0003.tex

\begin{exercise}{A2}{Standard matrices}{0003}
\begin{exerciseStatement}

\begin{enumerate}[(a)]
\item Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 3  \to \mathbb{R}^ 3 \) given by \[S\left(  \left[\begin{array}{c}
x \\
y \\
z
\end{array}\right]  \right) =  \left[\begin{array}{c}
x \\
y - 4 \, z \\
-y + 5 \, z
\end{array}\right] .\]
\item Let \(T:\mathbb{R}^ 2  \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix \[ \left[\begin{array}{cc}
-5 & 3 \\
3 & -2 \\
0 & 2
\end{array}\right] .\] Compute \(T\left( \left[\begin{array}{c}
-5 \\
3
\end{array}\right]  \right)\). 
\end{enumerate}

    \end{exerciseStatement}
\begin{exerciseAnswer}

\begin{enumerate}[(a)]
\item \[ \left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & -4 \\
0 & -1 & 5
\end{array}\right] \]
\item \[T\left( \left[\begin{array}{c}
-5 \\
3
\end{array}\right]  \right)= \left[\begin{array}{c}
34 \\
-21 \\
6
\end{array}\right] \]
\end{enumerate}

    \end{exerciseAnswer}
\end{exercise}