0011.tex

\begin{exercise}{A2}{Standard matrices}{0011}
\begin{exerciseStatement}

\begin{enumerate}[(a)]
\item Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 3  \to \mathbb{R}^ 2 \) given by \[S\left(  \left[\begin{array}{c}
x \\
y \\
z
\end{array}\right]  \right) =  \left[\begin{array}{c}
-x + 4 \, y + 2 \, z \\
2 \, x - 8 \, y - 5 \, z
\end{array}\right] .\]
\item Let \(T:\mathbb{R}^ 3  \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \[ \left[\begin{array}{ccc}
0 & -2 & -5 \\
-1 & 6 & 4 \\
0 & -5 & -4 \\
-1 & 3 & -2
\end{array}\right] .\] Compute \(T\left( \left[\begin{array}{c}
-6 \\
5 \\
4
\end{array}\right]  \right)\). 
\end{enumerate}

    \end{exerciseStatement}
\begin{exerciseAnswer}

\begin{enumerate}[(a)]
\item \[ \left[\begin{array}{ccc}
-1 & 4 & 2 \\
2 & -8 & -5
\end{array}\right] \]
\item \[T\left( \left[\begin{array}{c}
-6 \\
5 \\
4
\end{array}\right]  \right)= \left[\begin{array}{c}
-30 \\
52 \\
-41 \\
13
\end{array}\right] \]
\end{enumerate}

    \end{exerciseAnswer}
\end{exercise}