%% Homework template by Henrik Bachmann
%% Please feel free to change anything you want or to contact me for questions
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\newcounter{exercisecnt}
\newenvironment{exer}[1][]{\refstepcounter{exercisecnt}\par\medskip\noindent    \textbf{Exercise~\theexercisecnt #1.}\, \rmfamily}{\medskip}


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\setlength\parindent{0pt}
\fancypagestyle{titlepage}{
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    \lhead{{\bf Linear Algebra I \& Mathematics Tutorial 1b}\\Nagoya University, G30 Program, Fall 2020} 
    \chead{} 
    \rhead{{\bf STUDENT NAME} \\ \today} 
    \cfoot{- \thepage \,\,-} 
}
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}


\begin{document}
\thispagestyle{titlepage}


\begin{center}
{\bf \large  Solution to homework X:  Topic ... or whatever you want to write}
\end{center}
\vspace{-0.2cm}
\PRLsep % This you can also remove
\vspace{-0.5cm}


\begin{exer} Here is the solution to the first exercise. If I you want to write down a linear system you can use the following code:
	\begin{align*}
	\left\{
	\begin{array}{rrrrrrr}
	2x_1&+&12x_2&+&7x_3&=& 12a+7\\
	2x_1&+&4 x_2&+&2 x_3 &=& 12a\\
	x_1&+&10x_2&+&6x_3 &=& 7a+8
	\end{array}
	\right. \,.
	\end{align*}
	If you want to write a matrix you can use $\pmatr{1 & 2 & 3 \\ 4 & 5 & 6}$.
\end{exer}

\begin{exer}  Second exercise. Maybe at some point you need to talk about a linear map $F : \R^n \rightarrow \R^m$ and about its kernel $\ker(F)$ or image $\im(F)$. Maybe you also need to write down how this linear map is defined:
	\begin{align*}
	F : \R^3 &\longrightarrow \R^3\,,\\
	\pmatr{x_1\\x_2 \\x_3} &\longmapsto \pmatr{2x_2+2x_3\\x_1-4x_2+6x_3\\x_2+x_3} = \pmatr{0 & 2 & 3\\1 & -4 & 6\\0  & 1 & 1} \pmatr{x_1 \\ x_2 \\ x_3}\,.
	\end{align*}
	You can enumerate things..
	\begin{enumerate}[i)]
		\item One
		\item Two
		\item Three
	\end{enumerate}
	A subspace can be for example written as 
	\begin{align*}
	U &= \left\{  \pmatr{x_1 \\ x_2 \\ x_3} \in \R^3 \,\Big\vert\, x_1+x_2+x_3=0 \right\}\,.
	\end{align*}
	or you could have something like
		\begin{align*}
	U &= \left\{  x \in \R^n \mid  Ax = 0 \right\}\,,
	\end{align*}
	where $A \in \R^{n \times n}$ is a matrix.
\end{exer}

\begin{exer} Later we will also use the dot-product $u \bullet v$ and the norm $\norm{u}$. We will also talk about the span of vectors: 
	\begin{align*}
	U = \spa\left\{ \pmatr{1\\2\\3}\,,\,\pmatr{3\\3\\8}\,,\,\pmatr{1\\3\\2} \right\}\,.
	\end{align*}
\end{exer}
\end{document}

