0007.tex

\begin{exercise}{A4}{Injectivity and surjectivity}{0007}
\begin{exerciseStatement} Let \(T:\mathbb{R}^ 4  \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix \( \left[\begin{array}{cccc}
0 & 1 & -1 & 3 \\
-2 & -3 & 4 & -5 \\
-1 & -1 & 2 & -3 \\
0 & 0 & 1 & -3
\end{array}\right] .\)

\begin{enumerate}[(a)]
\item Explain why \(T\) is or is not injective.
\item Explain why \(T\) is or is not surjective.
\end{enumerate}

    \end{exerciseStatement}
\begin{exerciseAnswer}

\[\operatorname{RREF} \left[\begin{array}{cccc}
0 & 1 & -1 & 3 \\
-2 & -3 & 4 & -5 \\
-1 & -1 & 2 & -3 \\
0 & 0 & 1 & -3
\end{array}\right] = \left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right] \]



\begin{enumerate}[(a)]
\item \(T\) is injective.
\item \(T\) is surjective.
\end{enumerate}

    \end{exerciseAnswer}
\end{exercise}