Chapter 5.2, Problem 14E

Let A be an $m\times n$ matrix, B an $n\times r$ matrix, and $C=AB.$ Show that
(a) $N\left(B\right)$ isasubspaceof $N\left(C\right).$
(b) $N{\left(C\right)}^{\perp }$ is a subspace of $N{\left(B\right)}^{\perp }$ and, consequently, $R\left(C^T\right)$ is a subspace of $R\left(B^T\right).$