7901:
7896:
7895:
7878:
7873:
7866:
7861:
7854:
7849:
7842:
7837:
7830:
7829:Halmos (1974)
7825:
7818:
7817:Halmos (1974)
7813:
7806:
7801:
7794:
7789:
7782:
7777:
7771:, p. 22)
7770:
7765:
7756:
7750:, p. 21)
7749:
7744:
7738:, p. 20)
7737:
7732:
7725:
7720:
7714:, p. 21)
7713:
7708:
7701:
7696:
7689:
7684:
7677:
7672:
7666:, p. 20)
7665:
7660:
7653:
7648:
7641:
7636:
7629:
7624:
7617:
7612:
7605:
7604:Halmos (1974)
7600:
7596:
7581:
7571:
7564:
7557:
7552:
7548:
7545:
7535:
7529:
7523:
7519:
7512:
7506:
7499:
7495:
7491:
7487:
7481:
7474:
7470:
7466:
7462:
7458:
7452:
7448:
7439:
7436:
7434:
7431:
7429:
7426:
7424:
7421:
7419:
7416:
7414:
7411:
7410:
7401:
7397:
7396:
7375:
7370:
7366:
7362:
7359:
7354:
7350:
7346:
7343:
7341:
7334:
7330:
7320:
7316:
7312:
7309:
7304:
7300:
7296:
7291:
7287:
7283:
7280:
7278:
7271:
7267:
7257:
7253:
7249:
7246:
7241:
7237:
7233:
7230:
7227:
7225:
7218:
7214:
7202:
7201:
7200:
7199:
7195:
7191:
7190:
7167:
7157:
7154:
7149:
7139:
7136:
7134:
7127:
7111:
7101:
7098:
7093:
7083:
7078:
7068:
7065:
7063:
7056:
7040:
7030:
7027:
7022:
7012:
7009:
7006:
7004:
6997:
6979:
6978:
6977:
6976:
6968:
6961:
6956:
6955:
6937:
6928:
6917:
6906:
6895:
6884:
6873:
6866:
6863:
6857:
6851:
6845:
6839:
6833:
6826:
6820:
6817:
6811:
6805:
6799:
6793:
6786:
6780:
6774:
6768:
6765:
6759:
6753:
6745:
6737:
6736:
6735:
6734:
6730:
6726:
6723:
6722:
6716:
6711:
6707:
6703:
6696:
6692:
6685:
6681:
6677:
6673:
6668:
6664:
6657:
6650:
6645:
6642:
6638:
6633:
6629:
6622:
6615:
6610:
6606:
6605:
6603:
6599:
6595:
6592: −
6591:
6587:
6584:
6582:
6578:
6573:
6569:
6562:
6555:
6551:
6548:
6547:
6541:
6539:
6523:
6520:
6517:
6497:
6494:
6491:
6465:
6463:
6453:
6447:
6439:
6432:
6428:
6418:
6411:
6406:
6403:
6399:
6398:
6397:
6393:
6390:
6387:
6383:
6380: ×
6379:
6375:
6372:
6371:
6365:
6363:
6359:
6349:
6343:
6339:
6332:
6325:
6320:
6316:
6313:
6309:
6306:
6301:
6297:
6290:
6286:
6283:
6279:
6278:
6275:
6271:
6267:
6263:
6256:
6250:
6246:
6240:
6236:
6229:
6222:
6218:
6215:
6212:
6208:
6203:
6199:
6194:
6190:
6183:
6176:
6172:
6169:
6168:
6162:
6158:
6156:
6146:
6142:
6139:
6135:
6134:
6133:
6129:
6126:
6124:
6120:
6117: ×
6116:
6112:
6109:
6108:
6096:
6092:
6085:
6080:
6076:
6069:
6063:
6060:
6056:
6053:
6048:
6044:
6037:
6033:
6029:
6025:
6021:
6020:
6019:
6015:
6011:
6008:
6005:
6001:
5997:
5993:
5988:
5984:
5977:
5970:
5966:
5963:
5962:
5956:
5954:
5949:
5944:
5942:
5938:
5928:
5924:
5921:
5917:
5916:
5914:
5910:
5907:
5904:
5900:
5897: ×
5896:
5892:
5889:
5888:
5879:
5876:
5873:
5872:
5871:
5869:
5865:
5861:
5857:
5856:row reduction
5847:
5845:
5826:
5820:
5815:
5811:
5807:
5804:
5784:
5776:
5772:
5768:
5764:
5759:
5757:
5733:
5728:
5714:
5694:
5691:
5686:
5676:
5672:
5646:
5642:
5621:
5613:
5597:
5574:
5568:
5563:
5559:
5555:
5552:
5529:
5523:
5520:
5517:
5509:
5505:
5498:
5495:
5492:
5486:
5480:
5477:
5455:
5451:
5430:
5410:
5390:
5368:
5364:
5343:
5335:
5319:
5299:
5291:
5275:
5261:
5259:
5255:
5251:
5250:least element
5247:
5243:
5239:
5235:
5225:
5209:
5203:
5200:
5197:
5191:
5185:
5182:
5179:
5173:
5170:
5167:
5161:
5158:
5150:
5136:
5133:
5130:
5121:
5107:
5104:
5101:
5093:
5092:
5087:
5082:
5069:
5063:
5060:
5057:
5051:
5048:
5045:
5039:
5033:
5030:
5027:
5021:
5015:
5012:
5009:
5003:
5000:
4997:
4991:
4988:
4978:
4965:
4959:
4953:
4950:
4947:
4941:
4935:
4932:
4929:
4923:
4920:
4917:
4911:
4908:
4905:
4899:
4896:
4893:
4890:
4887:
4884:
4881:
4866:
4853:
4849:
4845:
4842:
4834:
4831:
4828:
4820:
4812:
4803:
4799:
4796:
4793:
4790:
4782:
4778:
4774:
4764:
4762:
4758:
4754:
4752:
4746:
4738:
4735: ∩
4734:
4730:
4726:
4722:
4718:
4714:
4711:
4707:
4703:
4700:
4696:
4692:
4688:
4684:
4680:
4676:
4672:
4669: ∩
4668:
4664:
4660:
4657:
4654: ∩
4653:
4649:
4646: +
4645:
4641:
4637:
4634: +
4633:
4629:
4625:
4621:
4618: +
4617:
4613:
4609:
4605:
4601:
4597:
4593:
4590: ∩
4589:
4585:
4581:
4577:
4576:
4575:
4574:
4570:
4568:
4564:
4560:
4556:
4552:
4549: ∈
4548:
4544:
4541: ∩
4540:
4537:
4534:, then their
4533:
4529:
4525:
4517:
4512:
4503:
4501:
4498: =
4497:
4493:
4490: =
4489:
4485:
4482: ⊂
4481:
4477:
4474: =
4473:
4468:
4466:
4465:partial order
4462:
4444:
4440:
4437:The subspace
4436:
4435:
4418:
4412:
4409:
4406:
4403:
4400:
4397:
4394:
4386:
4382:
4378:
4372:
4369:
4366:
4363:
4360:
4357:
4354:
4346:
4342:
4338:
4330:
4326:
4322:
4317:
4313:
4309:
4306:
4301:
4297:
4293:
4288:
4284:
4280:
4270:
4269:
4268:
4267:
4263:
4259:
4243:
4238:
4234:
4230:
4227:
4222:
4218:
4199:
4195:
4191:
4188:
4183:
4179:
4171:
4170:
4168:
4164:
4160:
4157:
4156:
4155:
4153:
4149:
4145:
4141:
4136:
4133:
4123:
4119:
4113:
4109:
4102:
4097:
4087:
4080:
4076:
4066:
4059:
4038:
4026:
4022:
4018:
4015:
4012:
4007:
3995:
3991:
3986:
3980:
3968:
3964:
3960:
3957:
3954:
3949:
3937:
3933:
3925:
3924:
3923:
3921:
3916:
3912:
3905:
3900:
3894:
3887:
3880:
3875:
3871:
3867:
3863:
3859:
3855:
3847:
3843:
3839:
3834:
3829:
3825:
3821:
3811:
3809:
3804:
3802:
3798:
3794:
3790:
3786:
3761:
3751:
3745:
3742:
3734:
3731:
3723:
3715:
3709:
3701:
3697:
3694:
3673:
3670:
3658:
3657:
3656:
3652:
3639:
3635:
3634:
3609:
3606:
3603:
3600:
3597:
3589:
3585:
3581:
3575:
3572:
3569:
3566:
3563:
3555:
3551:
3547:
3539:
3535:
3531:
3528:
3525:
3520:
3516:
3505:
3504:
3503:
3502:
3498:
3494:
3490:
3489:
3472:
3467:
3463:
3459:
3456:
3450:
3447:
3444:
3441:
3435:
3430:
3426:
3422:
3419:
3412:
3411:
3409:
3405:
3401:
3398:
3397:
3396:
3393:
3389:
3382:
3378:
3374:
3370:
3365:
3361:
3354:
3335:
3331:
3327:
3324:
3319:
3315:
3311:
3308:
3305:
3300:
3296:
3292:
3287:
3275:
3271:
3267:
3264:
3261:
3256:
3244:
3240:
3235:
3231:
3223:
3213:
3210:
3207:
3202:
3180:
3179:
3178:
3176:
3157:
3152:
3140:
3136:
3132:
3129:
3126:
3121:
3109:
3105:
3097:
3096:
3095:
3092:
3088:
3081:
3074:
3070:
3065:
3063:
3044:
3039:
3033:
3026:
3023:
3016:
3010:
3002:
2998:
2994:
2989:
2983:
2980:
2973:
2966:
2960:
2952:
2948:
2943:
2937:
2931:
2924:
2917:
2911:
2902:
2901:
2900:
2896:
2886:
2884:
2880:
2876:
2855:
2851:
2847:
2844:
2839:
2835:
2831:
2828:
2825:
2809:
2804:
2800:
2796:
2793:
2788:
2784:
2780:
2777:
2774:
2767:
2762:
2758:
2754:
2751:
2746:
2742:
2738:
2735:
2732:
2725:
2724:
2723:
2721:
2717:
2713:
2694:
2690:
2686:
2683:
2678:
2674:
2670:
2667:
2664:
2659:
2655:
2639:
2635:
2629:
2626:
2622:
2614:
2611:
2608:
2599:
2595:
2589:
2586:
2582:
2574:
2565:
2561:
2555:
2552:
2548:
2540:
2531:
2527:
2506:
2494:
2490:
2484:
2481:
2477:
2469:
2466:
2463:
2454:
2450:
2444:
2440:
2432:
2423:
2419:
2413:
2409:
2401:
2392:
2388:
2377:
2373:
2367:
2364:
2360:
2352:
2349:
2346:
2337:
2333:
2327:
2323:
2315:
2306:
2302:
2296:
2292:
2284:
2275:
2271:
2263:
2258:
2254:
2250:
2246:
2234:
2230:
2222:
2213:
2209:
2199:
2195:
2184:
2179:
2171:
2170:
2169:
2167:
2163:
2153:
2151:
2147:
2128:
2123:
2117:
2112:
2109:
2104:
2097:
2092:
2087:
2081:
2076:
2073:
2066:
2065:
2064:
2062:
2043:
2035:
2027:
2020:
2019:
2018:
2014:
2004:
2002:
1998:
1994:
1990:
1986:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1933:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1901:
1895:
1891:
1887:
1880:
1867:
1863:
1854:
1851:
1842:
1838:
1832:
1829:
1825:
1817:
1814:
1811:
1802:
1798:
1792:
1789:
1785:
1777:
1768:
1764:
1758:
1755:
1751:
1741:
1723:
1720:
1711:
1707:
1701:
1698:
1694:
1686:
1683:
1680:
1671:
1667:
1661:
1657:
1649:
1640:
1636:
1630:
1626:
1617:
1614:
1605:
1601:
1595:
1592:
1588:
1580:
1577:
1574:
1565:
1561:
1555:
1551:
1543:
1534:
1530:
1524:
1520:
1512:
1507:
1503:
1499:
1495:
1483:
1479:
1471:
1462:
1458:
1448:
1444:
1433:
1428:
1419:
1416:
1412:
1408:
1398:
1396:
1395:the remaining
1392:
1367:
1356:
1353:
1348:
1330:
1327:
1323:
1314:
1311:
1308:
1305:
1295:
1294:
1293:
1291:
1287:
1283:
1279:
1275:
1271:
1267:
1262:
1260:
1256:
1252:
1248:
1223:
1212:
1209:
1204:
1186:
1183:
1179:
1170:
1167:
1164:
1161:
1151:
1150:
1149:
1147:
1143:
1139:
1134:
1132:
1128:
1124:
1120:
1116:
1112:
1108:
1104:
1100:
1090:
1088:
1084:
1080:
1076:
1072:
1069:, a subspace
1068:
1065:
1060:
1058:
1054:
1050:
1046:
1042:
1038:
1028:
1026:
1021:
1019:
1015:
1002:
999:
995:
991:
985:
984:
983:
982:
978:
976:
972:
968:
964:
960:
956:
952:
948:
944:
940:
930:
928:
925: =
924:
920:
916:
912:
904:
900:
897:
890:
883:
876:
869:
860:
853:
849:
846:
841:
837:
830:
823:
819:
812:
805:
801:
797:
794:
790:
786:
779:
772:
765:
758:
751:
744:
737:
730:
721:
714:
707:
700:
696:
692:
684:
677:
670:
663:
659:
650:
643:
639:
629:
622:
618:
613:
612:
611:
610:
602:
598:
596:
592:
588:
584:
580:
576:
572:
568:
564:
561:
557:
553:
540:
536:
533:
524:
517:
513:
506:
499:
495:
492:
482:
475:
471:
466:
462:
459:and a scalar
458:
454:
450:
447:
443:
439:
430:
423:
416:
409:
402:
395:
388:
381:
377:
373:
363:
356:
352:
342:
335:
331:
326:
322:
318:
314:
313:
312:
311:
307:
305:
301:
297:
293:
289:
285:
281:
277:
273:
258:
256:
252:
248:
243:
241:
233:
226:
221:
216:
212:
207:
199:
192:
188:if, whenever
187:
183:
179:
175:
171:
167:
163:
159:
155:
151:
147:
143:
139:
136:
132:
122:
120:
116:
112:
108:
104:
100:
96:
92:
79:
75:
71:
64:
61:
57:
56:
52:
47:
41:
36:
32:
31:
19:
8665:
8637:Vector space
8369:Vector space
8287:– via
8281:. Retrieved
8259:– via
8253:. Retrieved
8222:. Retrieved
8216:
8197:. Retrieved
8191:
8159:
8138:
8129:the original
8113:
8103:
8085:
8061:
8033:
8008:
7989:
7963:
7936:
7911:
7899:
7877:Axler (2015)
7872:
7865:Axler (2015)
7860:
7853:Axler (2015)
7848:
7841:Axler (2015)
7836:
7824:
7812:
7800:
7795:p. 21 § 1.40
7793:Axler (2015)
7788:
7776:
7769:Nering (1970
7764:
7755:
7748:Nering (1970
7743:
7736:Nering (1970
7731:
7719:
7712:Nering (1970
7707:
7695:
7683:
7671:
7664:Nering (1970
7659:
7647:
7635:
7623:
7611:
7599:
7576:
7569:
7562:
7555:
7550:
7546:
7540:
7533:
7527:
7521:
7517:
7510:
7505:
7485:
7480:
7468:
7456:
7451:
7399:
7193:
6966:
6959:
6728:
6714:
6709:
6705:
6701:
6694:
6690:
6683:
6679:
6675:
6666:
6662:
6655:
6648:
6640:
6631:
6627:
6620:
6613:
6608:
6601:
6597:
6593:
6589:
6585:
6580:
6576:
6571:
6567:
6560:
6553:
6549:
6471:
6459:
6451:
6442:
6434:
6423:
6416:
6409:
6401:
6395:
6391:
6385:
6381:
6377:
6373:
6361:
6357:
6355:
6347:
6341:
6337:
6330:
6323:
6318:
6311:
6304:
6299:
6295:
6288:
6284:
6273:
6269:
6265:
6261:
6254:
6248:
6244:
6238:
6234:
6227:
6220:
6216:
6210:
6206:
6201:
6197:
6192:
6188:
6181:
6174:
6170:
6159:
6152:
6137:
6131:
6127:
6122:
6118:
6114:
6110:
6094:
6090:
6083:
6078:
6074:
6067:
6058:
6051:
6046:
6042:
6035:
6031:
6027:
6023:
6017:
6013:
6009:
6003:
5999:
5995:
5991:
5986:
5982:
5975:
5968:
5964:
5952:
5947:
5945:
5934:
5926:
5919:
5912:
5908:
5902:
5898:
5894:
5890:
5853:
5775:null vectors
5760:
5729:
5267:
5257:
5246:{0} subspace
5244:, where the
5234:intersection
5231:
5151:
5122:
5089:
5085:
5083:
4979:
4867:
4780:
4776:
4772:
4770:
4760:
4756:
4750:
4744:
4742:
4736:
4732:
4728:
4724:
4720:
4716:
4709:
4705:
4701:
4698:
4694:
4690:
4686:
4682:
4678:
4674:
4670:
4666:
4662:
4655:
4651:
4647:
4643:
4639:
4635:
4631:
4627:
4623:
4619:
4615:
4611:
4607:
4603:
4599:
4595:
4591:
4587:
4583:
4579:
4572:
4571:
4566:
4562:
4558:
4554:
4550:
4546:
4542:
4538:
4536:intersection
4531:
4527:
4523:
4521:
4515:
4506:Intersection
4499:
4495:
4491:
4487:
4483:
4479:
4475:
4471:
4469:
4458:
4442:
4438:
4261:
4166:
4162:
4152:§ Algorithms
4147:
4143:
4139:
4137:
4128:
4121:
4117:
4111:
4107:
4100:
4092:
4085:
4078:
4071:
4064:
4057:
4055:
3919:
3914:
3910:
3903:
3901:
3892:
3885:
3878:
3873:
3869:
3865:
3861:
3857:
3853:
3851:
3845:
3841:
3837:
3836:The vectors
3805:
3800:
3796:
3792:
3784:
3782:
3654:
3637:
3496:
3492:
3407:
3403:
3391:
3387:
3380:
3376:
3372:
3368:
3363:
3359:
3352:
3350:
3174:
3172:
3090:
3086:
3079:
3072:
3068:
3066:
3061:
3059:
2898:
2883:number field
2878:
2874:
2872:
2719:
2715:
2711:
2709:
2161:
2159:
2145:
2143:
2058:
2016:
2000:
1996:
1988:
1984:
1893:
1889:
1885:
1881:
1417:
1410:
1404:
1388:
1285:
1277:
1263:
1258:
1254:
1244:
1145:
1135:
1130:
1115:column space
1096:
1093:Descriptions
1070:
1066:
1061:
1056:
1052:
1040:
1034:
1022:
1013:
1011:
993:
989:
980:
979:
974:
970:
962:
958:
954:
946:
942:
938:
936:
926:
922:
918:
910:
908:
902:
898:
895:
888:
881:
874:
867:
858:
851:
847:
844:
839:
835:
828:
821:
817:
810:
803:
799:
792:
788:
784:
777:
770:
763:
756:
749:
742:
735:
728:
719:
712:
705:
698:
694:
690:
682:
675:
668:
661:
657:
648:
641:
637:
627:
620:
616:
608:
607:
594:
590:
586:
582:
578:
574:
570:
566:
562:
555:
551:
549:
538:
534:
531:
522:
515:
511:
504:
497:
493:
490:
488:again, then
480:
473:
469:
464:
460:
456:
452:
445:
441:
437:
428:
421:
414:
407:
400:
393:
386:
379:
375:
371:
361:
354:
350:
340:
333:
329:
324:
320:
316:
309:
308:
303:
299:
295:
291:
288:real numbers
283:
275:
271:
269:
254:
244:
239:
231:
224:
219:
214:
210:
205:
197:
190:
185:
181:
173:
170:vector space
165:
161:
157:
153:
149:
145:
141:
137:
130:
128:
114:
107:vector space
102:
98:
88:
77:
62:
60:finite field
8617:Multivector
8582:Determinant
8539:Dot product
8384:Linear span
8011:, Waltham:
7616:Anton (2005
7484:Generally,
6006:components.
5955:are equal.
5332:, then the
5086:independent
4731:belongs to
4650:belongs to
4638:belongs to
4622:belongs to
4118:coordinates
3918:are called
3071:of vectors
2895:Linear span
2003:functions.
1247:linear span
1140:of one non-
1083:codimension
945:be the set
933:Example III
251:zero vector
91:mathematics
8742:Categories
8651:Direct sum
8486:Invertible
8389:Linear map
8188:"Subspace"
7941:, Boston:
6280:Create an
6242:such that
6022:Create a (
5850:Algorithms
5797:such that
5612:direct sum
5356:, denoted
5091:direct sum
4673:, and let
4665:belong to
4610:. Because
4545: := {
4077:) ≠ (
3406:-plane in
2061:null space
2013:Null space
1290:dual space
1111:null space
1008:Example IV
834:, and let
581:such that
546:Example II
406:, 0+0) = (
125:Definition
109:that is a
74:dimensions
8681:Numerical
8444:Transpose
8193:MathWorld
7962:(1974) .
7690:Subspace.
7591:Citations
7473:manifolds
7455:The term
7360:−
7297:−
7231:−
7155:−
7084:−
7010:−
6864:−
6818:−
6766:−
6552:A basis {
6521:∩
6173:A basis {
5967:A basis {
5937:row space
5821:≠
5816:⊥
5808:∩
5742:¬
5687:⊥
5677:⊥
5647:⊥
5564:⊥
5556:∩
5524:
5510:⊥
5499:
5481:
5456:⊥
5369:⊥
5204:
5186:
5171:⊕
5162:
5134:⊕
5105:⊕
5061:∩
5052:
5046:−
5034:
5016:
4992:
4954:
4936:
4930:≤
4912:
4906:≤
4897:
4885:
4843:∈
4829:∈
4821::
4708:and
4561:and
4455:Inclusion
4016:⋯
3987:≠
3958:⋯
3743:−
3732:−
3325:∈
3309:…
3265:⋯
3211:…
3130:⋯
3024:−
2981:−
2832:−
2794:−
2684:∈
2668:…
2612:⋯
2507:⋮
2467:⋯
2350:⋯
2251:∈
2223:⋮
2110:−
1950:−
1815:⋯
1742:⋮
1684:⋯
1578:⋯
1500:∈
1472:⋮
1309:∈
1303:∃
1284:subspace
1165:∈
1159:∃
1119:row space
1016:) of all
969:. Then C(
951:functions
266:Example I
119:subspaces
8727:Category
8666:Subspace
8661:Quotient
8612:Bivector
8526:Bilinear
8468:Matrices
8343:Glossary
8277:Archived
8249:Archived
8152:76091646
8058:(1972),
7988:(2020).
7968:Springer
7916:Springer
7910:(2015).
7892:Textbook
7539:+ ··· +
7498:integers
7407:See also
6260:+ ··· +
6219:Numbers
5756:infinite
5732:negation
5252:, is an
4689:. Since
4642:. Thus,
4553: :
1993:null set
1368:′
1324:′
1251:equality
1224:′
1180:′
1077:, but a
1049:finitely
961:. Let C(
866:; since
727:; since
530:. Thus,
436:. Thus,
290:), take
261:Examples
178:nonempty
115:subspace
8338:Outline
8289:YouTube
8261:YouTube
7887:Sources
7575:, ...,
7516:, ...,
6965:, ...,
6724:Example
6661:, ...,
6626:, ...,
6566:, ...,
6462:example
6422:, ...,
6384:matrix
6336:, ...,
6233:, ...,
6187:, ...,
6155:example
6121:matrix
6073:, ...,
6030:matrix
5981:, ...,
5941:example
5939:for an
5901:matrix
5610:is the
4594:. Then
4158:Example
4127:, ...,
4106:, ...,
4084:,
4063:,
3399:Example
3078:,
2718:,
2714:,
1144:vector
949:of all
880:, then
842:. Then
755:, then
688:. Then
589:. Then
565:. Take
558:be the
369:. Then
180:subset
148:, then
140:and if
8622:Tensor
8434:Kernel
8364:Vector
8359:Scalar
8283:17 Feb
8255:17 Feb
8224:17 Feb
8199:16 Feb
8166:
8150:
8121:
8092:
8074:
8044:
8019:
7996:
7974:
7949:
7922:
6698:. The
6586:Output
6392:Output
6217:Output
6145:pivots
6128:Output
6010:Output
5909:Output
5288:is an
5248:, the
4747:, the
4715:Since
4573:Proof:
4098:). If
3826:, and
1282:kernel
1129:in an
1123:matrix
1117:, and
1105:, the
1075:closed
1037:closed
988:0 ∈ C(
981:Proof:
609:Proof:
514:0) = (
451:Given
448:, too.
315:Given
310:Proof:
238:is in
111:subset
76:. All
70:origin
68:. The
8491:Minor
8476:Block
8414:Basis
8213:(PDF)
8144:Wiley
8068:Wiley
7461:flats
7444:Notes
6689:,...,
6550:Input
6374:Input
6294:,...,
6171:Input
6111:Input
6002:with
5965:Input
5891:Input
4749:set {
4140:basis
4056:for (
3789:image
3687:where
3367:have
2881:is a
2877:, if
1409:with
1121:of a
1062:In a
953:from
894:, so
783:, so
577:) of
467:, if
278:(the
172:over
168:is a
152:is a
135:field
105:is a
8646:Dual
8501:Rank
8285:2021
8257:2021
8226:2021
8201:2021
8164:ISBN
8148:LCCN
8119:ISBN
8090:ISBN
8072:ISBN
8042:ISBN
8017:ISBN
7994:ISBN
7972:ISBN
7947:ISBN
7920:ISBN
7559:for
7494:rank
7463:and
6708:) ×
6646:Let
6588:An (
6476:and
6050:and
5769:and
5634:and
5590:and
5443:and
5292:and
5236:and
4775:and
4755:and
4719:and
4693:and
4685:and
4661:Let
4606:and
4598:and
4582:and
4578:Let
4526:and
4459:The
4161:Let
3840:and
3402:The
3189:Span
3175:span
3062:span
1274:zero
1266:dual
1142:zero
1127:flat
1107:span
992:) ⊂
798:Let
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674:and
635:and
614:Let
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486:, 0)
434:, 0)
367:, 0)
348:and
346:, 0)
319:and
208:and
97:, a
8177:Web
6579:of
6480:of
6448:= 1
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6113:An
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5893:An
5866:or
5846:).
5614:of
5521:dim
5496:dim
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5336:of
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5238:sum
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4989:dim
4951:dim
4933:dim
4909:dim
4894:dim
4882:dim
4876:max
4781:sum
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4767:Sum
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4210:and
3922:if
2818:and
1995:of
1939:and
1292:):
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1089:).
957:to
929:.)
850:= (
802:= (
697:= (
640:= (
619:= (
528:,0)
496:= (
472:= (
463:in
455:in
378:= (
353:= (
332:= (
323:in
286:of
156:of
129:If
101:or
89:In
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8215:.
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7568:,
7554:≠
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6540:.
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6364:.
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887:=
882:cp
873:=
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852:cp
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762:+
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681:=
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626:,
597:.
585:=
573:,
523:cu
521:,
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510:,
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232:βw
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7219:3
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5905:.
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5899:n
5895:m
5830:}
5827:0
5824:{
5812:N
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5785:N
5734:(
5715:N
5695:N
5692:=
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5598:V
5578:}
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5572:{
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5506:N
5502:(
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5490:)
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5004:W
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4419:.
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3767:.
3762:]
3752:2
3746:1
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3698:=
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3678:t
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3671:=
3667:x
3617:.
3613:)
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3607:,
3604:0
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3332:}
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2826:z
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2775:y
2768:,
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2691:}
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2679:m
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2665:,
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2541:=
2532:n
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2393:2
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2180:{
2162:K
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2129:.
2124:]
2118:5
2113:4
2105:2
2098:2
2093:3
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2077:=
2074:A
2044:.
2040:0
2036:=
2032:x
2028:A
2001:n
1997:A
1989:K
1985:n
1971:0
1968:=
1965:z
1962:5
1959:+
1956:y
1953:4
1947:x
1944:2
1934:0
1931:=
1928:z
1925:2
1922:+
1919:y
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1886:x
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1868:.
1864:}
1855:0
1852:=
1843:n
1839:x
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1818:+
1812:+
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1790:m
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1769:1
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1721:=
1712:n
1708:x
1702:n
1699:2
1695:a
1687:+
1681:+
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1668:x
1658:a
1650:+
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1627:a
1618:0
1615:=
1606:n
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1429:{
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1354:1
1349:=
1345:F
1335:F
1331:c
1328:=
1320:F
1315::
1312:K
1306:c
1286:F
1278:F
1259:k
1255:k
1229:)
1220:v
1213:c
1210:1
1205:=
1201:v
1191:v
1187:c
1184:=
1176:v
1171::
1168:K
1162:c
1146:v
1131:n
1071:W
1067:X
1057:W
1053:W
1041:W
1014:R
997:.
994:R
990:R
975:R
971:R
963:R
959:R
955:R
947:R
943:V
939:R
927:y
923:x
919:z
911:R
905:.
903:W
899:p
896:c
892:2
885:1
878:2
875:p
871:1
868:p
864:)
862:2
855:1
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840:R
836:c
832:2
829:p
825:1
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818:W
814:2
811:p
807:1
804:p
800:p
795:.
793:W
789:q
785:p
781:2
778:q
774:2
771:p
767:1
764:q
760:1
757:p
753:2
750:q
746:1
743:q
739:2
736:p
732:1
729:p
725:)
723:2
720:q
718:+
716:2
713:p
709:1
706:q
704:+
702:1
699:p
695:q
691:p
686:2
683:q
679:1
676:q
672:2
669:p
665:1
662:p
658:W
654:)
652:2
649:q
645:1
642:q
638:q
633:)
631:2
628:p
624:1
621:p
617:p
595:R
591:W
587:y
583:x
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575:y
571:x
567:W
563:R
556:V
552:R
539:W
535:u
532:c
526:2
519:1
512:c
508:2
501:1
494:u
491:c
484:2
481:u
477:1
474:u
470:u
465:R
461:c
457:W
453:u
446:W
442:v
438:u
432:2
429:v
427:+
425:2
422:u
418:1
415:v
413:+
411:1
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404:2
401:v
399:+
397:2
394:u
390:1
387:v
385:+
383:1
380:u
376:v
372:u
365:2
362:v
358:1
355:v
351:v
344:2
341:u
337:1
334:u
330:u
325:W
321:v
317:u
304:V
300:W
296:V
292:W
284:R
276:R
272:V
240:W
235:2
228:1
220:K
215:β
211:α
206:W
201:2
198:w
194:1
191:w
186:V
182:W
174:K
166:W
162:V
158:V
150:W
146:V
142:W
138:K
131:V
81:5
78:F
66:5
63:F
20:)
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