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

\textbf{Due:} Friday, September 17.

\begin{prob}
  Determine whether the following sets are linearly dependent or linearly
  independent. If they are linearly dependent, find a subset that is linearly
  independent and has the same span.
  \begin{enumerate}
    \item $\left\{
	\begin{pmatrix}
	  1 & 1\\
	  0 & 0\\
	  0 & 0
	  \end{pmatrix},  \begin{pmatrix}
	  0 & 0\\
	  1 & 1\\
	  0 & 0
	  \end{pmatrix},  \begin{pmatrix}
	  0 & 0\\
	  0 & 0\\
	  1 & 1
	  \end{pmatrix},  \begin{pmatrix}
	  1 & 0\\
	  1 & 0\\
	  1 & 0
	  \end{pmatrix},  \begin{pmatrix}
	  0 & 1\\
	  0 & 1\\
	  0 & 1
      \end{pmatrix}\right\}$ in $M_{3\times 2}(\Q)$.
    \item $\{(1,3,2),(2,-1,2),(1,2,4)\}$ in $\R^3$.
    \item $\{(1,1,0),(1,0,1),(0,1,1)\}$ in $(\F_2)^3$ (recall that $\F_2=\Z/2\Z$, the field with two elements).
  \end{enumerate}
\end{prob}

\begin{prob}
    Let $V$ be a vector space over $\R$. Let $u$ and $v$ be distinct vectors in~$V$.
    Prove that $\{u,v\}$ is linearly independent if and only if $\{u+v,u-v\}$ is
    linearly independent.
    
      {\sc Bonus:} Would the same be true over $\F_2$?
\end{prob}

\begin{prob}
  \begin{enumerate}
    \item For any field $F$, we have defined the vector space $F^n$ of
      $n$-tuples with components in $F$.  List all elements of (i) $F^2$ and
      (ii) $F^3$ in the case that $F =\F_2$.
    \item Let $S = \{u_1,...,u_n\}$ be a set of linearly independent vectors in
      a vector space over $\F_2$.  How many elements are in $\Span(S)$? Justify
      your solution.
  \end{enumerate}
\end{prob}
 
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