quiz3.tex

\documentclass[11pt]{article}
\usepackage[margin=0.75in]{geometry}
\usepackage{amssymb}
\pagestyle{empty}
\begin{document}

%\maketitle
\begin{center}
\Large
\rm{Math 221-AB1:  Quiz 3}
\\
\end{center}
\vspace{0.1in}

The transformations $T:\mathbb{R}^2\to\mathbb{R}^4$ and $S:\mathbb{R}^3\to\mathbb{R}^2$ 
are defined as follows:
\begin{displaymath}
T\left(\left[ \begin{array}{r} x \\ y \end{array} \right] \right) =  \left[ \begin{array}{c} x \\ x+3y \\ 2x-4y \\ y \end{array} \right] \quad \quad \quad 
S\left(\left[ \begin{array}{r} x \\ y \\ z  \end{array} \right] \right) =  \left[ \begin{array}{c} 2x \\ y+z \end{array} \right] \quad
\end{displaymath}
\begin{enumerate}
%\item{(\emph{2 pts}) Give a general formula for $T\circ S$.}
\item{(\emph{2 pts}) Find a matrix $B$ so that $T(v)=Bv$ for all $v\in\mathbb{R}^2$.}
\item{(\emph{3 pts}) Is $T$ a \textbf{one-to-one} transformation?  Explain your answer clearly.}
\item{(\emph{3 pts}) Show that $S$ a \textbf{linear} transformation.}
\item{(\emph{2 pts}) Give a basis for Ker($S$).}
\end{enumerate}


\vspace{4.5in}



\vspace{0.1in}
\end{document}