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
(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.