5066:
4510:
1915:
5061:{\displaystyle {\begin{aligned}{\begin{bmatrix}1&3&2\\2&7&4\\1&5&2\end{bmatrix}}&\xrightarrow {\mathbf {r} _{2}-2\mathbf {r} _{1}\to \mathbf {r} _{2}} {\begin{bmatrix}1&3&2\\0&1&0\\1&5&2\end{bmatrix}}\xrightarrow {\mathbf {r} _{3}-\,\,\mathbf {r} _{1}\to \mathbf {r} _{3}} {\begin{bmatrix}1&3&2\\0&1&0\\0&2&0\end{bmatrix}}\\&\xrightarrow {\mathbf {r} _{3}-2\mathbf {r} _{2}\to \mathbf {r} _{3}} {\begin{bmatrix}1&3&2\\0&1&0\\0&0&0\end{bmatrix}}\xrightarrow {\mathbf {r} _{1}-3\mathbf {r} _{2}\to \mathbf {r} _{1}} {\begin{bmatrix}1&0&2\\0&1&0\\0&0&0\end{bmatrix}}.\end{aligned}}}
1356:
2906:
1910:{\displaystyle {\begin{array}{rcl}A{\begin{bmatrix}c_{1}\\\vdots \\c_{n}\end{bmatrix}}&=&{\begin{bmatrix}a_{11}&\cdots &a_{1n}\\\vdots &\ddots &\vdots \\a_{m1}&\cdots &a_{mn}\end{bmatrix}}{\begin{bmatrix}c_{1}\\\vdots \\c_{n}\end{bmatrix}}={\begin{bmatrix}c_{1}a_{11}+\cdots +c_{n}a_{1n}\\\vdots \\c_{1}a_{m1}+\cdots +c_{n}a_{mn}\end{bmatrix}}=c_{1}{\begin{bmatrix}a_{11}\\\vdots \\a_{m1}\end{bmatrix}}+\cdots +c_{n}{\begin{bmatrix}a_{1n}\\\vdots \\a_{mn}\end{bmatrix}}\\&=&c_{1}\mathbf {v} _{1}+\cdots +c_{n}\mathbf {v} _{n}\end{array}}}
2464:
6446:
2901:{\displaystyle {\begin{bmatrix}1&3&1&4\\2&7&3&9\\1&5&3&1\\1&2&0&8\end{bmatrix}}\sim {\begin{bmatrix}1&3&1&4\\0&1&1&1\\0&2&2&-3\\0&-1&-1&4\end{bmatrix}}\sim {\begin{bmatrix}1&0&-2&1\\0&1&1&1\\0&0&0&-5\\0&0&0&5\end{bmatrix}}\sim {\begin{bmatrix}1&0&-2&0\\0&1&1&0\\0&0&0&1\\0&0&0&0\end{bmatrix}}.}
6710:
43:
31:
3522:
5278:
5592:
4275:
665:
2232:
3085:
2441:
3396:
5119:
5475:
4123:
2071:
4458:
5327:
of the matrix. This is the same as the maximum number of linearly independent rows that can be chosen from the matrix, or equivalently the number of pivots. For example, the 3 × 3 matrix in the example above has rank two.
3918:
1259:
816:
958:
887:
739:
521:
5837:
Linear algebra, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in Lay 2005, Meyer 2001, and Strang
4074:
2940:
3318:
2032:
5403:
3687:
5657:
to the row space. For example, if the row space is a plane through the origin in three dimensions, then the null space will be the perpendicular line through the origin. This provides a proof of the
2320:
3101:
The above algorithm can be used in general to find the dependence relations between any set of vectors, and to pick out a basis from any spanning set. Also finding a basis for the column space of
5093:
of a matrix is equal to its rank). Since row operations can affect linear dependence relations of the row vectors, such a basis is instead found indirectly using the fact that the column space of
3203:
4515:
3517:{\displaystyle A^{\mathsf {T}}\mathbf {x} ={\begin{bmatrix}\mathbf {v} _{1}\cdot \mathbf {x} \\\mathbf {v} _{2}\cdot \mathbf {x} \\\vdots \\\mathbf {v} _{n}\cdot \mathbf {x} \end{bmatrix}},}
5089:
It is sometimes convenient to find a basis for the row space from among the rows of the original matrix instead (for example, this result is useful in giving an elementary proof that the
5273:{\displaystyle A^{\mathrm {T} }={\begin{bmatrix}1&2&1\\3&7&5\\2&4&2\end{bmatrix}}\sim {\begin{bmatrix}1&2&1\\0&1&2\\0&0&0\end{bmatrix}}.}
2911:
At this point, it is clear that the first, second, and fourth columns are linearly independent, while the third column is a linear combination of the first two. (Specifically,
5587:{\displaystyle A\mathbf {x} ={\begin{bmatrix}\mathbf {r} _{1}\cdot \mathbf {x} \\\mathbf {r} _{2}\cdot \mathbf {x} \\\vdots \\\mathbf {r} _{m}\cdot \mathbf {x} \end{bmatrix}},}
3232:
1084:
991:
349:
320:
3150:, and is the maximum number of linearly independent columns that can be chosen from the matrix. For example, the 4 × 4 matrix in the example above has rank three.
4270:{\displaystyle c_{1}{\begin{bmatrix}1&0&2\end{bmatrix}}+c_{2}{\begin{bmatrix}0&1&0\end{bmatrix}}={\begin{bmatrix}c_{1}&c_{2}&2c_{1}\end{bmatrix}}.}
195:
4374:
6272:
3825:
1166:
744:
238:
158:
124:
892:
821:
673:
660:{\displaystyle J={\begin{bmatrix}2&4&1&3&2\\-1&-2&1&0&5\\1&6&2&2&2\\3&6&2&5&1\end{bmatrix}}}
2227:{\displaystyle c_{1}{\begin{bmatrix}1\\0\\2\end{bmatrix}}+c_{2}{\begin{bmatrix}0\\1\\0\end{bmatrix}}={\begin{bmatrix}c_{1}\\c_{2}\\2c_{1}\end{bmatrix}}}
6304:
4007:
3080:{\displaystyle {\begin{bmatrix}1\\2\\1\\1\end{bmatrix}},\;\;{\begin{bmatrix}3\\7\\5\\2\end{bmatrix}},\;\;{\begin{bmatrix}4\\9\\1\\8\end{bmatrix}}.}
6695:
3263:
1966:
6199:
6158:
6129:
6085:
5859:
5349:
5760:, then the orthogonal complement to the kernel can be thought of as a generalization of the row space. This is sometimes called the
6181:
6111:
6067:
6049:
3249:, and is equal to the number of columns in the reduced row echelon form that do not have pivots. The rank and nullity of a matrix
2436:{\displaystyle A={\begin{bmatrix}1&3&1&4\\2&7&3&9\\1&5&3&1\\1&2&0&8\end{bmatrix}}.}
3632:
6685:
5082:
This algorithm can be used in general to find a basis for the span of a set of vectors. If the matrix is further simplified to
6647:
6583:
5796:
6148:
6258:
3164:
6425:
6297:
3119:
6530:
6380:
6277:
6267:
6435:
6329:
5932:
5669:
5318:
4329:
3137:
198:
6734:
6675:
6324:
4354:
2300:
165:
6550:
5658:
5417:
5071:
Once the matrix is in echelon form, the nonzero rows are a basis for the row space. In this case, the basis is
3324:
6667:
6103:
6013:
5083:
3713:
3147:
2455:
2934:.) Therefore, the first, second, and fourth columns of the original matrix are a basis for the column space:
6739:
6713:
6420:
6290:
5705:
4322:
2276:
1050:
6477:
6410:
6400:
4362:
2308:
2292:
6637:
6492:
6487:
6482:
6360:
5953:
5709:
5654:
5323:
5312:
5090:
4314:
3720:
3589:
3158:
3142:
3131:
2268:
1952:
427:
291:
202:
100:
3208:
1060:
967:
325:
296:
6502:
6467:
6454:
6345:
6041:
5919:
4464:
3773:
2447:
2296:
1114:
73:
47:
35:
6680:
6560:
6535:
6385:
5777:
5757:
4318:
3769:
3161:, the rank of a matrix is the same as the dimension of the image. For example, the transformation
3154:
2272:
1948:
1944:
1110:
1029:
431:
127:
96:
92:
67:
3094:. This makes it possible to determine which columns are linearly independent by reducing only to
2303:
do not affect the dependence relations between the column vectors. This makes it possible to use
6390:
6262:
5818:
3816:
3620:
3235:
3090:
Note that the independent columns of the reduced row echelon form are precisely the columns with
1157:
218:
84:
3603:, the column space, row space, null space, and left null space are sometimes referred to as the
6744:
6588:
6545:
6472:
6365:
6235:
6216:
6195:
6177:
6154:
6125:
6107:
6081:
6063:
6045:
5922:
row-reduction algorithm. Each of the shown steps involves multiple elementary row operations.
5865:
5855:
6593:
6497:
6350:
5949:
4472:
4453:{\displaystyle A={\begin{bmatrix}1&3&2\\2&7&4\\1&5&2\end{bmatrix}}.}
3095:
1032:, the row space is 4-dimensional. Moreover, in this case it can be seen that they are all
173:
6652:
6445:
6405:
6395:
6005:
5681:
3913:{\displaystyle c_{1}\mathbf {r} _{1}+c_{2}\mathbf {r} _{2}+\cdots +c_{m}\mathbf {r} _{m},}
3363:
3336:
3246:
1254:{\displaystyle c_{1}\mathbf {v} _{1}+c_{2}\mathbf {v} _{2}+\cdots +c_{n}\mathbf {v} _{n},}
811:{\displaystyle \mathbf {r} _{2}={\begin{bmatrix}-1&-2&1&0&5\end{bmatrix}}}
161:
6098:
A First Course In Linear
Algebra: with Optional Introduction to Groups, Rings, and Fields
953:{\displaystyle \mathbf {r} _{4}={\begin{bmatrix}3&6&2&5&1\end{bmatrix}}}
882:{\displaystyle \mathbf {r} _{3}={\begin{bmatrix}1&6&2&2&2\end{bmatrix}}}
734:{\displaystyle \mathbf {r} _{1}={\begin{bmatrix}2&4&1&3&2\end{bmatrix}}}
6657:
6642:
6578:
6313:
6253:
6096:
6024:
4332:, the row space consists of all linear equations that follow from those in the system.
223:
143:
109:
55:
6164:
2450:, in which case some subset of them will form a basis. To find this basis, we reduce
1361:
6728:
6690:
6613:
6573:
6540:
6520:
5749:. The kernel of a linear transformation is analogous to the null space of a matrix.
5701:
4358:
3091:
2304:
1033:
401:
88:
50:. The column space of this matrix is the vector space spanned by the column vectors.
6623:
6512:
6462:
6355:
6009:
5677:
5332:
6238:
17:
6603:
6568:
6525:
6370:
6001:
5773:
5466:
3977:
3716:
3387:
1318:
993:
287:
80:
5854:(Fifth ed.). Wellesley, MA: Wellesley-Cambridge Press. pp. 128, 168.
503:
as coefficients; that is, the columns of the matrix generate the column space.
6632:
6375:
6219:
5931:
Columns without pivots represent free variables in the associated homogeneous
5639:
5429:
5336:
3572:
3528:
38:. The row space of this matrix is the vector space spanned by the row vectors.
5869:
4069:{\displaystyle A={\begin{bmatrix}1&0&2\\0&1&0\end{bmatrix}},}
6430:
6243:
6224:
3367:
3106:
42:
5849:
4467:, in which case the rows will not be a basis. To find a basis, we reduce
6598:
3313:{\displaystyle \operatorname {rank} (A)+\operatorname {nullity} (A)=n.\,}
3118:
To find the basis in a practical setting (e.g., for large matrices), the
2027:{\displaystyle A={\begin{bmatrix}1&0\\0&1\\2&0\end{bmatrix}}}
30:
5761:
5398:{\displaystyle \operatorname {rank} (A)+\operatorname {nullity} (A)=n,}
2446:
The columns of this matrix span the column space, but they may not be
1057:), so it can be deduced that the row space consists of all vectors in
6608:
5343:
of the matrix, and is related to the rank by the following equation:
6012:. It is not necessarily correct over fields and rings with non-zero
5086:, then the resulting basis is uniquely determined by the row space.
4940:
4825:
4704:
4589:
5299:(before any row reductions) also form a basis of the row space of
41:
29:
6282:
3146:
of the matrix. The rank is equal to the number of pivots in the
2238:. In this case, the column space is precisely the set of vectors
4463:
The rows of this matrix span the row space, but they may not be
418:= the maximum number of linearly independent rows or columns of
6286:
4284:. In this case, the row space is precisely the set of vectors
6273:
Lecture on column space and nullspace by
Gilbert Strang of MIT
3946:
are scalars. The set of all possible linear combinations of
1287:
are scalars. The set of all possible linear combinations of
6259:
MIT Linear
Algebra Lecture on the Four Fundamental Subspaces
3682:{\displaystyle \sum \limits _{k=1}^{n}\mathbf {v} _{k}c_{k}}
5331:
The rank of a matrix is also equal to the dimension of the
3105:
is equivalent to finding a basis for the row space of the
6153:, Society for Industrial and Applied Mathematics (SIAM),
1342:
Any linear combination of the column vectors of a matrix
441:
is the set of all linear combinations of the columns in
3615:
Similarly the column space (sometimes disambiguated as
3582:
It follows that the left null space (the null space of
5495:
5209:
5143:
4993:
4878:
4756:
4642:
4523:
4389:
4216:
4184:
4142:
4022:
3425:
3039:
2994:
2949:
2797:
2688:
2576:
2473:
2335:
2172:
2136:
2090:
1981:
1792:
1723:
1576:
1526:
1426:
1372:
916:
845:
768:
697:
536:
5478:
5352:
5122:
4513:
4377:
4126:
4010:
3828:
3635:
3399:
3266:
3211:
3167:
2943:
2467:
2323:
2074:
1969:
1359:
1169:
1063:
970:
895:
824:
747:
676:
524:
328:
299:
226:
176:
146:
112:
3198:{\displaystyle \mathbb {R} ^{4}\to \mathbb {R} ^{4}}
6666:
6622:
6559:
6511:
6453:
6338:
5772:is one-to-one on its coimage, and the coimage maps
2234:The set of all such vectors is the column space of
6095:
6094:Beauregard, Raymond A.; Fraleigh, John B. (1973),
5906:
5586:
5397:
5283:The pivots indicate that the first two columns of
5272:
5060:
4452:
4269:
4068:
3912:
3681:
3516:
3312:
3226:
3197:
3079:
2900:
2435:
2226:
2026:
1909:
1253:
1078:
985:
952:
881:
810:
733:
659:
343:
314:
290:. The row and column spaces are subspaces of the
232:
189:
152:
118:
6078:Linear Algebra and Matrix Analysis for Statistics
3619:column space) can be defined for matrices over a
3575:(perpendicular) to each of the column vectors of
6076:Banerjee, Sudipto; Roy, Anindya (June 6, 2014),
4280:The set of all such vectors is the row space of
2291:span the column space, but they may not form a
493:returns a linear combination of the columns of
253:The row space and the column space of a matrix
5791:is not an inner product space, the coimage of
5642:(perpendicular) to each of the row vectors of
6298:
240:
8:
5668:The row space and null space are two of the
3743:such that it is written in an unusual order
430:, the column space of the matrix equals the
4328:For a matrix that represents a homogeneous
3819:of these vectors is any vector of the form
1160:of these vectors is any vector of the form
6305:
6291:
6283:
6150:Matrix Analysis and Applied Linear Algebra
3205:described by the matrix above maps all of
3033:
3032:
2988:
2987:
5996:
5994:
5568:
5559:
5554:
5537:
5528:
5523:
5513:
5504:
5499:
5490:
5482:
5477:
5351:
5204:
5138:
5128:
5127:
5121:
4988:
4980:
4975:
4965:
4960:
4947:
4942:
4873:
4865:
4860:
4850:
4845:
4832:
4827:
4751:
4743:
4738:
4728:
4723:
4721:
4720:
4711:
4706:
4637:
4629:
4624:
4614:
4609:
4596:
4591:
4518:
4514:
4512:
4384:
4376:
4250:
4235:
4223:
4211:
4179:
4173:
4137:
4131:
4125:
4017:
4009:
3901:
3896:
3889:
3870:
3865:
3858:
3845:
3840:
3833:
3827:
3673:
3663:
3658:
3651:
3640:
3634:
3498:
3489:
3484:
3467:
3458:
3453:
3443:
3434:
3429:
3420:
3412:
3405:
3404:
3398:
3309:
3265:
3218:
3214:
3213:
3210:
3189:
3185:
3184:
3174:
3170:
3169:
3166:
3034:
2989:
2944:
2942:
2792:
2683:
2571:
2468:
2466:
2330:
2322:
2210:
2193:
2179:
2167:
2131:
2125:
2085:
2079:
2073:
1976:
1968:
1897:
1892:
1885:
1866:
1861:
1854:
1823:
1799:
1787:
1781:
1751:
1730:
1718:
1712:
1688:
1678:
1656:
1646:
1622:
1612:
1593:
1583:
1571:
1554:
1533:
1521:
1504:
1484:
1450:
1433:
1421:
1400:
1379:
1367:
1360:
1358:
1242:
1237:
1230:
1211:
1206:
1199:
1186:
1181:
1174:
1168:
1070:
1066:
1065:
1062:
977:
973:
972:
969:
911:
902:
897:
894:
840:
831:
826:
823:
763:
754:
749:
746:
692:
683:
678:
675:
531:
523:
335:
331:
330:
327:
306:
302:
301:
298:
225:
181:
175:
145:
111:
5830:
5412:is the number of columns of the matrix
6696:Comparison of linear algebra libraries
5416:. The equation above is known as the
3406:
1028:. Since these four row vectors are
91:. The column space of a matrix is the
6174:Linear Algebra: A Modern Introduction
6142:(7th ed.), Pearson Prentice Hall
6025:Linear algebra § Further reading
5894:
5882:
3257:columns are related by the equation:
7:
6147:Meyer, Carl D. (February 15, 2001),
5289:form a basis of the column space of
3245:of a matrix is the dimension of the
27:Vector spaces associated to a matrix
6192:Linear Algebra and Its Applications
6122:Linear Algebra and Its Applications
5295:. Therefore, the first two rows of
5113:and reduce it to row echelon form:
3637:
286:This article considers matrices of
217:. A definition for matrices over a
5952:. Actually, this form is merely a
5649:It follows that the null space of
5129:
3140:of the column space is called the
1924:consists of all possible products
201:of the column space is called the
25:
6190:Strang, Gilbert (July 19, 2005),
6120:Lay, David C. (August 22, 2005),
6062:(2nd ed.), Springer-Verlag,
4368:For example, consider the matrix
4357:. This makes it possible to use
4353:The row space is not affected by
3708:, with replacement of the vector
3537:are transposes of column vectors
2314:For example, consider the matrix
1346:can be written as the product of
960:. Consequently, the row space of
6709:
6708:
6686:Basic Linear Algebra Subprograms
6444:
6140:Linear Algebra With Applications
6124:(3rd ed.), Addison Wesley,
5569:
5555:
5538:
5524:
5514:
5500:
5483:
5079:comes from a further reduction.
4976:
4961:
4943:
4861:
4846:
4828:
4739:
4724:
4707:
4625:
4610:
4592:
3897:
3866:
3841:
3659:
3499:
3485:
3468:
3454:
3444:
3430:
3413:
3227:{\displaystyle \mathbb {R} ^{4}}
3153:Because the column space is the
1893:
1862:
1313:. That is, the column space of
1238:
1207:
1182:
1079:{\displaystyle \mathbb {R} ^{5}}
986:{\displaystyle \mathbb {R} ^{5}}
898:
827:
750:
679:
344:{\displaystyle \mathbb {R} ^{m}}
315:{\displaystyle \mathbb {R} ^{n}}
6584:Seven-dimensional cross product
5985:
5907:Beauregard & Fraleigh (1973
5465:can be written in terms of the
5321:of the row space is called the
3386:can be written in terms of the
3331:Relation to the left null space
1920:Therefore, the column space of
434:of this linear transformation.
6000:The example is valid over the
5980:where the order of factors is
5851:Introduction to linear algebra
5662:
5383:
5377:
5365:
5359:
4971:
4856:
4734:
4620:
3719:", which changes the order of
3297:
3291:
3279:
3273:
3180:
2295:if the column vectors are not
2034:, then the column vectors are
1:
6194:(4th ed.), Brooks Cole,
6176:(2nd ed.), Brooks/Cole,
5455:. The product of the matrix
5099:is equal to the row space of
4339:is equal to the row space of
3972:. That is, the row space of
437:The column space of a matrix
205:of the matrix and is at most
6426:Eigenvalues and eigenvectors
3374:. The product of the matrix
3120:singular-value decomposition
1131:matrix, with column vectors
140:matrix with components from
6268:Khan Academy video tutorial
6080:(1st ed.), CRC Press,
6058:Axler, Sheldon Jay (1997),
5103:. Using the example matrix
4099:. A linear combination of
1943:. This is the same as the
483:, the action of the matrix
426:If the matrix represents a
6761:
6278:Row Space and Column Space
6040:(5th ed.), New York:
6022:
5933:system of linear equations
5918:This computation uses the
5670:four fundamental subspaces
5436:is the set of all vectors
5424:Relation to the null space
5310:
4330:system of linear equations
4117:is any vector of the form
3605:four fundamental subspaces
3343:is the set of all vectors
3234:to some three-dimensional
3129:
2068:is any vector of the form
2054:. A linear combination of
6704:
6442:
6320:
6060:Linear Algebra Done Right
6038:Elementary Linear Algebra
5676:(the other two being the
5672:associated with a matrix
5075:. Another possible basis
4355:elementary row operations
4079:then the row vectors are
3790:matrix, with row vectors
3362:. It is the same as the
2301:elementary row operations
257:are sometimes denoted as
130:. The column space of an
6138:Leon, Steven J. (2006),
6104:Houghton Mifflin Company
5848:Strang, Gilbert (2016).
5335:. The dimension of the
5084:reduced row echelon form
4303:satisfying the equation
3611:For matrices over a ring
3148:reduced row echelon form
2456:reduced row echelon form
2257:satisfying the equation
497:with the coordinates of
46:The column vectors of a
5615:are the row vectors of
4323:three-dimensional space
3592:to the column space of
2277:three-dimensional space
1951:) of the corresponding
1086:that are orthogonal to
404:in any echelon form of
6411:Row and column vectors
6261:at Google Video, from
6036:Anton, Howard (1987),
5825:References & Notes
5795:can be defined as the
5768:. The transformation
5725:is the set of vectors
5588:
5399:
5274:
5062:
4454:
4321:through the origin in
4271:
4070:
3914:
3683:
3656:
3518:
3314:
3228:
3199:
3081:
2902:
2437:
2311:for the column space.
2275:through the origin in
2228:
2028:
1911:
1350:with a column vector:
1255:
1080:
987:
954:
883:
812:
735:
661:
345:
316:
250:is defined similarly.
234:
191:
154:
120:
51:
39:
6416:Row and column spaces
6361:Scalar multiplication
6172:Poole, David (2006),
5968:to the column vector
5710:linear transformation
5655:orthogonal complement
5589:
5400:
5313:Rank (linear algebra)
5275:
5063:
4504:represents the rows.
4455:
4315:Cartesian coordinates
4272:
4071:
3915:
3721:scalar multiplication
3684:
3636:
3590:orthogonal complement
3519:
3323:This is known as the
3315:
3229:
3200:
3159:matrix transformation
3157:of the corresponding
3132:Rank (linear algebra)
3082:
2903:
2438:
2269:Cartesian coordinates
2229:
2029:
1953:matrix transformation
1912:
1256:
1081:
1049:is an element of the
988:
955:
884:
813:
736:
662:
428:linear transformation
346:
317:
235:
192:
190:{\displaystyle F^{m}}
155:
121:
101:matrix transformation
99:of the corresponding
83:(set of all possible
45:
34:The row vectors of a
33:
6551:Gram–Schmidt process
6503:Gaussian elimination
5659:rank–nullity theorem
5476:
5418:rank–nullity theorem
5350:
5120:
4511:
4465:linearly independent
4375:
4335:The column space of
4124:
4008:
3826:
3633:
3397:
3325:rank–nullity theorem
3264:
3209:
3165:
2941:
2465:
2448:linearly independent
2321:
2297:linearly independent
2072:
1967:
1357:
1167:
1061:
1030:linearly independent
968:
893:
822:
745:
674:
522:
326:
297:
224:
174:
144:
110:
6681:Numerical stability
6561:Multilinear algebra
6536:Inner product space
6386:Linear independence
5758:inner product space
5688:Relation to coimage
4986:
4871:
4749:
4635:
4365:for the row space.
3122:is typically used.
964:is the subspace of
85:linear combinations
6391:Linear combination
6263:MIT OpenCourseWare
6236:Weisstein, Eric W.
6217:Weisstein, Eric W.
5944:Important only if
5819:Euclidean subspace
5584:
5575:
5395:
5270:
5261:
5195:
5091:determinantal rank
5058:
5056:
5045:
4930:
4808:
4694:
4575:
4450:
4441:
4267:
4258:
4202:
4160:
4066:
4057:
3910:
3817:linear combination
3679:
3514:
3505:
3310:
3224:
3195:
3077:
3068:
3023:
2978:
2898:
2889:
2783:
2674:
2562:
2433:
2424:
2224:
2218:
2158:
2112:
2024:
2018:
1907:
1905:
1834:
1762:
1699:
1562:
1515:
1408:
1251:
1158:linear combination
1076:
983:
950:
944:
879:
873:
808:
802:
731:
725:
657:
651:
341:
312:
230:
187:
150:
116:
52:
40:
6722:
6721:
6589:Geometric algebra
6546:Kronecker product
6381:Linear projection
6366:Vector projection
6201:978-0-03-010567-8
6160:978-0-89871-454-8
6131:978-0-321-28713-7
6087:978-1-42-009538-8
5986:the formula above
5861:978-0-9802327-7-6
4987:
4872:
4750:
4636:
233:{\displaystyle R}
153:{\displaystyle F}
119:{\displaystyle F}
62:(also called the
18:Range of a matrix
16:(Redirected from
6752:
6735:Abstract algebra
6712:
6711:
6594:Exterior algebra
6531:Hadamard product
6448:
6436:Linear equations
6307:
6300:
6293:
6284:
6256:
6249:
6248:
6230:
6229:
6204:
6186:
6168:
6167:on March 1, 2001
6163:, archived from
6143:
6134:
6116:
6101:
6090:
6072:
6054:
6017:
6006:rational numbers
5998:
5989:
5979:
5973:
5967:
5963:
5947:
5942:
5936:
5929:
5923:
5916:
5910:
5904:
5898:
5892:
5886:
5880:
5874:
5873:
5845:
5839:
5835:
5808:
5794:
5790:
5783:
5771:
5767:
5755:
5748:
5734:
5724:
5699:
5695:
5675:
5652:
5645:
5637:
5631:
5618:
5614:
5593:
5591:
5590:
5585:
5580:
5579:
5572:
5564:
5563:
5558:
5541:
5533:
5532:
5527:
5517:
5509:
5508:
5503:
5486:
5464:
5458:
5454:
5441:
5435:
5415:
5411:
5404:
5402:
5401:
5396:
5302:
5298:
5294:
5288:
5279:
5277:
5276:
5271:
5266:
5265:
5200:
5199:
5134:
5133:
5132:
5112:
5106:
5102:
5098:
5078:
5074:
5067:
5065:
5064:
5059:
5057:
5050:
5049:
4985:
4984:
4979:
4970:
4969:
4964:
4952:
4951:
4946:
4936:
4935:
4934:
4870:
4869:
4864:
4855:
4854:
4849:
4837:
4836:
4831:
4821:
4817:
4813:
4812:
4748:
4747:
4742:
4733:
4732:
4727:
4716:
4715:
4710:
4700:
4699:
4698:
4634:
4633:
4628:
4619:
4618:
4613:
4601:
4600:
4595:
4585:
4580:
4579:
4503:
4494:
4485:
4473:row echelon form
4470:
4459:
4457:
4456:
4451:
4446:
4445:
4344:
4338:
4317:, this set is a
4312:
4302:
4283:
4276:
4274:
4273:
4268:
4263:
4262:
4255:
4254:
4240:
4239:
4228:
4227:
4207:
4206:
4178:
4177:
4165:
4164:
4136:
4135:
4116:
4107:
4098:
4088:
4075:
4073:
4072:
4067:
4062:
4061:
4001:For example, if
3997:
3975:
3971:
3963:
3945:
3919:
3917:
3916:
3911:
3906:
3905:
3900:
3894:
3893:
3875:
3874:
3869:
3863:
3862:
3850:
3849:
3844:
3838:
3837:
3814:
3789:
3779:
3767:
3742:
3733:
3711:
3707:
3688:
3686:
3685:
3680:
3678:
3677:
3668:
3667:
3662:
3655:
3650:
3625:
3602:
3595:
3587:
3578:
3570:
3564:
3551:
3547:
3536:
3523:
3521:
3520:
3515:
3510:
3509:
3502:
3494:
3493:
3488:
3471:
3463:
3462:
3457:
3447:
3439:
3438:
3433:
3416:
3411:
3410:
3409:
3385:
3379:
3373:
3361:
3348:
3342:
3319:
3317:
3316:
3311:
3256:
3252:
3233:
3231:
3230:
3225:
3223:
3222:
3217:
3204:
3202:
3201:
3196:
3194:
3193:
3188:
3179:
3178:
3173:
3114:
3104:
3086:
3084:
3083:
3078:
3073:
3072:
3028:
3027:
2983:
2982:
2933:
2907:
2905:
2904:
2899:
2894:
2893:
2788:
2787:
2679:
2678:
2567:
2566:
2453:
2442:
2440:
2439:
2434:
2429:
2428:
2299:. Fortunately,
2290:
2271:, this set is a
2266:
2256:
2237:
2233:
2231:
2230:
2225:
2223:
2222:
2215:
2214:
2198:
2197:
2184:
2183:
2163:
2162:
2130:
2129:
2117:
2116:
2084:
2083:
2053:
2043:
2033:
2031:
2030:
2025:
2023:
2022:
1942:
1932:
1923:
1916:
1914:
1913:
1908:
1906:
1902:
1901:
1896:
1890:
1889:
1871:
1870:
1865:
1859:
1858:
1843:
1839:
1838:
1831:
1830:
1807:
1806:
1786:
1785:
1767:
1766:
1759:
1758:
1735:
1734:
1717:
1716:
1704:
1703:
1696:
1695:
1683:
1682:
1664:
1663:
1651:
1650:
1630:
1629:
1617:
1616:
1598:
1597:
1588:
1587:
1567:
1566:
1559:
1558:
1538:
1537:
1520:
1519:
1512:
1511:
1492:
1491:
1458:
1457:
1438:
1437:
1413:
1412:
1405:
1404:
1384:
1383:
1349:
1345:
1338:
1316:
1312:
1304:
1286:
1260:
1258:
1257:
1252:
1247:
1246:
1241:
1235:
1234:
1216:
1215:
1210:
1204:
1203:
1191:
1190:
1185:
1179:
1178:
1155:
1130:
1120:
1108:
1091:
1085:
1083:
1082:
1077:
1075:
1074:
1069:
1056:
1048:
1042:
1027:
992:
990:
989:
984:
982:
981:
976:
963:
959:
957:
956:
951:
949:
948:
907:
906:
901:
888:
886:
885:
880:
878:
877:
836:
835:
830:
817:
815:
814:
809:
807:
806:
759:
758:
753:
740:
738:
737:
732:
730:
729:
688:
687:
682:
666:
664:
663:
658:
656:
655:
514:
502:
496:
492:
486:
482:
475:
451:
444:
440:
421:
417:
407:
399:
389:
370:
366:
362:
350:
348:
347:
342:
340:
339:
334:
321:
319:
318:
313:
311:
310:
305:
282:
269:
256:
241:is also possible
239:
237:
236:
231:
216:
196:
194:
193:
188:
186:
185:
159:
157:
156:
151:
139:
125:
123:
122:
117:
21:
6760:
6759:
6755:
6754:
6753:
6751:
6750:
6749:
6725:
6724:
6723:
6718:
6700:
6662:
6618:
6555:
6507:
6449:
6440:
6406:Change of basis
6396:Multilinear map
6334:
6316:
6311:
6252:
6234:
6233:
6215:
6214:
6211:
6202:
6189:
6184:
6171:
6161:
6146:
6137:
6132:
6119:
6114:
6093:
6088:
6075:
6070:
6057:
6052:
6035:
6032:
6030:Further reading
6027:
6021:
6020:
5999:
5992:
5975:
5969:
5965:
5956:
5945:
5943:
5939:
5930:
5926:
5917:
5913:
5905:
5901:
5893:
5889:
5881:
5877:
5862:
5847:
5846:
5842:
5836:
5832:
5827:
5815:
5799:
5792:
5788:
5781:
5769:
5765:
5753:
5736:
5726:
5712:
5697:
5693:
5690:
5682:left null space
5673:
5650:
5643:
5633:
5632:if and only if
5620:
5616:
5613:
5604:
5598:
5574:
5573:
5553:
5550:
5549:
5543:
5542:
5522:
5519:
5518:
5498:
5491:
5474:
5473:
5460:
5459:and the vector
5456:
5443:
5437:
5433:
5426:
5413:
5409:
5348:
5347:
5315:
5309:
5300:
5296:
5290:
5284:
5260:
5259:
5254:
5249:
5243:
5242:
5237:
5232:
5226:
5225:
5220:
5215:
5205:
5194:
5193:
5188:
5183:
5177:
5176:
5171:
5166:
5160:
5159:
5154:
5149:
5139:
5123:
5118:
5117:
5108:
5104:
5100:
5094:
5076:
5072:
5055:
5054:
5044:
5043:
5038:
5033:
5027:
5026:
5021:
5016:
5010:
5009:
5004:
4999:
4989:
4974:
4959:
4941:
4929:
4928:
4923:
4918:
4912:
4911:
4906:
4901:
4895:
4894:
4889:
4884:
4874:
4859:
4844:
4826:
4815:
4814:
4807:
4806:
4801:
4796:
4790:
4789:
4784:
4779:
4773:
4772:
4767:
4762:
4752:
4737:
4722:
4705:
4693:
4692:
4687:
4682:
4676:
4675:
4670:
4665:
4659:
4658:
4653:
4648:
4638:
4623:
4608:
4590:
4581:
4574:
4573:
4568:
4563:
4557:
4556:
4551:
4546:
4540:
4539:
4534:
4529:
4519:
4509:
4508:
4502:
4496:
4493:
4487:
4484:
4478:
4468:
4440:
4439:
4434:
4429:
4423:
4422:
4417:
4412:
4406:
4405:
4400:
4395:
4385:
4373:
4372:
4351:
4340:
4336:
4304:
4285:
4281:
4257:
4256:
4246:
4241:
4231:
4229:
4219:
4212:
4201:
4200:
4195:
4190:
4180:
4169:
4159:
4158:
4153:
4148:
4138:
4127:
4122:
4121:
4115:
4109:
4106:
4100:
4096:
4090:
4086:
4080:
4056:
4055:
4050:
4045:
4039:
4038:
4033:
4028:
4018:
4006:
4005:
3996:
3987:
3981:
3980:of the vectors
3973:
3969:
3962:
3953:
3947:
3943:
3937:
3930:
3924:
3895:
3885:
3864:
3854:
3839:
3829:
3824:
3823:
3813:
3804:
3797:
3791:
3781:
3777:
3765:
3762:
3757:
3740:
3735:
3732:
3724:
3709:
3705:
3699:
3693:
3669:
3657:
3631:
3630:
3623:
3613:
3600:
3593:
3583:
3576:
3566:
3565:if and only if
3553:
3549:
3546:
3538:
3532:
3504:
3503:
3483:
3480:
3479:
3473:
3472:
3452:
3449:
3448:
3428:
3421:
3400:
3395:
3394:
3381:
3380:and the vector
3375:
3371:
3350:
3344:
3340:
3337:left null space
3333:
3262:
3261:
3254:
3250:
3212:
3207:
3206:
3183:
3168:
3163:
3162:
3134:
3128:
3110:
3102:
3067:
3066:
3060:
3059:
3053:
3052:
3046:
3045:
3035:
3022:
3021:
3015:
3014:
3008:
3007:
3001:
3000:
2990:
2977:
2976:
2970:
2969:
2963:
2962:
2956:
2955:
2945:
2939:
2938:
2932:
2925:
2918:
2912:
2888:
2887:
2882:
2877:
2872:
2866:
2865:
2860:
2855:
2850:
2844:
2843:
2838:
2833:
2828:
2822:
2821:
2816:
2808:
2803:
2793:
2782:
2781:
2776:
2771:
2766:
2760:
2759:
2751:
2746:
2741:
2735:
2734:
2729:
2724:
2719:
2713:
2712:
2707:
2699:
2694:
2684:
2673:
2672:
2667:
2659:
2651:
2645:
2644:
2636:
2631:
2626:
2620:
2619:
2614:
2609:
2604:
2598:
2597:
2592:
2587:
2582:
2572:
2561:
2560:
2555:
2550:
2545:
2539:
2538:
2533:
2528:
2523:
2517:
2516:
2511:
2506:
2501:
2495:
2494:
2489:
2484:
2479:
2469:
2463:
2462:
2451:
2423:
2422:
2417:
2412:
2407:
2401:
2400:
2395:
2390:
2385:
2379:
2378:
2373:
2368:
2363:
2357:
2356:
2351:
2346:
2341:
2331:
2319:
2318:
2288:
2287:The columns of
2285:
2258:
2239:
2235:
2217:
2216:
2206:
2200:
2199:
2189:
2186:
2185:
2175:
2168:
2157:
2156:
2150:
2149:
2143:
2142:
2132:
2121:
2111:
2110:
2104:
2103:
2097:
2096:
2086:
2075:
2070:
2069:
2067:
2060:
2051:
2045:
2041:
2035:
2017:
2016:
2011:
2005:
2004:
1999:
1993:
1992:
1987:
1977:
1965:
1964:
1961:
1934:
1925:
1921:
1904:
1903:
1891:
1881:
1860:
1850:
1848:
1841:
1840:
1833:
1832:
1819:
1816:
1815:
1809:
1808:
1795:
1788:
1777:
1761:
1760:
1747:
1744:
1743:
1737:
1736:
1726:
1719:
1708:
1698:
1697:
1684:
1674:
1652:
1642:
1639:
1638:
1632:
1631:
1618:
1608:
1589:
1579:
1572:
1561:
1560:
1550:
1547:
1546:
1540:
1539:
1529:
1522:
1514:
1513:
1500:
1498:
1493:
1480:
1477:
1476:
1471:
1466:
1460:
1459:
1446:
1444:
1439:
1429:
1422:
1419:
1414:
1407:
1406:
1396:
1393:
1392:
1386:
1385:
1375:
1368:
1355:
1354:
1347:
1343:
1337:
1328:
1322:
1321:of the vectors
1314:
1310:
1303:
1294:
1288:
1284:
1278:
1271:
1265:
1236:
1226:
1205:
1195:
1180:
1170:
1165:
1164:
1154:
1145:
1138:
1132:
1122:
1118:
1106:
1103:
1098:
1087:
1064:
1059:
1058:
1054:
1044:
1037:
1025:
1018:
1011:
1004:
997:
971:
966:
965:
961:
943:
942:
937:
932:
927:
922:
912:
896:
891:
890:
872:
871:
866:
861:
856:
851:
841:
825:
820:
819:
801:
800:
795:
790:
785:
777:
764:
748:
743:
742:
724:
723:
718:
713:
708:
703:
693:
677:
672:
671:
650:
649:
644:
639:
634:
629:
623:
622:
617:
612:
607:
602:
596:
595:
590:
585:
580:
572:
563:
562:
557:
552:
547:
542:
532:
520:
519:
512:
511:Given a matrix
509:
498:
494:
488:
484:
480:
479:Given a matrix
473:
464:
453:
446:
442:
438:
419:
411:
405:
393:
384:)) = dim(colsp(
375:
368:
364:
360:
357:
329:
324:
323:
300:
295:
294:
271:
258:
254:
222:
221:
206:
177:
172:
171:
162:linear subspace
142:
141:
131:
108:
107:
28:
23:
22:
15:
12:
11:
5:
6758:
6756:
6748:
6747:
6742:
6740:Linear algebra
6737:
6727:
6726:
6720:
6719:
6717:
6716:
6705:
6702:
6701:
6699:
6698:
6693:
6688:
6683:
6678:
6676:Floating-point
6672:
6670:
6664:
6663:
6661:
6660:
6658:Tensor product
6655:
6650:
6645:
6643:Function space
6640:
6635:
6629:
6627:
6620:
6619:
6617:
6616:
6611:
6606:
6601:
6596:
6591:
6586:
6581:
6579:Triple product
6576:
6571:
6565:
6563:
6557:
6556:
6554:
6553:
6548:
6543:
6538:
6533:
6528:
6523:
6517:
6515:
6509:
6508:
6506:
6505:
6500:
6495:
6493:Transformation
6490:
6485:
6483:Multiplication
6480:
6475:
6470:
6465:
6459:
6457:
6451:
6450:
6443:
6441:
6439:
6438:
6433:
6428:
6423:
6418:
6413:
6408:
6403:
6398:
6393:
6388:
6383:
6378:
6373:
6368:
6363:
6358:
6353:
6348:
6342:
6340:
6339:Basic concepts
6336:
6335:
6333:
6332:
6327:
6321:
6318:
6317:
6314:Linear algebra
6312:
6310:
6309:
6302:
6295:
6287:
6281:
6280:
6275:
6270:
6265:
6254:Gilbert Strang
6250:
6239:"Column Space"
6231:
6210:
6209:External links
6207:
6206:
6205:
6200:
6187:
6182:
6169:
6159:
6144:
6135:
6130:
6117:
6112:
6091:
6086:
6073:
6068:
6055:
6050:
6031:
6028:
6019:
6018:
6014:characteristic
5990:
5964:of the matrix
5937:
5924:
5911:
5909:, p. 254)
5899:
5897:, p. 183)
5887:
5885:, p. 179)
5875:
5860:
5840:
5829:
5828:
5826:
5823:
5822:
5821:
5814:
5811:
5797:quotient space
5774:isomorphically
5689:
5686:
5609:
5602:
5595:
5594:
5583:
5578:
5571:
5567:
5562:
5557:
5552:
5551:
5548:
5545:
5544:
5540:
5536:
5531:
5526:
5521:
5520:
5516:
5512:
5507:
5502:
5497:
5496:
5494:
5489:
5485:
5481:
5425:
5422:
5406:
5405:
5394:
5391:
5388:
5385:
5382:
5379:
5376:
5373:
5370:
5367:
5364:
5361:
5358:
5355:
5339:is called the
5311:Main article:
5308:
5305:
5281:
5280:
5269:
5264:
5258:
5255:
5253:
5250:
5248:
5245:
5244:
5241:
5238:
5236:
5233:
5231:
5228:
5227:
5224:
5221:
5219:
5216:
5214:
5211:
5210:
5208:
5203:
5198:
5192:
5189:
5187:
5184:
5182:
5179:
5178:
5175:
5172:
5170:
5167:
5165:
5162:
5161:
5158:
5155:
5153:
5150:
5148:
5145:
5144:
5142:
5137:
5131:
5126:
5069:
5068:
5053:
5048:
5042:
5039:
5037:
5034:
5032:
5029:
5028:
5025:
5022:
5020:
5017:
5015:
5012:
5011:
5008:
5005:
5003:
5000:
4998:
4995:
4994:
4992:
4983:
4978:
4973:
4968:
4963:
4958:
4955:
4950:
4945:
4939:
4933:
4927:
4924:
4922:
4919:
4917:
4914:
4913:
4910:
4907:
4905:
4902:
4900:
4897:
4896:
4893:
4890:
4888:
4885:
4883:
4880:
4879:
4877:
4868:
4863:
4858:
4853:
4848:
4843:
4840:
4835:
4830:
4824:
4820:
4818:
4816:
4811:
4805:
4802:
4800:
4797:
4795:
4792:
4791:
4788:
4785:
4783:
4780:
4778:
4775:
4774:
4771:
4768:
4766:
4763:
4761:
4758:
4757:
4755:
4746:
4741:
4736:
4731:
4726:
4719:
4714:
4709:
4703:
4697:
4691:
4688:
4686:
4683:
4681:
4678:
4677:
4674:
4671:
4669:
4666:
4664:
4661:
4660:
4657:
4654:
4652:
4649:
4647:
4644:
4643:
4641:
4632:
4627:
4622:
4617:
4612:
4607:
4604:
4599:
4594:
4588:
4584:
4582:
4578:
4572:
4569:
4567:
4564:
4562:
4559:
4558:
4555:
4552:
4550:
4547:
4545:
4542:
4541:
4538:
4535:
4533:
4530:
4528:
4525:
4524:
4522:
4517:
4516:
4500:
4491:
4482:
4461:
4460:
4449:
4444:
4438:
4435:
4433:
4430:
4428:
4425:
4424:
4421:
4418:
4416:
4413:
4411:
4408:
4407:
4404:
4401:
4399:
4396:
4394:
4391:
4390:
4388:
4383:
4380:
4350:
4347:
4278:
4277:
4266:
4261:
4253:
4249:
4245:
4242:
4238:
4234:
4230:
4226:
4222:
4218:
4217:
4215:
4210:
4205:
4199:
4196:
4194:
4191:
4189:
4186:
4185:
4183:
4176:
4172:
4168:
4163:
4157:
4154:
4152:
4149:
4147:
4144:
4143:
4141:
4134:
4130:
4113:
4104:
4094:
4084:
4077:
4076:
4065:
4060:
4054:
4051:
4049:
4046:
4044:
4041:
4040:
4037:
4034:
4032:
4029:
4027:
4024:
4023:
4021:
4016:
4013:
3992:
3985:
3964:is called the
3958:
3951:
3941:
3935:
3928:
3921:
3920:
3909:
3904:
3899:
3892:
3888:
3884:
3881:
3878:
3873:
3868:
3861:
3857:
3853:
3848:
3843:
3836:
3832:
3809:
3802:
3795:
3761:
3758:
3756:
3753:
3738:
3734:to the scalar
3728:
3723:of the vector
3703:
3697:
3690:
3689:
3676:
3672:
3666:
3661:
3654:
3649:
3646:
3643:
3639:
3612:
3609:
3542:
3525:
3524:
3513:
3508:
3501:
3497:
3492:
3487:
3482:
3481:
3478:
3475:
3474:
3470:
3466:
3461:
3456:
3451:
3450:
3446:
3442:
3437:
3432:
3427:
3426:
3424:
3419:
3415:
3408:
3403:
3332:
3329:
3321:
3320:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3221:
3216:
3192:
3187:
3182:
3177:
3172:
3130:Main article:
3127:
3124:
3088:
3087:
3076:
3071:
3065:
3062:
3061:
3058:
3055:
3054:
3051:
3048:
3047:
3044:
3041:
3040:
3038:
3031:
3026:
3020:
3017:
3016:
3013:
3010:
3009:
3006:
3003:
3002:
2999:
2996:
2995:
2993:
2986:
2981:
2975:
2972:
2971:
2968:
2965:
2964:
2961:
2958:
2957:
2954:
2951:
2950:
2948:
2930:
2923:
2916:
2909:
2908:
2897:
2892:
2886:
2883:
2881:
2878:
2876:
2873:
2871:
2868:
2867:
2864:
2861:
2859:
2856:
2854:
2851:
2849:
2846:
2845:
2842:
2839:
2837:
2834:
2832:
2829:
2827:
2824:
2823:
2820:
2817:
2815:
2812:
2809:
2807:
2804:
2802:
2799:
2798:
2796:
2791:
2786:
2780:
2777:
2775:
2772:
2770:
2767:
2765:
2762:
2761:
2758:
2755:
2752:
2750:
2747:
2745:
2742:
2740:
2737:
2736:
2733:
2730:
2728:
2725:
2723:
2720:
2718:
2715:
2714:
2711:
2708:
2706:
2703:
2700:
2698:
2695:
2693:
2690:
2689:
2687:
2682:
2677:
2671:
2668:
2666:
2663:
2660:
2658:
2655:
2652:
2650:
2647:
2646:
2643:
2640:
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2621:
2618:
2615:
2613:
2610:
2608:
2605:
2603:
2600:
2599:
2596:
2593:
2591:
2588:
2586:
2583:
2581:
2578:
2577:
2575:
2570:
2565:
2559:
2556:
2554:
2551:
2549:
2546:
2544:
2541:
2540:
2537:
2534:
2532:
2529:
2527:
2524:
2522:
2519:
2518:
2515:
2512:
2510:
2507:
2505:
2502:
2500:
2497:
2496:
2493:
2490:
2488:
2485:
2483:
2480:
2478:
2475:
2474:
2472:
2444:
2443:
2432:
2427:
2421:
2418:
2416:
2413:
2411:
2408:
2406:
2403:
2402:
2399:
2396:
2394:
2391:
2389:
2386:
2384:
2381:
2380:
2377:
2374:
2372:
2369:
2367:
2364:
2362:
2359:
2358:
2355:
2352:
2350:
2347:
2345:
2342:
2340:
2337:
2336:
2334:
2329:
2326:
2284:
2281:
2221:
2213:
2209:
2205:
2202:
2201:
2196:
2192:
2188:
2187:
2182:
2178:
2174:
2173:
2171:
2166:
2161:
2155:
2152:
2151:
2148:
2145:
2144:
2141:
2138:
2137:
2135:
2128:
2124:
2120:
2115:
2109:
2106:
2105:
2102:
2099:
2098:
2095:
2092:
2091:
2089:
2082:
2078:
2065:
2058:
2049:
2039:
2021:
2015:
2012:
2010:
2007:
2006:
2003:
2000:
1998:
1995:
1994:
1991:
1988:
1986:
1983:
1982:
1980:
1975:
1972:
1960:
1957:
1918:
1917:
1900:
1895:
1888:
1884:
1880:
1877:
1874:
1869:
1864:
1857:
1853:
1849:
1847:
1844:
1842:
1837:
1829:
1826:
1822:
1818:
1817:
1814:
1811:
1810:
1805:
1802:
1798:
1794:
1793:
1791:
1784:
1780:
1776:
1773:
1770:
1765:
1757:
1754:
1750:
1746:
1745:
1742:
1739:
1738:
1733:
1729:
1725:
1724:
1722:
1715:
1711:
1707:
1702:
1694:
1691:
1687:
1681:
1677:
1673:
1670:
1667:
1662:
1659:
1655:
1649:
1645:
1641:
1640:
1637:
1634:
1633:
1628:
1625:
1621:
1615:
1611:
1607:
1604:
1601:
1596:
1592:
1586:
1582:
1578:
1577:
1575:
1570:
1565:
1557:
1553:
1549:
1548:
1545:
1542:
1541:
1536:
1532:
1528:
1527:
1525:
1518:
1510:
1507:
1503:
1499:
1497:
1494:
1490:
1487:
1483:
1479:
1478:
1475:
1472:
1470:
1467:
1465:
1462:
1461:
1456:
1453:
1449:
1445:
1443:
1440:
1436:
1432:
1428:
1427:
1425:
1420:
1418:
1415:
1411:
1403:
1399:
1395:
1394:
1391:
1388:
1387:
1382:
1378:
1374:
1373:
1371:
1366:
1363:
1362:
1333:
1326:
1305:is called the
1299:
1292:
1282:
1276:
1269:
1262:
1261:
1250:
1245:
1240:
1233:
1229:
1225:
1222:
1219:
1214:
1209:
1202:
1198:
1194:
1189:
1184:
1177:
1173:
1150:
1143:
1136:
1102:
1099:
1097:
1094:
1073:
1068:
1036:to the vector
1023:
1016:
1009:
1002:
980:
975:
947:
941:
938:
936:
933:
931:
928:
926:
923:
921:
918:
917:
915:
910:
905:
900:
876:
870:
867:
865:
862:
860:
857:
855:
852:
850:
847:
846:
844:
839:
834:
829:
805:
799:
796:
794:
791:
789:
786:
784:
781:
778:
776:
773:
770:
769:
767:
762:
757:
752:
728:
722:
719:
717:
714:
712:
709:
707:
704:
702:
699:
698:
696:
691:
686:
681:
668:
667:
654:
648:
645:
643:
640:
638:
635:
633:
630:
628:
625:
624:
621:
618:
616:
613:
611:
608:
606:
603:
601:
598:
597:
594:
591:
589:
586:
584:
581:
579:
576:
573:
571:
568:
565:
564:
561:
558:
556:
553:
551:
548:
546:
543:
541:
538:
537:
535:
530:
527:
508:
505:
469:
462:
424:
423:
409:
391:
380:) = dim(rowsp(
356:
353:
351:respectively.
338:
333:
309:
304:
283:respectively.
229:
184:
180:
149:
115:
89:column vectors
56:linear algebra
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6757:
6746:
6743:
6741:
6738:
6736:
6733:
6732:
6730:
6715:
6707:
6706:
6703:
6697:
6694:
6692:
6691:Sparse matrix
6689:
6687:
6684:
6682:
6679:
6677:
6674:
6673:
6671:
6669:
6665:
6659:
6656:
6654:
6651:
6649:
6646:
6644:
6641:
6639:
6636:
6634:
6631:
6630:
6628:
6626:constructions
6625:
6621:
6615:
6614:Outermorphism
6612:
6610:
6607:
6605:
6602:
6600:
6597:
6595:
6592:
6590:
6587:
6585:
6582:
6580:
6577:
6575:
6574:Cross product
6572:
6570:
6567:
6566:
6564:
6562:
6558:
6552:
6549:
6547:
6544:
6542:
6541:Outer product
6539:
6537:
6534:
6532:
6529:
6527:
6524:
6522:
6521:Orthogonality
6519:
6518:
6516:
6514:
6510:
6504:
6501:
6499:
6498:Cramer's rule
6496:
6494:
6491:
6489:
6486:
6484:
6481:
6479:
6476:
6474:
6471:
6469:
6468:Decomposition
6466:
6464:
6461:
6460:
6458:
6456:
6452:
6447:
6437:
6434:
6432:
6429:
6427:
6424:
6422:
6419:
6417:
6414:
6412:
6409:
6407:
6404:
6402:
6399:
6397:
6394:
6392:
6389:
6387:
6384:
6382:
6379:
6377:
6374:
6372:
6369:
6367:
6364:
6362:
6359:
6357:
6354:
6352:
6349:
6347:
6344:
6343:
6341:
6337:
6331:
6328:
6326:
6323:
6322:
6319:
6315:
6308:
6303:
6301:
6296:
6294:
6289:
6288:
6285:
6279:
6276:
6274:
6271:
6269:
6266:
6264:
6260:
6255:
6251:
6246:
6245:
6240:
6237:
6232:
6227:
6226:
6221:
6218:
6213:
6212:
6208:
6203:
6197:
6193:
6188:
6185:
6183:0-534-99845-3
6179:
6175:
6170:
6166:
6162:
6156:
6152:
6151:
6145:
6141:
6136:
6133:
6127:
6123:
6118:
6115:
6113:0-395-14017-X
6109:
6105:
6100:
6099:
6092:
6089:
6083:
6079:
6074:
6071:
6069:0-387-98259-0
6065:
6061:
6056:
6053:
6051:0-471-84819-0
6047:
6043:
6039:
6034:
6033:
6029:
6026:
6015:
6011:
6010:number fields
6007:
6003:
5997:
5995:
5991:
5987:
5983:
5978:
5972:
5962:
5959:
5955:
5951:
5941:
5938:
5934:
5928:
5925:
5921:
5915:
5912:
5908:
5903:
5900:
5896:
5891:
5888:
5884:
5879:
5876:
5871:
5867:
5863:
5857:
5853:
5852:
5844:
5841:
5834:
5831:
5824:
5820:
5817:
5816:
5812:
5810:
5806:
5802:
5798:
5785:
5779:
5775:
5763:
5759:
5750:
5747:
5743:
5739:
5733:
5729:
5723:
5719:
5715:
5711:
5707:
5703:
5702:vector spaces
5687:
5685:
5683:
5679:
5671:
5666:
5664:
5660:
5656:
5647:
5641:
5636:
5630:
5626:
5623:
5612:
5608:
5601:
5581:
5576:
5565:
5560:
5546:
5534:
5529:
5510:
5505:
5492:
5487:
5479:
5472:
5471:
5470:
5468:
5463:
5453:
5449:
5446:
5440:
5431:
5423:
5421:
5419:
5392:
5389:
5386:
5380:
5374:
5371:
5368:
5362:
5356:
5353:
5346:
5345:
5344:
5342:
5338:
5334:
5329:
5326:
5325:
5320:
5314:
5306:
5304:
5293:
5287:
5267:
5262:
5256:
5251:
5246:
5239:
5234:
5229:
5222:
5217:
5212:
5206:
5201:
5196:
5190:
5185:
5180:
5173:
5168:
5163:
5156:
5151:
5146:
5140:
5135:
5124:
5116:
5115:
5114:
5111:
5097:
5092:
5087:
5085:
5080:
5051:
5046:
5040:
5035:
5030:
5023:
5018:
5013:
5006:
5001:
4996:
4990:
4981:
4966:
4956:
4953:
4948:
4937:
4931:
4925:
4920:
4915:
4908:
4903:
4898:
4891:
4886:
4881:
4875:
4866:
4851:
4841:
4838:
4833:
4822:
4819:
4809:
4803:
4798:
4793:
4786:
4781:
4776:
4769:
4764:
4759:
4753:
4744:
4729:
4717:
4712:
4701:
4695:
4689:
4684:
4679:
4672:
4667:
4662:
4655:
4650:
4645:
4639:
4630:
4615:
4605:
4602:
4597:
4586:
4583:
4576:
4570:
4565:
4560:
4553:
4548:
4543:
4536:
4531:
4526:
4520:
4507:
4506:
4505:
4499:
4490:
4481:
4476:
4474:
4466:
4447:
4442:
4436:
4431:
4426:
4419:
4414:
4409:
4402:
4397:
4392:
4386:
4381:
4378:
4371:
4370:
4369:
4366:
4364:
4360:
4359:row reduction
4356:
4348:
4346:
4343:
4333:
4331:
4326:
4324:
4320:
4316:
4311:
4307:
4301:
4297:
4293:
4289:
4264:
4259:
4251:
4247:
4243:
4236:
4232:
4224:
4220:
4213:
4208:
4203:
4197:
4192:
4187:
4181:
4174:
4170:
4166:
4161:
4155:
4150:
4145:
4139:
4132:
4128:
4120:
4119:
4118:
4112:
4103:
4093:
4083:
4063:
4058:
4052:
4047:
4042:
4035:
4030:
4025:
4019:
4014:
4011:
4004:
4003:
4002:
3999:
3995:
3991:
3984:
3979:
3967:
3961:
3957:
3950:
3944:
3934:
3927:
3907:
3902:
3890:
3886:
3882:
3879:
3876:
3871:
3859:
3855:
3851:
3846:
3834:
3830:
3822:
3821:
3820:
3818:
3812:
3808:
3801:
3794:
3788:
3784:
3775:
3771:
3759:
3754:
3752:
3750:
3746:
3741:
3731:
3727:
3722:
3718:
3715:
3712:-space with "
3706:
3696:
3674:
3670:
3664:
3652:
3647:
3644:
3641:
3629:
3628:
3627:
3622:
3618:
3610:
3608:
3606:
3599:For a matrix
3597:
3591:
3586:
3580:
3574:
3569:
3563:
3559:
3556:
3545:
3541:
3535:
3530:
3511:
3506:
3495:
3490:
3476:
3464:
3459:
3440:
3435:
3422:
3417:
3401:
3393:
3392:
3391:
3389:
3384:
3378:
3369:
3365:
3360:
3356:
3353:
3347:
3338:
3330:
3328:
3326:
3306:
3303:
3300:
3294:
3288:
3285:
3282:
3276:
3270:
3267:
3260:
3259:
3258:
3248:
3244:
3239:
3237:
3219:
3190:
3175:
3160:
3156:
3151:
3149:
3145:
3144:
3139:
3133:
3125:
3123:
3121:
3116:
3113:
3108:
3099:
3097:
3093:
3074:
3069:
3063:
3056:
3049:
3042:
3036:
3029:
3024:
3018:
3011:
3004:
2997:
2991:
2984:
2979:
2973:
2966:
2959:
2952:
2946:
2937:
2936:
2935:
2929:
2922:
2915:
2895:
2890:
2884:
2879:
2874:
2869:
2862:
2857:
2852:
2847:
2840:
2835:
2830:
2825:
2818:
2813:
2810:
2805:
2800:
2794:
2789:
2784:
2778:
2773:
2768:
2763:
2756:
2753:
2748:
2743:
2738:
2731:
2726:
2721:
2716:
2709:
2704:
2701:
2696:
2691:
2685:
2680:
2675:
2669:
2664:
2661:
2656:
2653:
2648:
2641:
2638:
2633:
2628:
2623:
2616:
2611:
2606:
2601:
2594:
2589:
2584:
2579:
2573:
2568:
2563:
2557:
2552:
2547:
2542:
2535:
2530:
2525:
2520:
2513:
2508:
2503:
2498:
2491:
2486:
2481:
2476:
2470:
2461:
2460:
2459:
2457:
2449:
2430:
2425:
2419:
2414:
2409:
2404:
2397:
2392:
2387:
2382:
2375:
2370:
2365:
2360:
2353:
2348:
2343:
2338:
2332:
2327:
2324:
2317:
2316:
2315:
2312:
2310:
2306:
2305:row reduction
2302:
2298:
2294:
2282:
2280:
2278:
2274:
2270:
2265:
2261:
2255:
2251:
2247:
2243:
2219:
2211:
2207:
2203:
2194:
2190:
2180:
2176:
2169:
2164:
2159:
2153:
2146:
2139:
2133:
2126:
2122:
2118:
2113:
2107:
2100:
2093:
2087:
2080:
2076:
2064:
2057:
2048:
2038:
2019:
2013:
2008:
2001:
1996:
1989:
1984:
1978:
1973:
1970:
1958:
1956:
1954:
1950:
1946:
1941:
1937:
1931:
1928:
1898:
1886:
1882:
1878:
1875:
1872:
1867:
1855:
1851:
1845:
1835:
1827:
1824:
1820:
1812:
1803:
1800:
1796:
1789:
1782:
1778:
1774:
1771:
1768:
1763:
1755:
1752:
1748:
1740:
1731:
1727:
1720:
1713:
1709:
1705:
1700:
1692:
1689:
1685:
1679:
1675:
1671:
1668:
1665:
1660:
1657:
1653:
1647:
1643:
1635:
1626:
1623:
1619:
1613:
1609:
1605:
1602:
1599:
1594:
1590:
1584:
1580:
1573:
1568:
1563:
1555:
1551:
1543:
1534:
1530:
1523:
1516:
1508:
1505:
1501:
1495:
1488:
1485:
1481:
1473:
1468:
1463:
1454:
1451:
1447:
1441:
1434:
1430:
1423:
1416:
1409:
1401:
1397:
1389:
1380:
1376:
1369:
1364:
1353:
1352:
1351:
1340:
1336:
1332:
1325:
1320:
1308:
1302:
1298:
1291:
1285:
1275:
1268:
1248:
1243:
1231:
1227:
1223:
1220:
1217:
1212:
1200:
1196:
1192:
1187:
1175:
1171:
1163:
1162:
1161:
1159:
1153:
1149:
1142:
1135:
1129:
1125:
1116:
1112:
1100:
1095:
1093:
1090:
1071:
1052:
1047:
1040:
1035:
1031:
1022:
1015:
1008:
1001:
995:
978:
945:
939:
934:
929:
924:
919:
913:
908:
903:
874:
868:
863:
858:
853:
848:
842:
837:
832:
803:
797:
792:
787:
782:
779:
774:
771:
765:
760:
755:
726:
720:
715:
710:
705:
700:
694:
689:
684:
670:the rows are
652:
646:
641:
636:
631:
626:
619:
614:
609:
604:
599:
592:
587:
582:
577:
574:
569:
566:
559:
554:
549:
544:
539:
533:
528:
525:
518:
517:
516:
506:
504:
501:
491:
477:
472:
468:
461:
457:
449:
435:
433:
429:
415:
410:
403:
397:
392:
387:
383:
379:
374:
373:
372:
371:matrix. Then
354:
352:
336:
307:
293:
289:
284:
280:
276:
275:
267:
263:
262:
251:
249:
244:
242:
227:
220:
214:
210:
204:
200:
182:
178:
170:
168:
163:
147:
138:
134:
129:
113:
104:
102:
98:
94:
90:
86:
82:
78:
75:
71:
70:
65:
61:
57:
49:
44:
37:
32:
19:
6624:Vector space
6415:
6356:Vector space
6242:
6223:
6191:
6173:
6165:the original
6149:
6139:
6121:
6097:
6077:
6059:
6037:
6008:, and other
6002:real numbers
5981:
5976:
5970:
5960:
5957:
5940:
5927:
5920:Gauss–Jordan
5914:
5902:
5890:
5878:
5850:
5843:
5833:
5804:
5800:
5786:
5751:
5745:
5741:
5737:
5731:
5727:
5721:
5717:
5713:
5691:
5678:column space
5667:
5648:
5634:
5628:
5624:
5621:
5610:
5606:
5599:
5596:
5469:of vectors:
5461:
5451:
5447:
5444:
5438:
5427:
5407:
5340:
5333:column space
5330:
5322:
5316:
5291:
5285:
5282:
5109:
5107:above, find
5095:
5088:
5081:
5070:
4497:
4488:
4479:
4477:
4462:
4367:
4352:
4341:
4334:
4327:
4309:
4305:
4299:
4295:
4291:
4287:
4279:
4110:
4101:
4091:
4081:
4078:
4000:
3993:
3989:
3982:
3965:
3959:
3955:
3948:
3939:
3932:
3925:
3922:
3810:
3806:
3799:
3792:
3786:
3782:
3763:
3748:
3744:
3736:
3729:
3725:
3701:
3694:
3691:
3616:
3614:
3604:
3598:
3584:
3581:
3567:
3561:
3557:
3554:
3543:
3539:
3533:
3526:
3390:of vectors:
3382:
3376:
3358:
3354:
3351:
3345:
3334:
3322:
3242:
3240:
3152:
3141:
3135:
3117:
3111:
3109:matrix
3100:
3096:echelon form
3089:
2927:
2920:
2913:
2910:
2445:
2313:
2286:
2263:
2259:
2253:
2249:
2245:
2241:
2062:
2055:
2046:
2036:
1962:
1939:
1935:
1929:
1926:
1919:
1341:
1334:
1330:
1323:
1307:column space
1306:
1300:
1296:
1289:
1280:
1273:
1266:
1263:
1151:
1147:
1140:
1133:
1127:
1123:
1104:
1096:Column space
1088:
1045:
1038:
1020:
1013:
1006:
999:
669:
510:
499:
489:
487:on a vector
478:
470:
466:
459:
455:
447:
436:
425:
413:
400:= number of
395:
385:
381:
377:
358:
288:real numbers
285:
278:
273:
272:
265:
260:
259:
252:
247:
245:
212:
208:
166:
136:
132:
105:
76:
68:
63:
60:column space
59:
53:
6604:Multivector
6569:Determinant
6526:Dot product
6371:Linear span
6220:"Row Space"
5950:commutative
5895:Anton (1987
5883:Anton (1987
5704:, then the
5467:dot product
3717:free module
3529:row vectors
3388:dot product
292:real spaces
6729:Categories
6638:Direct sum
6473:Invertible
6376:Linear map
6102:, Boston:
6023:See also:
5735:for which
5640:orthogonal
5442:for which
5432:of matrix
5430:null space
5337:null space
4361:to find a
3760:Definition
3573:orthogonal
3364:null space
3349:such that
3247:null space
2307:to find a
1101:Definition
1034:orthogonal
458:) = span({
6668:Numerical
6431:Transpose
6244:MathWorld
6225:MathWorld
5984:, unlike
5982:preserved
5870:956503593
5776:onto the
5663:dimension
5566:⋅
5547:⋮
5535:⋅
5511:⋅
5375:
5357:
5319:dimension
5307:Dimension
5202:∼
4972:→
4954:−
4857:→
4839:−
4735:→
4718:−
4621:→
4603:−
3966:row space
3880:⋯
3755:Row space
3638:∑
3588:) is the
3496:⋅
3477:⋮
3465:⋅
3441:⋅
3368:transpose
3289:
3271:
3181:→
3138:dimension
3126:Dimension
3107:transpose
2811:−
2790:∼
2754:−
2702:−
2681:∼
2662:−
2654:−
2639:−
2569:∼
1876:⋯
1813:⋮
1772:⋯
1741:⋮
1669:⋯
1636:⋮
1603:⋯
1544:⋮
1496:⋯
1474:⋮
1469:⋱
1464:⋮
1442:⋯
1390:⋮
1221:⋯
780:−
772:−
575:−
567:−
248:row space
199:dimension
87:) of its
6745:Matrices
6714:Category
6653:Subspace
6648:Quotient
6599:Bivector
6513:Bilinear
6455:Matrices
6330:Glossary
5813:See also
5665:above).
5619:. Thus
4938:→
4823:→
4702:→
4587:→
3692:for any
3552:. Thus
3527:because
3236:subspace
355:Overview
6325:Outline
5954:product
5948:is not
5762:coimage
5653:is the
5605:, ...,
5372:nullity
5341:nullity
4313:(using
3988:, ...,
3976:is the
3954:, ...,
3938:, ...,
3805:, ...,
3774:scalars
3700:, ...,
3366:of the
3286:nullity
3243:nullity
2267:(using
1959:Example
1329:, ...,
1317:is the
1295:, ...,
1279:, ...,
1146:, ...,
1115:scalars
994:spanned
507:Example
465:, ...,
452:, then
197:. The
164:of the
79:is the
72:) of a
6609:Tensor
6421:Kernel
6351:Vector
6346:Scalar
6198:
6180:
6157:
6128:
6110:
6084:
6066:
6048:
6004:, the
5868:
5858:
5803:/ ker(
5756:is an
5706:kernel
5597:where
5408:where
5077:{ , }
5073:{ , }
3923:where
3780:be an
3776:. Let
3749:scalar
3745:vector
3092:pivots
1933:, for
1264:where
1121:be an
1117:. Let
1051:kernel
454:colsp(
445:. If
402:pivots
363:be an
169:-space
74:matrix
58:, the
48:matrix
36:matrix
6478:Minor
6463:Block
6401:Basis
6042:Wiley
5974:from
5838:2005.
5787:When
5778:image
5708:of a
5661:(see
4363:basis
4349:Basis
4319:plane
3815:. A
3770:field
3768:be a
3714:right
3617:right
3253:with
3155:image
2309:basis
2293:basis
2283:Basis
2273:plane
1949:range
1945:image
1156:. A
1111:field
1109:be a
432:image
412:rank(
394:rank(
376:rank(
160:is a
128:field
126:be a
97:range
93:image
69:image
64:range
6633:Dual
6488:Rank
6196:ISBN
6178:ISBN
6155:ISBN
6126:ISBN
6108:ISBN
6082:ISBN
6064:ISBN
6046:ISBN
5866:OCLC
5856:ISBN
5744:) =
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2014:0
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