0005.tex

\begin{exercise}{A2}{Standard matrices}{0005}
\begin{exerciseStatement}

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

    \end{exerciseStatement}
\begin{exerciseAnswer}

\begin{enumerate}[(a)]
\item \[ \left[\begin{array}{cc}
1 & -1 \\
3 & -2 \\
-3 & 3 \\
-1 & -1
\end{array}\right] \]
\item \[T\left( \left[\begin{array}{c}
7 \\
8 \\
0
\end{array}\right]  \right)= \left[\begin{array}{c}
-15 \\
-38 \\
-38 \\
15
\end{array}\right] \]
\end{enumerate}

    \end{exerciseAnswer}
\end{exercise}