\documentclass[12pt]{amsart}

\newcommand{\defi}[1]{\textsf{#1}} 

\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{amsxtra}
\usepackage{mathrsfs}
\usepackage[all]{xy}

\setlength{\hfuzz}{4pt}

\newenvironment{problab}[1]
{\noindent\textbf{Problem #1}.}
{\vskip 6pt}

\theoremstyle{remark}
\newtheorem*{solu}{Solution}

\newenvironment{enumalph}
{\begin{enumerate}\renewcommand{\labelenumi}{\textnormal{(\alph{enumi})}}}
{\end{enumerate}}

\newenvironment{enumroman}
{\begin{enumerate}\renewcommand{\labelenumi}{\textnormal{(\roman{enumi})}}}
{\end{enumerate}}

\newcommand{\C}{\mathbb C}
\newcommand{\F}{\mathbb F}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\newcommand{\Z}{\mathbb Z}

\DeclareMathOperator{\lcm}{lcm}
\DeclareMathOperator{\img}{img}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\ann}{ann}
\DeclareMathOperator{\ev}{ev}

\usepackage{fullpage}

\begin{document}

\title{Math 101: Graduate Linear Algebra \\ Daily Homework \#8}
\date{Assigned Friday, 29 September 2017; due Monday, 2 October 2017}

\maketitle

\begin{problab}{8.1}
Let $F=\R$ or $F=\C$ and let $V,W$ be finite-dimensional inner product spaces over $F$ (i.e., vector spaces equipped with an inner product).  Let $T\colon V \to W$ be an $F$-linear map, and let $T^* \colon W \to V$ be the adjoint with respect to the inner product, so that $\langle T(v),w \rangle_W = \langle v,T^*(w) \rangle_V$ for all $v \in V$ and $w \in W$.

Let $v \in V$.  Show that $(T^*T)(v)=0$ if and only if $T(v)=0$. 
\end{problab}

\end{document}