0004.tex

\begin{exercise}{A2}{Standard matrices}{0004}
\begin{exerciseStatement}

\begin{enumerate}[(a)]
\item Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 4  \to \mathbb{R}^ 2 \) given by \[S\left(  \left[\begin{array}{c}
x \\
y \\
z \\
{w}
\end{array}\right]  \right) =  \left[\begin{array}{c}
2 \, x - 3 \, y + z + 6 \, {w} \\
x - y + 2 \, {w}
\end{array}\right] .\]
\item Let \(T:\mathbb{R}^ 4  \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix \[ \left[\begin{array}{cccc}
3 & 0 & 4 & 5 \\
0 & 1 & 1 & 3 \\
-1 & -1 & -2 & -4
\end{array}\right] .\] Compute \(T\left( \left[\begin{array}{c}
5 \\
8 \\
7 \\
-2
\end{array}\right]  \right)\). 
\end{enumerate}

    \end{exerciseStatement}
\begin{exerciseAnswer}

\begin{enumerate}[(a)]
\item \[ \left[\begin{array}{cccc}
2 & -3 & 1 & 6 \\
1 & -1 & 0 & 2
\end{array}\right] \]
\item \[T\left( \left[\begin{array}{c}
5 \\
8 \\
7 \\
-2
\end{array}\right]  \right)= \left[\begin{array}{c}
33 \\
9 \\
-19
\end{array}\right] \]
\end{enumerate}

    \end{exerciseAnswer}
\end{exercise}