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 
.
