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\begin{document}
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\begin{flushright}
{\Large \textmd{MA434: Spring 2019}}

\noindent
{\large \textmd{Homework 2}}

\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 20th, 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 Further linear algebra review}} 

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

\begin{enumerate}
\item Problem 2.1 in the textbook.
\item Problem 2.5 in the textbook.
\item Consider the matrix
	\[A =\begin{bmatrix} 0&0&1&0\\0&0&0&1\\1&0&0&0\\0&1&0&0\end{bmatrix}\]
Determine an invertible matrix $P$ and a diagonal matrix $D$ such that $P^{-1}AP=D$.
\item Let $V$ be a $k$-dimensional vector space. Define the {\sl endomorphism ring of $V$} to be
	\[\End(V) =\{T:V\rightarrow V\;|\; T\text{ linear}\}\]

\begin{itemize}
\item $\End(V)$ is a vector space over $\C$: for $T,S\in \End(V)$, we define $T+S\in \End(V)$ to be the linear map $(T+S)(v) = T(v)+S(v)$; for any scalar $c\in \C$ and linear map $T\in \End(V)$, define $cT\in \End(V)$ to be the linear map $(cT)	(v)=cT(v)$. 
\item $\End(V)$ is a ring: for $T,S\in \End(V)$ we define a multiplication $\ast$ on $\End(V)$ as follows: $T\ast S = T\circ S$ is composition of functions.
\end{itemize}

\begin{enumerate}
\item Prove that $\End(V)$ is isomorphic (as a vector space) to $M_{k}(\C)$, the vector space of $k\times k$ matrices with entries in $\C$. ({\sl Hint: choose a basis $B\subset V$})
\item (Problem 2.2 in textbook) Let $B\subset V$ be a basis. Prove that the function
	\[\End(V) \rightarrow M_{k}(\C)\;,\; T\mapsto [T]_{B}\]
is an isomorphism of rings.
\end{enumerate}

 
	
\item Denote the polynomial algebra in one variable with complex coefficients by $\C[t]$. For example, 
 $$\sqrt{-1}t^{5}-\pi t^{2}+2\in \C[t].$$

In your previous linear algebra/abstract algebra courses you will have obtained the following properties of $\C[t]$:
\begin{itemize}
\item $\C[t]$ is a vector space over $\C$;
\item $\C[t]$ is a ring;
\item $\C[t]$ is a principal ideal domain: if $I\subset \C[t]$ is an ideal then there exists $f\in I$ so that
	\[I=(f) \stackrel{def}{=} \{fp\;|\; p\in \C[t]\}\]
In words: elements of $I$ are multiples of $f$.

{\sl Note: since $\C[t]$ is a commutative ring, left ideals are right ideals and vice versa.}
\end{itemize}

\begin{enumerate}
\item Let $I\subset \C[t]$ be an nonzero ideal. Show that there exists a {\sl unique} polynomial $g\in \C[t]$ having leading coefficient $1$ so that $I=(g)$. We call $g$ the {\bf monic generator of $I$}.
\item Let $T\in \End(V)$ be a linear map, $T\neq \id_{V}$. Show that the function
	\[\rho_{T}:\C[t] \rightarrow \End(V)\;,\; p(t) \mapsto p(T)\]
is 
\begin{enumerate}
\item a linear map;
\item a ring homomorphism.
\end{enumerate}
\item Prove that $\ker \rho_{T} \neq \{0\}\subset \C[t]$. ({\sl Hint: injective linear maps take linearly independent sets to linearly independent sets.})

Define $m_{T}\in \C[t]$ to be the monic generator of $\ker \rho_{T}$: we call $m_{T}$ the {\bf minimal polynomial of $T$}. 
\item Suppose that $T\in \End(V)$ is diagonalisable. Prove that $m_{T}$ is a product of a distinct linear factors. ({\sl Hint: choose a basis of $V$ consisting of eigenvectors of $T$})

\item In fact, {\bf the converse is also true:} if $m_{T}$ is a product of distinct linear factors then $T$ is diagonalisable (you do not need to show this). 

Using this fact, prove that if $T\in \End(V)$ satisfies $T^{m}=\id_{V}$, for some integer $m>0$, then $T$ is diagonalisable.
\end{enumerate}
	
\item {\color{red} Updated 2/17:} Problem 2.8 in the textbook using the following hint: {\sl Let $v\in \C^{n}$ be an eigenvector of $A$ and extend $(v)$ to a basis $B\subset \C^{n}$.}


Deduce the following: let $T\in \End(V)$. Then, there exists a basis $B\subset V$ such that $[T]_{B}$ is upper-triangular.




\end{enumerate}












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