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
F
{\displaystyle \mathbb {F} }
be a field. The column space of an m × n matrix with components from
F
{\displaystyle \mathbb {F} }
is a linear subspace of the m-space
F
m
{\displaystyle \mathbb {F} ^{m}}
. The dimension of the column space is called the rank of the matrix and is at most min(m, n). A definition for matrices over a ring
K
{\displaystyle \mathbb {K} }
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.This article considers matrices of real numbers. The row and column spaces are subspaces of the real spaces