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\noindent
\parbox{2.25in}{\bf Math 3510}
\hfill {\Large \bf Homework \#3} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Sept. $26^\mathrm{th}$, 2018}

\vspace{0.2in}

%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
%
%\vspace{0.2in}

\begin{description}
\item[\#1 Linear Correspondence:] Find $A$ and $B$ given\dots

     \[ A = \begin{bmatrix} 1 & ? & 1 & ? & ? &  3 & ? & ? \\ 
                            0 & ? & 2 & ? & ? &  1 & ? & ? \\
                            2 & ? & 2 & ? & ? & -1 & ? & ? \\
                           -1 & ? & 1 & ? & ? &  2 & ? & ? \end{bmatrix} 
     \quad \stackrel{\mathrm{RREF}}{\sim} \quad
            \begin{bmatrix} 1 & 3 & 0 & -1 & -1 & 0 & 0 &  2 \\
                            0 & 0 & 1 &  2 &  1 & 0 & 1 & -1 \\
                            0 & 0 & 0 &  0 &  0 & 1 & 1 &  1 \\
                            0 & 0 & 0 &  0 &  0 & 0 & 0 &  0 \end{bmatrix}
     \]  
     \[ B = \begin{bmatrix} ? & ? & -2 & ? & -2 & ? \\
                            ? & ? & -4 & ? &  5 & ? \\
                            ? & ? & -6 & ? &  7 & ? \\
                            ? & ? & -2 & ? &  0 & ? \end{bmatrix}
     \quad \stackrel{\mathrm{RREF}}{\sim} \quad
            \begin{bmatrix} 0 & 1 & -2 & 0 &  3 & 3 \\
                            0 & 0 &  0 & 1 & -1 & 2 \\
                            0 & 0 &  0 & 0 &  0 & 0 \\
                            0 & 0 &  0 & 0 &  0 & 0 \end{bmatrix} 
      \]  

\item[\#2 Basis Computations:] A few basis calculations.
\begin{enumerate}[(a)]
\item Let $\B v_1 = (2,-3,4,-5,2)$, $\B v_2 = (-6,9,-12,15,-6)$, $\B v_3 = (3,-2,7,-9,1)$, $\B v_4 = (2,-8,2,-2,6)$, $\B v_5 = (-1,1,2,1,-3)$, $\B v_6 = (0,-3,-18,9,12)$, $\B v_7 = (1,0,-2,3,-2)$, and $\B v_8 = (2,-1,1,-9,7)$.

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I would suggest using Maple (or some other software to do the heavy lifting\dots

{\color{red}
\verb&with(LinearAlgebra):&\\
\verb&A := <<2,-3,4,-5,2>|<-6,9,-12,15,-6>|<3,-2,7,-9,1>|<2,-8,2,-2,6>|&\\
\verb&     <-1,1,2,1,-3>|<0,-3,-18,9,12>|<1,0,-2,3,-2>|<2,-1,1,-9,7>>;&
}

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Let $W = \mathrm{span}(S)$ where $S = \{\B v_1, \B v_2,\dots, \B v_8 \}$. Find a subset $\beta \subseteq S$ which is a basis for $W$. What is $\dim(W)$? Write down $\beta$-coordinates for each vector in $S$. 

\item Let $W = \left\{ (x_1,x_2,x_3) \;\Bigg|\; \begin{array}{rcl} x_1-2x_2+x_3 & = & 0 \\ 2x_1-3x_2+x_3 & = & 0 \end{array} \right\}$. Invent a linear transformation $T:??? \to ???$ such that $W=\mathrm{Ker}(T)$. Find a coordinate matrix for your $T$. Find a basis for $W$. What is $\dim(W)$? 
\end{enumerate}

\item[\#3 More Basis Computations:] 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 $\mathbb{R}^{2 \times 3}$. Show this is the case by re-expressing each as either a span or kernel [thus it's a subspace
          since spans and kernels are always subspaces]. Pick the most convenient description.
\item Show $W \subseteq V$.
\item Find a basis for $V$.
\item Find a basis for $W$. Then extend this to a basis for all of $V$.
\end{enumerate}

\item[\#4 Dimensional Issues:] Let $W_1$ and $W_2$ be subspaces of a finite dimensional vector space $V$.
\begin{enumerate}[(a)]
\item Prove that $\dim(W_1 \cap W_2) \leq \mathrm{min} \{ \dim(W_1), \dim(W_2) \}$.
\item Prove that $\dim(W_1 + W_2) \geq \mathrm{max} \{ \dim(W_1), \dim(W_2) \}$.
\end{enumerate}

{\it Suggestion:} For problems like this assume (without loss of generality = WLOG) that $\dim(W_1) = m \leq \dim(W_2)=n$ so that $\mathrm{min} \{\dim(W_1),\dim(W_2) \} = \dim(W_1)=m$ and $\mathrm{max} \{\dim(W_1),\dim(W_2) \} = \dim(W_2)=n$. Always look for such {\it harmless} simplifying assumptions when proving a statement. 

{\it Hint:} It might be helpful to remember that any linearly independent set can be extended to a basis and that every spanning set contains a basis.

\item[\#5 Quotients:] Let $U$ and $W$ be subspaces of some vector space $V$. Show that $\diagquotient{U+W}{W} \cong \diagquotient{U}{U \cap W}$.

{\it Hint:} Apply the first isomorphism theorem to $T:U+W \to \diagquotient{U}{U \cap W}$ where $T(\B u+\B w) = \B u + (U \cap W)$. Things consider: Well-defined? Linear? Onto? Kernel?

\end{description}

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