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\begin{document}

\title{Math 101: Graduate Linear Algebra \\ Daily Homework \#11}
\date{Assigned Wednesday, 4 October 2017; due Friday, 6 October 2017}

\maketitle

\begin{problab}{11.1}
Let $V$ be an $F$-vector space, and let $\phi\colon V \to V$ be an $F$-linear map.  We say that $\phi$ is an \emph{involution} if $\phi^2=1$.

Suppose that $V$ is an inner product space and let $\phi$ be an involution on $V$.  Show that the following are equivalent:
\begin{enumalph}
\item $\phi$ is normal;
\item $\phi$ is unitary;
\item $\phi$ is self-adjoint; and
\item All singular values of $\phi$ are equal to $1$.
\end{enumalph}
\end{problab}

\end{document}