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\lhead{{\sf Summer Bridge Course (Algebra)}: Wednesday July 14}
\chead{}\rhead{\thepage}
\lfoot{Due Thursday, July 16 2026}
\cfoot{}\rfoot{Written by M.~Macauley}
\pagestyle{fancy}


\begin{document}

%\(\;\) \vspace{-4mm}


\noindent \textbf{Topics}: The isomorphism theorems \\

\noindent \textbf{Do}: Answer the following questions. \\

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

  %% HW 7, Problem 1
\begin{exercise}
  Let \(f\colon X\to U\) be a linear map between vector spaces. In this problem, you will prove the \emph{first isomorphism theorem}, which says that \(X/\Ker(f)\cong\Image(f)\), where \(\Image(f)=f(X)\).
\begin{enumerate}[label=(\roman*)]
    \item Let \(Z=\Ker(f)\), and \(x+Z=\{x+z\mid z\in Z\}=\overline{x}\)\, be the equivalence class containing \(x\); either notation is fine, feel free to use either. Define a linear map
    \[
    \iota\colon X/Z\longto \Image(f),\qquad x+Z\longmapsto f(x).
    \]
    Carefully give a formal definition of what it means for \(\iota\) to be well-defined, and then prove that it is. 
    \item Prove that \(\iota\) is linear, one-to-one, and onto, and conclude that \(X/\Ker(f)\cong \Image(f)\). \\
\end{enumerate}
\end{exercise}

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


  %% HW 7, Problem 2
\begin{exercise}
In this problem, you will prove the main part of the second isomorphism theorem, sometimes called the \emph{correspondence theorem}. Let \(Z\) be a subspace of \(X\). 
  \begin{enumerate}[label=(\roman*)]
  \item Show that if \(Z\leq Y\leq X\), then \(Y/Z\) is a subspace of \(X/Z\).
  \item Show that \emph{every} subspace of \(X/Z\) arises in this manner. 
  \end{enumerate}
  In this problem, it is easier to use ``coset notation'' \(x+Z\) instead of \(\overline{x}\), because you may need to speak of both equivalence classes \(x+Y\) and \(x+Z\).
\end{exercise}

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

  %% HW 7, Problem 2
\begin{exercise}
Prove the \emph{third isomorphism theorem}: if \(Z\leq Y\leq X\), then \((X/Z)/(Y/Z)\cong X/Y\). 
  [\emph{Hint}: Define a map \(X/Z\to X/Y\), show it is linear, onto, and has kernel \(Y/Z\). Then apply the first isomorphism theorem.]
\end{exercise}
  
%%---------------------------------------------------------------------  

\end{document}
