\documentclass[a4paper,12pt]{article}
\begin{document}


{\bf Question}

Sketch the effect of each of the following matrices on the square
whose vertices lie at the points (0,0), (0,1), (1,0), (1,1):

$${\rm(i)}\  A = \left( \begin{array}{rr} 1 & 1 \\ 0 & 1 \\
\end{array} \right);\ {\rm(ii)}\  B = \left ( \begin{array}{rr} {\cos (\frac{\pi}{4})} & {-\sin
(\frac{\pi}{4})} \\ {\sin (\frac{\pi}{4})} & {\cos
(\frac{\pi}{4})} \end{array} \right);\ {\rm(iii)}\  C = \left(
\begin{array}{rr} {\cos (\theta)} &{\sin (\theta)}
 \\ {\sin (\theta)} & {-\cos (\theta)}  \end{array} \right).$$


{\bf Answer}

\begin{description}
\item[(i)]
$$\left(\begin{array} {cc} {1} & {1}\\ {0} & {1}
\end{array} \right)\left(\begin{array} {c} {0}\\ {0}
\end{array} \right)=\left(\begin{array} {c} {0}\\ {0}
\end{array} \right),\ \left(\begin{array} {cc} {1} & {1}\\ {0} & {1}
\end{array} \right)\left(\begin{array} {c} {1}\\ {0}
\end{array} \right)=\left(\begin{array} {c} {1}\\ {0}
\end{array} \right)$$
$$\left(\begin{array} {cc} {1} & {1}\\ {0} & {1}
\end{array} \right)\left(\begin{array} {c} {0}\\ {1}
\end{array} \right)=\left(\begin{array} {c} {1}\\ {1}
\end{array} \right),\ \left(\begin{array} {cc} {1} & {1}\\ {0} & {1}
\end{array} \right)\left(\begin{array} {c} {1}\\ {1}
\end{array} \right)=\left(\begin{array} {c} {2}\\ {1}
\end{array} \right).$$

\setlength{\unitlength}{.5in}

\begin{center}
\begin{picture}(4,2)
\put(-0.5,0){\vector(1,0){4}} \put(0,-0.5){\vector(0,1){2}}
\put(0,0){\line(1,1){1}} \put(1,1){\line(1,0){1}}
\put(1,0){\line(1,1){1}} \put(3.6,0){$x$} \put(-0.3,1.5){$y$}
\put(4,1){Area $=1$}

\end{picture}
\end{center}

\vspace{0.2in}

\item[(ii)]
The matrix $\left(\begin{array} {cc} {\cos \frac{\pi}{4}} & {-\sin
\frac{\pi}{4}}\\ {\sin \frac{\pi}{4}} & {\cos \frac{\pi}{4}}
\end{array} \right)$ corresponds to an anticlockwise rotation of
$\frac{\pi}{4}$ radians ($45^{\circ}$) about the origin:

\setlength{\unitlength}{.5in}

\begin{center}
\begin{picture}(4,3)
\put(-0.5,0){\vector(1,0){4}} \put(1,-0.5){\vector(0,1){3}}
\put(1,0){\line(1,1){1}} \put(2,1){\line(-1,1){1}}
\put(1,0){\line(-1,1){1}} \put(0,1){\line(1,1){1}}
\put(3.6,0){$x$} \put(0.7,2.5){$y$} \put(4,1){Area $=1$}
\put(1.3,0.1){$\pi/4$}

\end{picture}
\end{center}

\vspace{0.5in}

\item[(iii)]
$\left(\begin{array} {cc} {\cos \theta} & {\sin \theta}\\ {\sin
\theta} & {-\cos \theta}
\end{array} \right)\left(\begin{array} {c} {1}\\ {0}
\end{array} \right)=\left(\begin{array} {c} {\cos \theta}\\ {\sin \theta}
\end{array} \right),$

$\left(\begin{array} {cc} {\cos \theta} & {\sin \theta}\\ {\sin
\theta} & {-\cos \theta}
\end{array} \right)\left(\begin{array} {c} {0}\\ {1}
\end{array} \right)=\left(\begin{array} {c} {\sin \theta}\\ {-\cos \theta}
\end{array} \right)$

\setlength{\unitlength}{.5in}

\begin{center}
\begin{picture}(4,4)
\setlength{\unitlength}{1in}

\put(-0.1,1){\vector(1,0){3}} \put(1,-0.2){\vector(0,1){2}}
\put(1,1){\line(2,1){1}} \put(2,1.5){\line(1,-2){0.5}}
\put(1,1){\line(1,-2){0.5}} \put(1.5,0){\line(2,1){1}}
\put(1.4,1.05){$\theta$} \put(1.05,0.6){$\theta$}

 \put(2.8,1.1){$x$} \put(0.9,1.8){$y$}
\put(2,1.8){Area $=1$}

\end{picture}
\end{center}

\vspace{.3in}

The matrix corresponds to a reflection followed by a rotation.
(See Question 5. (ii) below.)

\end{description}



\end{document}