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In linear algebra, the column space C(''A'') of a matrix ''A'' (sometimes called the range of a matrix) is the set of all possible linear combinations of its column vectors. Let ''K'' be a field (such as real or complex numbers). The column space of an ''m'' × ''n'' matrix with components from ''K'' is a linear subspace of the ''m''-space ''K''''m''. The dimension of the column space is called the rank of the matrix.〔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 2005.〕 A definition for matrices over a ring ''K'' (such as integers) is also possible. The column space of a matrix is the image or range of the corresponding matrix transformation. The row space and column space of an ''m''-by-''n'' matrix are the linear subspaces generated by row vectors and column vectors, respectively, of the matrix. Its dimension is equal to the rank of the matrix and is at most min(''m'', ''n'').〔http://www.cliffsnotes.com/WileyCDA/CliffsReviewTopic/Row-Space-and-Column-Space-of-a-Matrix.topicArticleId-20807,articleId-20793.html〕 This article will consider matrices of real numbers: the row and column spaces are subspaces of R''n'' and R''m'' real spaces respectively. Row and column spaces can be constructed from matrices with components in any field or ring. ==Overview== Let A be an ''m''-by-''n'' matrix. Then # rank(A) = dim(rowsp(A)) = dim(colsp(A)), # rank(A) = number of pivots in any echelon form of A, # rank(A) = the maximum number of linearly independent rows or columns of A. If one considers the matrix as a linear transformation from R''n'' to R''m'', then the column space of the matrix equals the image of this linear transformation. The column space of a matrix A is the set of all linear combinations of the columns in A. If A = (...., an ), then colsp(A) = span . The concept of row space generalises to matrices to C, the field of complex numbers, or to any field. Intuitively, given a matrix A, the action of the matrix A on a vector x will return a linear combination of the columns of A weighted by the coordinates of x as coefficients. Another way to look at this is that it will (1) first project x into the row space of A, (2) perform an invertible transformation, and (3) place the resulting vector y in the column space of A. Thus the result y =A x must reside in the column space of A. See the singular value decomposition for more details on this second interpretation. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「row and column spaces」の詳細全文を読む スポンサード リンク
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