Row space

[matqkks]

Why is the row space of a matrix important?

 

[genericusrnme]

The same reason the column space is important.
One example is basic multiplication;
Multiplying a matrix with a vector to the right gives linear combinations of the columns, multiplying with a vector to the left gives linear combinations of the rows.

 

[Bacle2]

The row space is used in the fundamental theorem of linear algebra:

http://en.wikipedia.org/wiki/Fundamental_theorem_of_linear_algebra

which relates row space, column space, kernel and cokernel.

Together, of course, with the rank-nullity theorem.