\documentclass[11pt]{article}
\usepackage{jeffe,handout,multicol,tikz}
\usetikzlibrary{positioning}

\newcommand{\kgmcomment}[1]{}

\begin{document}
\headers{Math343}{Homework 6}{Due Oct 1 2025}

%\makeatletter
%\def\@oddfoot{\hfill \tikz \node[draw] {over} ; }
%\makeatother

\makeatletter
\def\@oddfoot{}
\makeatother

%\noindent
%\hfill Name: \underline{\hspace{3.5in}} 

%\begin{flushright}
%Name: \underline{\hspace{3.5in}} \\
%\end{flushright}
%\hrule

\setlength{columnseprule}{0pt}

\newcommand{\vect}[1]{\mathbf{#1}}
\newcommand{\Span}{\mathrm{Span}}
\newcommand{\st}{\colon \:}

% Matrix Three By One
\newcommand{\MTBO}[3]%
{%
	\left[%
	\begin{array}{*{3}{r}}
	#1\\
	#2\\
	#3
	\end{array}%
	\right]%
}

% Matrix TWo By One
\newcommand{\MTWBO}[2]%
{%
	\left[%
	\begin{array}{*{3}{r}}
	#1\\
	#2
	\end{array}%
	\right]%
}

%\mtrix{num_cols}{data}
\newcommand{\mtrix}[2]%
{%
	\left[%
	\begin{array}{*{#1}{r}}
	#2
	\end{array}
	\right]
}


\newcommand{\RR}{\mathbb{R}}

\noindent
\textbf{Directions:} You may work to solve these problems in groups, but all written work must be your own.  Show all work; no credit for solutions without work..

\begin{enumerate}

\item {} [1.8.\{13-16\}] Use a rectangular coordinate system to plot $\vect{u}=\MTWBO{5}{2}$, $\vect{v}=\MTWBO{-2}{4}$ and their images under the given transformation $T$.  (Make a separate sketch for each.)  Describe geometrically what $T$ does to each vector in $\RR^2$.
\begin{multicols}{2}
\begin{enumerate}
\item $T(\vect{x}) = \mtrix{2}{-1 & 0\\ 0& -1} \mtrix{1}{x_1 \\ x_2}$
\item $T(\vect{x}) = \mtrix{2}{0.5 & 0\\ 0& 0.5} \mtrix{1}{x_1 \\ x_2}$
\item $T(\vect{x}) = \mtrix{2}{0 & 0\\ 0& 1} \mtrix{1}{x_1 \\ x_2}$
\item $T(\vect{x}) = \mtrix{2}{0 & 1\\ 1& 0} \mtrix{1}{x_1 \\ x_2}$
\end{enumerate}
\end{multicols}

\item {} [1.8.17] Let $T\st \RR^2 \to \RR^2$ be a linear transformation that maps $\vect{u}=\MTWBO{5}{2}$ to $\MTWBO{2}{1}$ and maps $\vect{v}=\MTWBO{1}{3}$ to $\MTWBO{-1}{3}$.  Use the fact that $T$ is linear to find the images under $T$ of $3\vect{u}$, $2\vect{v}$, and $3\vect{u}+2\vect{v}$.

\item {} [1.8.32] Suppose vectors $\vect{v}_1, \ldots, \vect{v}_p$ span $\RR^n$, and let $T\st \RR^n \to \RR^n$ be a linear transformation.  Suppose $T(\vect{v_i})=\vect{0}$ for $1\le i\le p$.  Show that $T$ is the zero transformation (i.e. $T(\vect{x})=\vect{0}$ for each $\vect{x}\in \RR^n$).

\item {} [1.8.\{40,41\}] Show that the following transformations are not linear.
\begin{multicols}{2}
\begin{enumerate}
	\item $\MTWBO{x_1}{x_2} \mapsto \MTWBO{4x_1 - 2x_2}{3|x_2|}$.
	\item $\MTWBO{x_1}{x_2} \mapsto \MTBO{2x_1-3x_2}{x_1+4}{5x_2}$.
\end{enumerate}
\end{multicols}

\item {} [1.9] True/False.  Justify your answer.
\begin{enumerate}
	\item A linear transformation $T\st \RR^n\to \RR^m$ is completely determined by its effect on the columns of the $n\times n$ identity matrix.
	\item If $T\st \RR^2 \to \RR^2$ rotates vectors about the origin through an angle $\phi$, then $T$ is a linear transformation.
	\item The columns of the standard matrix for a linear transformation from $\RR^n$ to $\RR^m$ are the images of the columns of the $n\times n$ identity matrix.
	\item When two linear transformations are performed one after another, the combined effect may not always be a linear transformation.
	\item Not every linear transformation from $\RR^n$ to $\RR^m$ is a matrix transformation.
\end{enumerate}


\end{enumerate}
\end{document}
