Chapter 5.8, Problem 18E

Let $S:U\to V$ and $T:V\to W$ be linear transformations.

a) Prove that if $S$ and $T$ are both one to one, then $T\circ S$ is one to one.

b) Prove that if $S$ and $T$ are both onto, then $T\circ S$ is onto.

c) Prove that if $S$ and $T$ are both invertible, then $T\circ S$ is invertible and ${\left(T\circ S\right)}^{-1}=S^{-1}\circ T^{-1}$.