Column Space

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 .