The column space of a matrix is the vector space generated by its columns 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 matrix 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 matrix rank of 
.
