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\lhead{{\sf Summer Bridge Course (Algebra)}: Monday July 13}
\chead{}\rhead{\thepage}
\lfoot{Due Tuesday, July 14 2026}
\cfoot{}\rfoot{Written by M.~Macauley}
\pagestyle{fancy}

\begin{document}


\noindent \textbf{Topics}: Quotients \\

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

  %%--------------------------------------------------------------------- 
  
  %% HW 6, Problem 1
  \begin{exercise}
    If \(Y\) is a subspace of \(X\), then two vectors \(x_1,x_2\in X\) are
    \emph{congruent modulo \(Y\)}, denoted \(x_1\equiv x_2\bmod Y\), if
    \(x_1-x_2\in Y\). This is an equivalence relation; denote the
    equivalence class containing \(x\in X\) by \(\{x\}\), and let \(X/Y\)
    denote the set of equivalence classes. We can make \(X/Y\) into a
    vector space by defining addition and scalar multiplication as
    follows:
    \[
    \{x\}+\{z\}:=\{x+z\},\qquad a\{x\}:=\{ax\}.
    \]
    Show that these operations are \emph{well-defined}. That is, they do not depend on the choice of congruence class representatives.
  \end{exercise}

  %\begin{solution}
  %\end{solution}

  \medskip
  
  %%--------------------------------------------------------------------- 
  
  %% HW 6, Problem 2
  \begin{exercise}  
    Let \(Y\) be a subspace of \(X\). Prove that \(X\) is isomorphic to \(Y\times X/Y\) by defining an explicit map and showing that it is linear and a bijection. 
  \end{exercise}

  %\begin{solution}
  %\end{solution}

  \medskip
    
%%--------------------------------------------------------------------- 

\end{document}
