Row and column spaces

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. 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).

This article will consider matrices of real numbers: the row and column spaces are subspaces of Rⁿ and R^mreal spaces respectively. Row and column spaces can be constructed from matrices with components in any field or ring.