0009.tex

\begin{exercise}{A2}{Standard matrices}{0009}
\begin{exerciseStatement}

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

    \end{exerciseStatement}
\begin{exerciseAnswer}

\begin{enumerate}[(a)]
\item \[ \left[\begin{array}{cccc}
4 & 1 & -7 & -6 \\
3 & 1 & -5 & -4
\end{array}\right] \]
\item \[T\left( \left[\begin{array}{c}
1 \\
-2 \\
7 \\
-6
\end{array}\right]  \right)= \left[\begin{array}{c}
-30
\end{array}\right] \]
\end{enumerate}

    \end{exerciseAnswer}
\end{exercise}