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\parbox{2.25in}{\bf Math 4010-101}
\hfill {\Large \bf Homework \#3} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Feb. $9^\mathrm{th}$, 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.] Coordinated Practice! Let $P_2=\{ax^2+bx+c \;|\; a,b,c \in \mathbb{R}\}$ be the subspace of $\mathbb{R}[x]$ consisting of quadratic and lower degree polynomials. Here we let $\alpha=\{1,x,x^2\}$ be our standard basis (note our intended order). Let $\mathbb{R}^{2 \times 2}$ be the space of $2 \times 2$ real matrices. Let $\beta=\{E_{11},E_{12},E_{21},E_{22}\}$ be our standard basis (note the ordering here -- go successively across each row).

Let $T:P_2 \to \mathbb{R}^{2 \times 2}$ be defined by $T(ax^2+bx+c) = \begin{bmatrix} a+c & a+b+2c \\ 2b+2c & b+c \end{bmatrix}$. 

\begin{enumerate}[(a)]
\item Prove $T$ is a linear transformation.
\item Compute the standard matrix of $T$ (i.e., $[T]_\alpha^\beta$). 
\item Find a basis for both $\mathrm{ker}(T)$ and $T(P_2)$ -- don't forget to translate back out of coordinates! What is the rank and nullity of $T$? Is $T$ one-to-one? Is it onto? Is it invertible?
\item Let $\gamma = \{x+1,x^2+x,x^2+1\}$ and $\delta =\left\{ \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \right\}$. Justify why these are bases of $P_2$ and $\mathbb{R}^{2 \times 2}$ respectively.
\item Calculate the change of basis matrices $[I]_\alpha^\gamma$ and $[I]_\beta^\delta$.
\item Calculate the standard coordinates of $f(x)=x^2+2x+3$ and then the $\gamma$-coordinates of $f(x)$. Also, compute the standard and $\delta$-coordinates of $T(f(x))$.
\item Find $[T]_\gamma^\delta$. Then verify that $[T(f(x))]_\delta = [T]_\gamma^\delta \cdot [f(x)]_\gamma$.
\end{enumerate}

\item[\#2.] Composing Coordinates.  Recall the definitions of $P_2$, $\mathbb{R}^{2 \times 2}$, $\alpha$, and $\beta$ from Problem \#1 above.

Let $T=\dfrac{d}{dx}:P_2 \to P_2$ be the derivative operator on $P_2$ (i.e., $T(ax^2+bx+c)=2ax+b$) and let $S:\mathbb{R}^{2 \times 2} \to P_2$ be defined by $$S\left(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\right) = (a+b)x^2+(a+d)x-c$$ Find $[T]_\alpha^\alpha$, $[S]_\beta^\alpha$. Then compute $[T \circ S]_\beta^\alpha$ using the previous two matrices. Verify this formula by giving a general formula for the linear transformation $T \circ S:\mathbb{R}^{2 \times 2} \to P_2$ and the computing $[T \circ S]_\beta^\alpha$ directly.

\item[\#3.] The trace map merely sums the diagonal entries of a matrix. In particular, $\mathrm{tr}:\mathbb{F}^{n \times n} \to \mathbb{F}$ is defined by $\mathrm{tr}(A)=\sum\limits_{i=1}^n a_{ii}=a_{11}+a_{22}+\cdots+a_{nn}$ if the $(i,j)$-entry of $A$ is denoted $a_{ij}$.

\begin{enumerate}[(a)]
\item Prove that the trace map is linear (for general $n$).
\item Consider the trace map on $3 \times 3$ matrices. Find its standard matrix (use a basis $\alpha$ of $E_{ij}$'s for $\mathbb{F}^{3 \times 3}$ and $\beta=\{1\}$ for $\mathbb{F}$).
\item Continuing part (b), find a basis for its kernel and range. What is its nullity and rank? Is it one-to-one, onto, and/or invertible?
\end{enumerate}

\item[\#4.] Standard Stuff. Let $T:V \to W$ be a linear transformation.
\begin{enumerate}[(a)]
\item Prove $\mathrm{ker}(T) = \{ {\bf v} \in V \;|\; T({\bf v})={\bf 0} \} = T^{-1}(\{{\bf 0}\})$ is a subspace of $V$ (using the subspace test -- no cheating here). Then show that $T$ is one-to-one if and only if $\mathrm{ker}(T) = \{ {\bf 0}\}$.
\item Suppose $V$ and $W$ are finite dimensional (e.g., $\dim(V)=n$ and $\dim(W)=m$) with bases $\alpha$ and $\beta$ respectively. Given $A=[T]_\alpha^\beta$, what about $A$ tells us whether $T$ is one-to-one or not? What would tell us whether $T$ is onto or not? 
\item Continuing part (b), explain why in the case that $\dim(V)=\dim(W)$ (for example, in the case of a linear operator) we have that $T$ is one-to-one if and only if $T$ is onto.
\end{enumerate}

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