0001.tex

\begin{exercise}{A2}{Standard matrices}{0001}
\begin{exerciseStatement}

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

    \end{exerciseStatement}
\begin{exerciseAnswer}

\begin{enumerate}[(a)]
\item \[ \left[\begin{array}{ccc}
-1 & 3 & 4 \\
-1 & 0 & -1 \\
2 & -2 & -1 \\
-3 & 1 & -6
\end{array}\right] \]
\item \[T\left( \left[\begin{array}{c}
-3 \\
-7 \\
6 \\
-5
\end{array}\right]  \right)= \left[\begin{array}{c}
-27 \\
-8
\end{array}\right] \]
\end{enumerate}

    \end{exerciseAnswer}
\end{exercise}