Row and column spaces

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 ${\displaystyle F}$ be a field. The column space of an m × n matrix with components from ${\displaystyle F}$ is a linear subspace of the m-space ${\displaystyle F^{m}}$. 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 ${\displaystyle R}$ 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(A^T) and C(A) respectively.^[2]

This article considers matrices of real numbers. The row and column spaces are subspaces of the real spaces ${\displaystyle \mathbb {R} ^{n}}$ and ${\displaystyle \mathbb {R} ^{m}}$ respectively.^[3]