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\begin{document}

\title{Math 101: Graduate Linear Algebra \\ Weekly Homework \#2}
\date{Assigned Friday, 22 September 2017; due Monday, 2 October 2017}

\maketitle

\begin{problab}{W2.1}
Let $\phi\colon V \to W$ be an $F$-linear map, and let $\phi^*\colon W^* \to V^*$ be the dual map, defined via pullback.  Show that
\[ \img \phi^* = \ann(\ker \phi). \]
\end{problab}

\begin{problab}{W2.2}
Let $V$ be a finite-dimensional vector space over a field $F$, and let $W_1,W_2$ be subspaces.
  \begin{enumalph}
  \item Prove that $W_1=W_2$ if and only if $\ann(W_1)=\ann(W_2)$.  
  \item Show $\ann(W_1+W_2)=\ann(W_1) \cap \ann(W_2)$ and $\ann(W_1 \cap W_2)=\ann(W_1)+\ann(W_2)$.
\end{enumalph}
For a heightened sense of self-satisfaction, you could make it clear in your argument where you actually use that $V$ is finite-dimensional.  Which of the statements are still true when $V$ is infinite-dimensional?
\end{problab}

\begin{problab}{W2.3}
Let $V,W$ be $F$-vector spaces, let $v_1,\dots,v_n \in V$ be linearly independent, and let $w_1,\dots,w_n \in W$ be arbitrary.  Suppose that 
\[ \sum_{i=1}^n v_i \otimes w_i = 0 \in V \otimes_F W. \]
Show that $w_i=0$ for all $i=1,\dots,n$.  Conclude that $v \in V$ and $w \in W$ have $v \otimes w =0$ if and only if $v=0$ or $w=0$.
\end{problab}

\begin{problab}{W2.4}
In class, we showed that the tensor product is characterized by a universal property.  Perhaps the simplest situation of a universal property is the following.

Let $X,Y$ be sets.  The cartesian product $X \times Y$ has its two projection maps:
\[ \xymatrix@R=10pt{
& X \\
X \times Y \ar[ur]_{\pi_X} \ar[rd]^{\pi_Y} \\
& Y} \]
Show that the product $X \times Y$ is \emph{universal} in this respect: for every set $Z$ and maps
\[ \xymatrix@R=10pt{
& X \\
Z \ar[ur]^{f_X} \ar[rd]^{f_Y} \\
& Y} \]
of sets, there exists a unique map $h\colon Z \to X \times Y$ such that the diagram
\[ \xymatrix@R=10pt@C=40pt{
& & X \\
Z \ar@/^1pc/[urr]^{f_X} \ar@/_1pc/[drr]_{f_Y} \ar[r]^{h} & X \times Y \ar[ur]_{\pi_X} \ar[rd]^{\pi_Y} \\
& & Y} \]
commutes.  
\end{problab}

\begin{problab}{W2.5}
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 subspace.  

Define 
\[ W^\perp = \{v \in V : T(v,W)=0\} = \{v \in V : T(v,w)=0 \text{ for all $w \in W$}\}. \]
\begin{enumalph}
\item Show that the map 
\begin{align*}
V &\to V^* \\ 
v &\mapsto T_v = T(v,-)
\end{align*} 
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 $W$ (as a linear operator on $V$).  Let $V=\bfR^3$ have the standard inner product and let 
\[ W=\{(x,y,z) \in \bfR^3 : x+y+z=0\}. \]  
Compute the matrix of the orthogonal projection onto $W$ with respect to the standard basis.
\end{enumalph}
\end{problab}

\end{document}