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\fancyhead[L]{\textbf{MATH 510 Homework 3}}
\fancyhead[R]{\textbf{Your-Name-Goes-Here}}
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{\small
\begin{itemize}
\item Homework 3 is due February 9 at the beginning of class.
\item Write the problem statement followed by a proof or solution.
\item List problems in the same order they were given.
\item If you skip a problem, include the problem statement with no solution. 
\end{itemize}}

\begin{enumerate}[{\rm 1.}]
\item Prove the \emph{Second Isomorphism Theorem} for vector spaces, stating that if $V$ is a vector space and $U,W\le V$, then \[(U+W)/W \cong U/(U\cap W).\]
(You may assume $V$ is finite-dimensional if you wish.)
\smallskip
\item Let $V$ be a vector space over a field $\mathbb{F}$. The \emph{dual space} of $V$ is defined to be \[V^\ast=\operatorname{Hom}(V,\mathbb{F}).\]
\begin{enumerate}[{\rm (a)}]
\item Define a function $\beta:V^\ast\times W\to \operatorname{Hom}(V,W)$ by $\beta(\xi,w)(v)=\xi(v)w$. Show $\beta$ is bilinear, hence induces a linear map \[B:V^\ast\otimes W\to \operatorname{Hom}(V,W)\] satisfying $B(\xi\otimes w)(v)=\xi(v)w$ for all $\xi\in V^\ast, w\in W, v\in V$. 
\item Show that the map $B$ is injective. (\emph{Hint:} Choose bases.)
\item Show that the image of $B$ consists of all linear maps $T:V\to W$ of finite rank. (\emph{Hint:} For $\supseteq$, choose a basis for $T(V)$.)
\end{enumerate}
\smallskip
\item Let $V$ be a (finite-dimensional, if you wish) vector space. Define $V^{\otimes k}$ for $k>0$ recursively by $V^{\otimes 1}=V$ and $V^{\otimes k}=V^{\otimes (k-1)}\otimes V$ for $k>0$. We put $v_1\otimes v_2\otimes v_3=(v_1\otimes v_2)\otimes v_3$ and similarly with more factors.
Let $J_k$ be the subspace of $V^{\otimes k}$ spanned by all vectors $v_1\otimes v_2\otimes\cdots\otimes v_k$ where $v_1,v_2,\ldots,v_k\in V$ and $v_i=v_j$ for some $i\neq j$. The \emph{$k$:th exterior power of $V$} is defined as \[\wedge^k V=V^{\otimes k}/J_k.\]
Notation: $v_1\wedge v_2\wedge\cdots\wedge v_k=v_1\otimes v_2\otimes\cdots\otimes v_k + J_k$.
\begin{enumerate}[{\rm (a)}]
\item If $T:V\to V$ is a linear map, show that $T^{\otimes k}:V^{\otimes k}\to V^{\otimes k}$ (defined recursively by $T^{\otimes 1}=T,\;T^{\otimes s}=T^{\otimes (s-1)}\otimes T$ for $s>0$) leaves the subspace $J_k$ invariant. Conclude that there is an induced linear map $\wedge^k T:\wedge^k V\to \wedge^k V$.
\item If $\dim V=n$, show that $\dim \wedge^k V=\binom{n}{k}$. (\emph{Hint:} By bilinearity, $(u+v)\otimes (u+v)\in J_2$ implies that $u\wedge v+v\wedge u=0$.)
\item Take $V=\mathbb{F}^2$ and $T=T_A$ for an arbitrary $A\in\mathbb{F}^{2\times 2}$. Find the matrix of $\wedge^2 T$ with respect to the basis $\{e_1\wedge e_2\}$.
\end{enumerate}
\smallskip
\item Let $A\in\mathbb{F}^{m\times n}$ and $B\in\mathbb{F}^{s\times t}$ be matrices. Let $A\otimes B$ be the Kronecker product of the matrices. Prove that $\operatorname{rank} (A\otimes B)=(\operatorname{rank} A)(\operatorname{rank} B)$.
\item The \emph{trace} of a square matrix $A=[A_{ij}]\in\mathbb{F}^{n\times n}$ is $\operatorname{Tr} A=\sum_i A_{ii} \in \mathbb{F}$ (sum of the diagonal elements).
Show that if $A\in\mathbb{F}^{n\times n}$ and $B\in\mathbb{F}^{m\times m}$ then $\operatorname{Tr} (A\otimes B)=(\operatorname{Tr} A)(\operatorname{Tr} B)$.
\end{enumerate}

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