0006.tex

\begin{exercise}{A2}{Standard matrices}{0006}
\begin{exerciseStatement}

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

    \end{exerciseStatement}
\begin{exerciseAnswer}

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

    \end{exerciseAnswer}
\end{exercise}