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\lhead{{\sf Summer Bridge Course (Algebra)}: Tuesday July 7}
\chead{}\rhead{\thepage}
\lfoot{Due Thursday, July 9 2026}
\cfoot{}\rfoot{Written by M.~Macauley}
\pagestyle{fancy}


\begin{document}

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

\noindent \textbf{Topics}: Vector spaces and subspaces \\

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


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

%% HW 4, Problem 1
\begin{exercise}
\item Let \(X\) be a vector space over a field \(K\). Let \(0\) be the zero element of \(K\), and \(\0\) the zero element of \(X\). Using only the definitions of a group, vector space, and field, carefully prove the following.
  \begin{enumerate}[label=(\roman*)]
  \item \((-1)x=-x\) for all \(x\in X\).
  \item For every \(k\in K\) and \(x\in X\), if \(kx=\0\), then \(k=0\) or \(x=\0\). \\
  \end{enumerate}
\end{exercise}

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

%% HW 4, Problem 2
\begin{exercise}
  Let \(X\) be a vector space.
  \begin{enumerate}[label=(\roman*)]
  \item Prove that for any collection \(\big\{Y_\alpha\mid \alpha\in
    A\big\}\) of subspaces, \(\bigcap_{\alpha} Y_\alpha\) is a
    subspace.
  \item Prove that for any subset \(S\subseteq X\) of a vector
    space,
    \[
    \Span(S)=\bigcap_{S\subseteq Y_\alpha\leq X}Y_\alpha.
    \]
    Here, the intersection is taken over all subspaces \(Y_\alpha\) of
      \(X\) that contain \(S\). \\
  \end{enumerate}
\end{exercise}
    
    %%---------------------------------------------------------------------

%% HW 4, Problem 3
\begin{exercise}
  \(Y\) and \(Z\) be subspaces of a vector space \(X\), with \(Y+Z=X\). That is,
  \[
  Y+Z:=\big\{y+z\mid y\in Y,\,z\in Z\big\}=X.
  \]
  Prove that the following are equivalent:
  \begin{enumerate}[label=(\roman*)]
  \item \(Y\cap Z=\{0\}\).
  \item Every \(x\in X\) can be written \emph{uniquely} as \(x=y+z\), for some \(y\in Y\) and \(z\in Z\). \\
  \end{enumerate} 
\end{exercise}

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

\end{document}