Suppose $T \in \mathcal{L}(U, V)$ and $S \in \mathcal{L}(V, W)$ are both invertible linear maps. Prove that $ST \in \mathcal{L}(U, W)$ is invertible and that $(ST)^{-1} = T^{-1}S^{-1}$. Note that $\begin{align*} (ST)(T^{-1}S^{-1}) &= S(TT^{-1})S^{-1} = SIS^{-1} = SS^{-1} = I \\ (T^{-1}S^{-1})(ST) &= T^{-1}(S^{-1}S)T = T^{-1}IT = T^{-1}T = I. \end{align*}$ Thus $ST$ is invertible and $(ST)^{-1} = T^{-1}S^{-1}$. --- $\textbf{Exercise 2.}$ Suppose $T \in \mathcal{L}(U, V)$ and $S \in \mathcal{L}(V, W)$ are both invertible linear maps. Prove that $ST \in \mathcal{L}(U, W)$ is invertible and that $(ST)^{-1} = T^{-1}S^{-1}$. $\textbf{Solution 2.}$ Note that $\begin{align*} (ST)(T^{-1}S^{-1}) &= S(TT^{-1})S^{-1} = SIS^{-1} = SS^{-1} = I \\ (T^{-1}S^{-1})(ST) &= T^{-1}(S^{-1}S)T = T^{-1}IT = T^{-1}T = I. \end{align*}$ Thus $ST$ is invertible and $(ST)^{-1} = T^{-1}S^{-1}$. --- ![[sol-5.pdf#page=2]]