0008.tex

\begin{exercise}{A2}{Standard matrices}{0008}
\begin{exerciseStatement}

\begin{enumerate}[(a)]
\item Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 4  \to \mathbb{R}^ 2 \) given by \[S\left(  \left[\begin{array}{c}
x \\
y \\
z \\
{w}
\end{array}\right]  \right) =  \left[\begin{array}{c}
z \\
-x - 3 \, y + 6 \, z + 3 \, {w}
\end{array}\right] .\]
\item Let \(T:\mathbb{R}^ 3  \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix \[ \left[\begin{array}{ccc}
-5 & -2 & -3 \\
-2 & -1 & -2 \\
0 & -2 & -7
\end{array}\right] .\] Compute \(T\left( \left[\begin{array}{c}
3 \\
-4 \\
-6
\end{array}\right]  \right)\). 
\end{enumerate}

    \end{exerciseStatement}
\begin{exerciseAnswer}

\begin{enumerate}[(a)]
\item \[ \left[\begin{array}{cccc}
0 & 0 & 1 & 0 \\
-1 & -3 & 6 & 3
\end{array}\right] \]
\item \[T\left( \left[\begin{array}{c}
3 \\
-4 \\
-6
\end{array}\right]  \right)= \left[\begin{array}{c}
11 \\
10 \\
50
\end{array}\right] \]
\end{enumerate}

    \end{exerciseAnswer}
\end{exercise}