\documentclass[12pt]{amsart}

\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{amsxtra}
\usepackage{mathrsfs}

\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}

\usepackage{fullpage}

\begin{document}

\title[101 Homework \#9]{Math 101: Algebra I \\ Worksheet, Day \#9}
\date{Assigned Monday, 26 September 2016; due Tuesday, 27 September 2016}

\maketitle

\begin{problab}{JV9.A}
Show that $\C \otimes_{\R} \C$ and $\C \otimes_\C \C$ are both $\R$-vector spaces, and compute their dimensions.  Are they isomorphic as $\R$-vector spaces?
\end{problab}

\begin{problab}{JV9.B}
Let $F$ be a field, let $V$ be a finite-dimensional $F$-vector space, and let $T:V \times V \to F$ be a nondegenerate symmetric bilinear form.  Let $W \subseteq V$ be a $F$-subspace.  

Define $W^\perp = \{v \in V : T(v,W)=0\}$.  
\begin{enumalph}
\item Show that the map $V \to V^*$ by $v \mapsto T(v,-)$ maps $W^\perp$ isomorphically to $\ann(W)$.
\item Deduce that $\dim V = \dim W + \dim W^\perp$.
\item Suppose that $T|_{W\times W}$ is nondegenerate (accordingly, we say that $W$ is a \emph{nondegenerate} subspace under $T$).  Show that $V=W \oplus W^\perp$.  In this case, we say $W^\perp$ is the \emph{orthogonal complement} of the nondegenerate subspace $W$.
\item Define the \emph{orthogonal projection} onto $W$.  Then let $V=\R^3$ have the standard inner product and let 
\[ W=\{(x,y,z) \in \R^3 : x+y+z=0\} \]  
and compute its matrix with respect to the standard basis.
\end{enumalph}
\end{problab}

\end{document}