Let $\bf A$ be an $m \times n$ matrix. If $\bf P$ and $\bf Q$ are invertible $m \times m$ and $n \times n$ matrices, respectively
prove $\operatorname{rank}(\mathbf{PA}) = \operatorname{rank}(\bf{A})$
"I know how to prove $\operatorname{rank}(\mathbf{AQ})= \operatorname{rank}(\bf{A})$, where I start with $R(L_{\mathbf{AQ}})=L_{\bf{A}} L_{\bf{Q}}(F^n)=\dots$" but it seems like I can not prove $\operatorname{rank}(\mathbf{PA})=\operatorname{rank}(\bf{A})$ by this approach"
any help from you guys would be great. Thanks
Hint: By the rank-nullity theorem, you can prove that two matrices (of the same sizes) have equal rank by instead proving that their null spaces have equal dimension.