The column space of a matrix **A** is the vector space spanned by the columns of **A**. The dimension of the column space is equal to that of the row space, which in turn is equal to the rank of **A**.

For instance, the matrix,

[ 1 1 2 ]
[ 2 2 4 ]
[ 3 5 6 ]

Performing elimination will show that this matrix has two pivot variables and one free variable, and thus the column space spans a two dimensional subspace of R^{3}. It is also plainly visible, as the third column is a direct multiple of the first column, and is thus dependent.