The vector space generated by the columns of a matrix viewed as vectors. The column space of an matrix with real entries is a subspace generated by elements of , hence its dimension is at most . It is equal to the dimension of the row space of and is called the rank of .

The matrix is associated with a linear transformation , defined by

for all vectors of , which we suppose written as column vectors. Note that is the product of an and an matrix, hence it is an matrix according to the rules of matrix multiplication. In this framework, the column vectors of are the vectors , where are the elements of the standard basis of . This shows that the column space of is the range of , and explains why the dimension of the latter is equal to the rank of .