The Column Space of a matrix is the span (set of all possible linear combinations) of its column vectors.

The column space is also called the image.

The row space is defined similarly.

Example

If , the column space of is a subspace (a plane through the origin) of containing and

Column space is crucial in linear algebra since to solve is the same as to express as a linear combination of columns of .

The system is solvable if and only if is in the column space of

References