The row space of a matrix is orthogonal to the null space of that matrix. If P is invertible, PAv= 0 if and only if Av= 0. That is, the null space of PA is the same as the null space of A and so the row spaces are also the same.
Let A be an mxn matrix with entries in F. Let Row(A) be a subset of M1xn(F) be the span of the rows of A.
Let
P ∈ Mm×m(F) be an m×m matrix. Show that Row(PA) ⊆ Row(A), with equality of P is invertible.