The vector space generated by the rows of a matrix viewed as vectors. The row space of a 
matrix 
with real entries is a subspace generated by 
elements of 
, hence its dimension is at most equal to 
. It is equal to the dimension of the column space of
(as will be shown below), and is called
the rank of 
.
The row vectors of 
are the coefficients of the unknowns 
in the linear equation system
 |
(1)
|
where
![x=[x_1; |; x_m],](/images/equations/RowSpace/NumberedEquation2.svg) |
(2)
|
and 
is the zero
vector in 
.
Hence, the solutions span the orthogonal complement 
to the row space 
in 
,
and
 |
(3)
|
On the other hand, the space of solutions also coincides with the kernel (or null space) of the linear transformation 
, defined by
 |
(4)
|
for all vectors 
of 
. And it also true that
 |
(5)
|
where 
denotes the kernel and 
the image, since the nullity and the rank always add up
to the dimension of the domain. It follows that the dimension of the row space is
 |
(6)
|
which is equal to the dimension of the column space.
