Column Space - Definition

Definition

Let A be an m × n matrix, with column vectors v₁, v₂, ..., v_n. A linear combination of these vectors is any vector of the form

where c₁, c₂, ..., c_n are scalars. The set of all possible linear combinations of v₁,...,v_n is called the column space of A. That is, the column space of A is the span of the vectors v₁,...,v_n.

Example

If, then the column vectors are v₁ = (1, 0, 2)T and v₂ = (0, 1, 0)T.

A linear combination of v₁ and v₂ is any vector of the form

The set of all such vectors is the column space of A. In this case, the column space is precisely the set of vectors (x, y, z) ∈ R3 satisfying the equation z = 2x (using Cartesian coordinates, this set is a plane through the origin in three-dimensional space).

Any linear combination of the column vectors of a matrix A can be written as the product of A with a column vector:

Therefore, the column space of A consists of all possible products Ax, for x ∈ Rn. This is the same as the image (or range) of the corresponding matrix transformation.