Linear algebra problem?

Question

Let A be and m x n matrix.

A. If B is a nonsingular m x m matrix, show that BA and A have the same nullspace.

B. If C is a nonsingular n x n matrix, show that AC and A have the same rank

Answer 1

HOMEWORK! These are easy enough that you need to see how they are done, though.

A. The point is that Bx=0 if and only if x=0 since B is non-singular. Thus, BAx=0 is the same as B(Ax)=0, so Ax=0. In other words, the null space of BA is the same as that for A.

B. The rank of A is the number of independent rows of the matrix A. But the k^th row of AC will be the k^th row of A multiplied by C. But since C is non-singular, it takes independent vectors to independent vectors and vice versa.

Answer 2

1) just write out. nullspace v :: Av=0
since B is non sigular Bv=0 has only v=0 as solution
Aw ; with w nullspace of A is Aw=0 by definition and BAw = 0 because B0=0.

AC is not defined ...