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{\Large \textmd{MA434: Spring 2019}}

\noindent
{\large \textmd{Homework 3}}

\noindent
{\bf Contact:} \texttt{gwmelvin\makeatletter@\makeatother colby.edu}


\end{flushright}
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\noindent
\textbf{Some thoughts and advice:} 
\begin{itemize}
\item Please submit solutions to the following problems by {\bf Wednesday, February 27th, 1.10pm}. You can either submit your solution in class or leave it outside my office.
\item You should expect to spend several hours on homework sets. A lot of practice problem-solving is essential to understand the material and skills covered in class. Be organised and do not leave problem sets until the last-minute. Instead, get a good start on the problems as soon as possible. 
\item When approaching a problem think about the following: \textsl{do you understand the words used to state the problem? what is the problem asking you to do? can you restate the problem in your own words? have you seen a similar problem worked out in class? is there a similar problem worked out in the textbook? what results/skills did you see in class that might be related to the problem?}

If you are stuck for inspiration come to office hours, or send me an email. However, don't just ask for the solution - provide your thought process, the difficulties you are having, and ask a coherent question in complete English sentences. 
\item Form study groups - get together and work through problem sets. \textbf{This will make your life easier!} You must write your solutions \textsl{on your own} and \textsl{in your own words}.
\item You {\bf are not allowed} to use any additional resources (e.g. \texttt{stackoverflow.com}).  If you are concerned then please ask.
\end{itemize}
\hrule

\medskip
{\large {\bf Inner product spaces; representations} }

\medskip
{\sc Unless otherwise specified, all vector spaces will have scalar field $\C$.}

\begin{enumerate}
\item Let $(V,\langle, \rangle)$ be an inner product space, $T\in \End(V)$, and let $T^{\ast}\in \End(V)$ be the adjoint of $T$. Show that $T^{\ast}\in \End(V)$ is linear.
\item Let $V$ be a finite dimensional vector space. Denote 
\[V^{\ast}=\Hom(V,\C)=\{L:V\rightarrow \C\;|\; \text{ $L$ linear}\},\]
the {\sl dual space of $V$}. $V^{\ast}$ is a vector space over $\C$: for any $K,L\in V^{\ast}$, $c\in \C$, define $K+L\in V^{\ast}$ and $cL\in V^{\ast}$ as follows
	\[ (K+L)(v) = K(v) +L(v),\quad v\in V,\]
	\[(cL)(v) = cL(v),\quad v\in V.\]

In class, when $(V,\langle, \rangle)$ is an inner product space, we used the inner product to determine a bijection $\alpha: V\to V^{\ast}$.
\begin{enumerate}
\item Show that $\alpha(cv) \neq c\alpha(v)$, for $c\in \C$, $v\in V$. Deduce that $\alpha$ is not linear. Hence, $\alpha$ is not an {\sl isomorphism of vector spaces}.
\item Given $v\in V$, define the {\sl evaluation at $v$} function
	\[ E_{v}:V^{\ast}\rightarrow \C\;,\; L\mapsto L(v)\]
Remember: $L:V\rightarrow \C$ is a linear function.
\begin{enumerate}
\item Show that $E_{v}$ is linear. Hence, $E_{v}\in\Hom(V^{\ast},\C)=(V^{\ast})^{\ast}$.
\item Consider the function
	\[ E: V\rightarrow (V^{\ast})^{\ast}\;,\; v\mapsto E_{v}\]
Show that $E$ is linear and that $E$ is injective. 

We say that $E$ is a {\sl natural embedding of $V$ in its double dual $(V^{\ast})^{\ast}$}: there is no arbitrary choice (of a basis, say) required to construct this embedding.
\end{enumerate}

\end{enumerate}



\item Let $G=\Z/3\Z$ and define the degree three representation $\rho$ on $\C^{3}$ by
	\[ \rho(\overline{0}) = \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix},\quad \rho(\overline{1}) = \begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix},\quad \rho(\overline{2}) = \begin{bmatrix} 0&0&1\\1&0&0\\0&1&0\end{bmatrix}\]
\begin{enumerate}
\item Find a one-dimensional subrepresentation $U\subset \C^{3}$.
\item Can you find another one-dimensional subrepresentation? ({\sl Hint: what is the relationship between one-dimensional subrepresentations and eigenvectors?})
\end{enumerate}


\item In this problem you will investigate a finite dimensional representation of an infinite group.
\begin{enumerate}
\item Show that 
	\[ \rho: \Z \rightarrow \GL_{2}(\C)\;,\; n\mapsto \begin{bmatrix} 1& n\\0& 1\end{bmatrix}\]
defines a representation of $(\Z,+)$.
\item Show that $U=\Span(e_{1})\subset \C^{2}$ is a subrepresentation equivalent to the trivial representation. ({\sl You will need to construct a linear isomorphism $f: U\rightarrow \C$ with the appropriate properties})
\item Is it possible to find a subrepresentation $W\subset \C^{2}$ such that
\begin{itemize}
\item $W\cap U =\{\underline{0}\}$, and 
\item $W+U = \C^{2}$?	
\end{itemize}
If so, give an example of such a subrepresentation $W$, making sure to verify that $W$ satisfies the required properties: if not, explain.

\end{enumerate}
\item Here is a {\sc fun fact:} {\sl Let $T\in \End(V)$ be a linear map, $V$ finite dimensional over $\C$. Then, the roots of $m_{T}$, the minimal polynomial of $T$, are precisely the eigenvalues of $T$.}

Let $\rho:G\rightarrow \GL(V)$ be a representation of finite degree of the finite group $G$. Using the {\sc fun fact}, show that the eigenvalues of $\rho(g)$, $g\in G$, are roots of unity.
\end{enumerate}












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