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\begin{document}

\title{Math 101: Graduate Linear Algebra \\ Daily Homework \#9}
\date{Assigned Monday, 2 October 2017; due Wednesday, 4 October 2017}

\maketitle

\begin{problab}{9.1}
Let $F=\R$ or $F=\C$, let $V$ be a finite-dimensional inner product space over $F$.  Let $\phi\colon V \to V$ be a linear operator.  
\begin{enumalph}
\item Let $\phi$ be self-adjoint, and suppose that $\langle x,\phi(x) \rangle = 0$ for all $x \in V$.  Show that $\phi=0$.
\item Suppose that $\| \phi(x) \|= \|x\|$ for all $x \in V$.  Show that $\phi\phi^*=\phi^*\phi=1$.  \emph{[Hint: Apply part \textup{(a)} to $\phi^*\phi-1$.]}
\end{enumalph}
\end{problab}

\end{document}