Row and column spaces

The row vectors of a matrix. The row space of this matrix is the vector space spanned by these row vectors.

The column vectors of a matrix. The column space of this matrix is the vector space spanned by these 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 $F$ be a field. The column space of an ⁠$m\times n$⁠ matrix with components from $F$ is a linear subspace of the m-space ⁠$F^{m}$⁠. 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 $R$ (i.e., with components from ⁠$R$⁠) is also possible.

The row space is defined similarly.

The row space and the column space of a matrix ⁠$A$⁠ are sometimes denoted as ⁠$\operatorname {C} (A^{\mathsf {T}})$⁠ and ⁠$\operatorname {C} (A)$⁠, respectively.

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