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\lhead{{\sf Bridge course (Algebra)}: Thursday July 17}
\chead{}\rhead{\thepage}
\lfoot{Due Friday, July 18 2024}
\cfoot{}\rfoot{Written by M.~Macauley}
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\begin{document}

%$\;$ \vspace{-4mm}

\noindent \textbf{Topics}: Linear maps. \\

\noindent \textbf{Do}: Answer the following questions. Throughout, assume that $X$ is a finite-dimensional vector space over a field $K$.

\begin{enumerate}

%%--------------------------------------------------------------------- 


\item Let $T\colon X\to U$ be a linear map.
\begin{enumerate}
    \item Show that the \emph{image} $T(X)$ (i.e., the \emph{range}, $R_T$) is a subspace of $U$.
    \item Show that the \emph{preimage} of a subspace $V\leq U$, denoted
\[
T^{-1}(V):=\big\{x\in X\mid Tx\in V\big\},
\]
is a subspace of $X$. \\
\end{enumerate}

 %%--------------------------------------------------------------------- 

\item Let $A\colon X\to X$ be a linear map.
  \begin{enumerate}
  \item Prove that if $\dim(X)<\infty$, then the following are equivalent:
    \begin{enumerate}[(i)]
    \item $A$ is bijective.
    \item $A$ is injective.
    \item $A$ is surjective.
    \end{enumerate}
  \item How can the results of Part~(a) fail if $\dim(X)=\infty$? Give
    an explicit example to demonstrate this. 
  \end{enumerate}

  %%--------------------------------------------------------------------- 
  
\end{enumerate}

\end{document}
