The row vectors of a matrix. The row space of this matrix is the vector space spanned by the row vectors.The column vectors of a matrix. The column space of this matrix is the vector space spanned by the column vectors.
In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation.
Let be a field. The column space of an m × n matrix with components from is a linear subspace of the m-space . The dimension of the column space is called the rank of the matrix and is at most min(m, n).[1] A definition for matrices over a ring is also possible.
The row space is defined similarly.
The row space and the column space of a matrix A are sometimes denoted as C(AT) and C(A) respectively.[2]
This article considers matrices of real numbers. The row and column spaces are subspaces of the real spaces and respectively.[3]
^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.
^Strang, Gilbert (2016). Introduction to linear algebra (Fifth ed.). Wellesley, MA: Wellesley-Cambridge Press. pp. 128, 168. ISBN 978-0-9802327-7-6. OCLC 956503593.
^Anton (1987, p. 179)
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