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

\noindent
{\large \textmd{Homework 1}}

\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 13th, 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 Some linear algebra review}} 

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

\begin{enumerate}
\item 
\begin{enumerate}
\item Verify that the set
	\[ S=\left\{ \begin{bmatrix}1\\0\\-1\end{bmatrix}, \begin{bmatrix}-1\\1\\0\end{bmatrix},\begin{bmatrix}1\\0\\1\end{bmatrix}\right\}\subset \C^{3}\]
is a basis.
\item Determine the $S$-coordinates of $v=\begin{bmatrix} 1\\1\\1\end{bmatrix}\in \C^{3}$.
\item Determine the $S$-coordinates of an arbitrary vector $\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\end{bmatrix}\in\C^{3}$. ({\sl Your solution will be in terms of $a_{1},a_{2},a_{3}$})
\item Determine an explicit formula for the $S$-coordinate map $[-]_{S}:\C^{3}\rightarrow \C^{3}$.
\end{enumerate}
\item Let $T:V\rightarrow W$ be a linear map.
\begin{enumerate}
\item Define the {\sl kernel of $T$} to be the subset 
	\[ \ker T = \{v\in V \;|\; T(v) = 0_{W}\in W\}\subset V\]
Prove that $\ker T$ is a subspace of $V$.
\item Define the {\sl image of $T$} to be the subset 
	\[\im T = \{w\in W\;|\; w=T(v) \text{ for some $v\in V$}\}\subset W\]
Prove that $\im T$ is a subspace of $W$.
\end{enumerate}
 
The following result is one of the most important theorems of linear algebra (it can be considered the Fundamental Theorem of Linear Maps).

{\bf Rank Theorem:} Let $T:V\rightarrow W$ be a linear map, $V$ finite dimensional. Then,
	\[\dim \ker T+ \dim \im T = \dim V\]

\item Consider the subset
	\[ U=\left\{ \begin{bmatrix}a_{1}\\a_{2}\\a_{3}\\a_{4}\end{bmatrix} \; \middle| \; a_{1}+a_{2}+a_{3}+a_{4}=0\right\} \subset \C^{4}\]
\begin{enumerate}
\item Determine a linear map $T:\C^{4}\rightarrow \C$ so that $U=\ker T$. Conclude that $U\subset \C^{4}$ is a subspace.
\item Show that $\im T =\C$. Deduce that $\dim U =3$.
\item Find a basis $\B$ of $U$. Make sure you verify that the set you give is, in fact, a basis. ({\sl There are an infinite number of possibilities here!})
\end{enumerate}

\item Consider the linear map
	\[ L:\C^{4} \rightarrow \C^{4}\;,\; \begin{bmatrix}a_{1}\\a_{2}\\a_{3}\\a_{4}\end{bmatrix} \mapsto \begin{bmatrix}a_{2}\\a_{3}\\a_{4}\\a_{1}\end{bmatrix} \]
Recall the subspace $U\subset \C^{4}$ and your basis $\B$ from the previous problem.
\begin{enumerate}
\item Show that $L(u)\in U$, for any $u\in U$. Deduce that $L$ {\sl restricts} to a linear map
	\[ L_{|U} :U\rightarrow U\;,\; u\mapsto L(u)\]
({\sl The last part is quite straightforward (i.e. one or two sentences of explanation) as there's only two things to consider: (1) $L_{|U}$ is well-defined, (2) $L_{|U}$ is linear})
\item Determine the matrix of $L_{|U}$ with respect to $\B$; your resulting matrix will be $3\times 3$.
\end{enumerate}
	



\end{enumerate}












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