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\begin{document}
\centerline{Math 201 Homework for Week 3, Tuesday}

\bigskip
\textbf{Due:} Tuesday, September 14.

\begin{prob}
In each of the following:
    \begin{itemize}
      \item Determine whether the given vector~$v$ is in the span of the
	set~$S$.
      \item If~$v$ is in the span of~$S$, then explicitly write~$v$ as a linear
	combination of the vectors in~$S$.
    \end{itemize} 
    \begin{enumerate}
      \item $V=\mathcal{P}_3(\Q)$, $v=x^3+8x^2+7x-18$, \\
	$S=\left\{ x^3+3x-2, x^3+4x^2-x+2,x^2-2x+3 \right\}$.
      \item $V=M_{2\times 2}(\R)$, $v=\left(\begin{array}{rrr}-1&3\\2& 4 \end{array}\right)$, \\
	$S=\left\{ 
	\left(\begin{array}{rrr}1&1\\1&-2 \end{array}\right),
	\left(\begin{array}{rrr}-1&2\\1& 2 \end{array}\right),
	\left(\begin{array}{rrr}2&4\\3& 5 \end{array}\right)
      \right\}$.
    \end{enumerate}
\end{prob}
    
\textbf{Definition.} Let $S$ be a subset of a vector space $V$. We say $S$ \emph{generates} $V$ if $\Span(S)=V$. 

\begin{prob}
  Let $F$ be a field and consider $S=\{(1, 1, 0), (1, 0, 1), (0, 1, 1)\}\subseteq F^3$. 
  \begin{enumerate}
    \item Prove that if $F=\Q$, then $S$ generates $\Q^3$. 
    \item Prove that if $F=\F_2$, then $S$ does not generate $\F_2^3$. 
  \end{enumerate}
\end{prob}

\begin{prob}
Let $V$ be a vector space over a field $F$, and let $S_1$ and $S_2$ be two subsets of $V$.
\begin{enumerate}
  \item Prove that $\Span(S_1 \cap S_2) \subseteq \Span(S_1) \cap \Span(S_2)$.
  \item Give an example in which  $\Span(S_1 \cap S_2)$ and $\Span(S_1) \cap \Span(S_2)$ are equal, and one in which they are not equal.
\end{enumerate}
\end{prob}

\end{document}
