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\noindent
\parbox{1.95in}{\bf Math 5230}
\hfill {\Large \bf Homework \#6} \hfill
\parbox{1.95in}{\bf \hfill Due: AFTER BREAK}

\vspace{0.2in}

\begin{description}
\item[\#1 Linear Independence and Spanning] Suppose $V$ has a basis $\beta = \{\B u, \B v, \B w\}$ (distinct vectors).\\ \force \hfill 
Let $S = \{\B u+\B v, \B v +\B w, \B w +\B u \}$ and $T = \{\B u+\B v, \B u - \B v, \B w \}$.

{\it Additional Assumption:} $\mathrm{char}(\mathbb{F}) \not=2$.

\begin{enumerate}[(a)]
    \item Prove $S$ is linearly independent (using the definition of independence).
    \item Prove $T$ spans $V$ (using the definition of span).
    \item Is $S \cup T$ linearly independent? Does it span $V$? Explain.
    \item Is $S \cap T$ linearly independent? Does it span $V$? Explain.
\end{enumerate}


\item[\#2 Bases] Consider $W = \left\{ \begin{bmatrix} -3a-4b+c & b-c & -4a-6b+2c \\ a+b & 2b-2c & a+b \end{bmatrix} \;\Big|\; a,b,c \in \mathbb{R} \right\}$ and 

$V = \left\{ \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix} \;\Bigg|\; \begin{array}{ccc} a+3b+2c+4d+e+7f & = & 0 \\ 2a+6b+c+5d-e+5f & = & 0 \\ 3a+9b+c+7d-2e+6f & = & 0 \end{array} \right\}.$
    \begin{enumerate}[(a)]
    \item Both $W$ and $V$ are subspaces of $\mathbb{R}^{2 \times 3}$. Prove that they are subspaces by re-expressing each as either a span or kernel (i.e., null space). Pick the most convenient description.
    
    {\it Note:} Re-expressing as a span or kernel would prove we have a subspace since spans and kernels are always subspaces. 
    \item Prove that $W \subseteq V$.
    \item Find a basis for $V$.
    \item Find a basis for $W$ and then extend this to a basis for all of $V$.
    \end{enumerate}

\end{description}

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