If one considers the matrix as a linear transformation from **R**^{n} to **R**^{m}, then the column space of the matrix equals the image of this linear transformation.

The column spaces of a matrix Z is the set of all linear combinations of the columns in Z. If Z = [**a**_{1}, .... , **a**_{n}], then Col Z = Span {**a**_{1}, ...., **a**_{n}}

See also row space.