- converted to LaTeX by Claude Problem 1: Suppose $T \in \mathcal{L}(V, W)$ is invertible. Show that $T^{-1}$ is invertible and $(T^{-1})^{-1} = T$. The desired result follows from the equations $\begin{equation*} TT^{-1} = I \text{ and } T^{-1}T = I, \end{equation*}$ which show that $T$ is an inverse of $T^{-1}$. --- Problem 1: Suppose $T \in \mathcal{L}(V, W)$ is invertible. Show that $T^{-1}$ is invertible and $(T^{-1})^{-1} = T$. The desired result follows from the equations$$\begin{equation} TT^{-1} = I \text{ and } T^{-1}T = I, \end{equation}$ which show that $T$ is an inverse of $T^{-1}$. try 4 $\textbf{Exercise 1.}$ Suppose $T \in \mathcal{L}(V, W)$ is invertible. Show that $T^{-1}$ is invertible and $(T^{-1})^{-1} = T$. $\textbf{Solution 1.}$ The desired result follows from the equations $\begin{equation*} TT^{-1} = I \text{ and } T^{-1}T = I, \end{equation*}$ which show that $T$ is an inverse of $T^{-1}$. --- ![[sol-5.pdf#page=1]]