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\noindent
\parbox{2.25in}{\bf Math 4010-101}
\hfill {\Large \bf Homework \#2} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Jan. $31^\mathrm{st}$, 2022}

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%\noindent {\bf\large Name:} \quad \underline{\large\qquad This is my Name \qquad}

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Please remember when submitting any work via email or in person to\dots\\
\force \hfill \fbox{\fbox{\bf\large\color{red}PUT YOUR NAME ON YOUR WORK!}} \hfill \force


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\begin{description}
\item[\#1.] Linear Correspondence practice. Find $ A $ and $ B $ where\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.] Define $ T:\mathbb{R}^{2 \times 2} \to \mathbb{R}[x] $ as follows: $ T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = (a+2b+3d)x^2+(-a-2b+c+d)x+(a+2b+c+7d) $

Note that $ T $ is a linear transformation (you don't need to prove this).

\begin{enumerate}[(a)]
\item Write down the standard coordinate matrix for $ T $ and find its RREF.
          
          Note that we use $\alpha=\{E_{11},E_{12},E_{21},E_{22}\}$ as our input basis and $\beta = \{1,x,x^2,\dots\}$ as our output basis.

         Also, you will need to truncate the infinitely many rows of zeros when doing Gaussian elimination.
         
\item Find a basis for the kernel and range of $T$.

\item What is the nullity and rank of $T$? Is $T$ 1-1, onto, both, neither?
\end{enumerate}

\item[\#3.] Consider the following subspaces of $\mathbb{R}^{2\times 3}$:
$$W = \left\{ \begin{bmatrix} -3a-4b+c & b-c & -4a-6b+2c \\ a+b & 2b-2c & a+b \end{bmatrix} \;\Big|\; a,b,c \in \mathbb{R} \right\}$$ 
$$V = \left\{ \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix} \;\Bigg|\; \begin{array}{ccc} a+3b+2c+4d+e+7f & = & 0 \\ 2a+6b+c+5d-e+5f & = & 0 \\ 3a+9b+c+7d-2e+6f & = & 0 \end{array} \right\}$$

\begin{enumerate}[(a)]
\item Show $W$ and $V$ are subspaces by re-expressing each as either a span or kernel. 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.] Let $V$ be a vector space over some field $\mathbb{F}$ and let $W$ be a subspace of $V$.

\begin{enumerate}[(a)]
\item Show that $\dim(W) \leq \dim(V)$. What can be said if $\dim(W)=\dim(V)$?
\item Show there exists some subspace $U$ such that $W \oplus U = V$.
\item Let $W = \mathbb{R} \times \{0\} = \{(x,0)\;|\; x \in \mathbb{R}\}$. Describe all possible subspaces $U$ such that $W \oplus U = \mathbb{R}^2$.
\end{enumerate}

\end{description}

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