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\noindent
\parbox{2.25in}{\bf Math 5230}
\hfill {\Large \bf Homework \#7} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Nov. $11^\mathrm{th}$, 2016}

\vspace{0.2in}

\begin{description}
\item[\#1 Linearity] Let $T:V \to W$ and $S:W \to U$ be linear transformations between vector spaces $V$, $W$, and $U$ (over some field $\mathbb{F}$). 

\begin{enumerate}[(a)]
   \item Prove that $S \circ T$ is linear.
   \item Assume that $T$ is a bijection. Show that $T^{-1}$ is linear (i.e. if $T$ is an isomorphism, then so is $T^{-1}$).
\end{enumerate}   

\item[\#2 Combating Coordinate Change] Let $T : P_2 \to \mathbb{R}^{2 \times 2}$ and $S: \mathbb{R}^{2 \times 2} \to \mathbb{R}^2$ be defined by $$T(ax^2+bx+c) = \begin{bmatrix} a+b+c & 2a-b+5c \\ b-c & a+2b \end{bmatrix} \quad \mbox{ and } \quad S\left(\begin{bmatrix} a & b \\ c & d \end{bmatrix} \right)= (a+b-c,b+c+d).$$

Let $\alpha = \{ 1,x,x^2 \}$, $\beta = \{ E_{11}, E_{12}, E_{21}, E_{22} \}$, and $\gamma = \{ (1,0), (0,1) \}$ be the standard bases for $P_2$, $\mathbb{R}^{2 \times 2}$, and $\mathbb{R}^2$ respectively.

\begin{enumerate}[(a)]
   \item Show $S$ and $T$ are linear. [So by the first problem, we have that $S \circ T$ is linear too.]
   \item Find $[T]_\alpha^\beta$, $[S]_\beta^\gamma$, and $[S \circ T]_\alpha^\gamma$.
   \item Let $\delta = \{ x,1+x,x+x^2 \}$. Explain why $\delta$ is a basis for $P_2$ and then find the change of basis matrices $[I]_\delta^\alpha$ and $[I]_\alpha^\delta$. 
   \item Let $\kappa = \{ (1,2), (3,4) \}$. Explain why $\kappa$ is a basis for $\mathbb{R}^2$ and then find the change of basis matrices $[I]_\kappa^\gamma$ and $[I]_\gamma^\kappa$. 
   \item Compute $[S \circ T]_\delta^\kappa$.   
\end{enumerate}

\item[\#3 Dual Basis] Let $\alpha = \{ (1,0,0), (1,-1,0), (2,0,1) \}$. 
\begin{enumerate}[(a)]
\item Explain why $\alpha$ is a basis for $\mathbb{R}^3$. 
\item Find $\alpha^*$ for $(\mathbb{R}^3)^*$ (i.e. find the basis dual to $\alpha$).
\item Explain why $f \in (\mathbb{R}^3)^*$ where $f(x,y,z)=3x+2y+z$. Then write $f$ as a linear combination of $\alpha^*$ elements (i.e. find its $\alpha^*$-coordinates).
\end{enumerate}

\item[\#4 Dual Proof] Let $W$ be a subspace of a vector space $V$ (over a field $\mathbb{F}$). We say that $f \in V^*$ {\bf annihilates} $W$ if $f(\B w)=0$ for all $\B w \in W$. Let $A(W) = \{ f \in V^* \;|\; f(\B w) = 0 \mbox{ for all } \B w \in W \}$ (the collection of all linear functionals which annihilate $W$). $A(W)$ is called the {\bf annihilator} of $W$. 

\begin{enumerate}[(a)]
\item Prove that $A(W)$ is a subspace of $V^*$.
\item Suppose that $V=U \oplus W$ for some subspaces $U$ and $W$. Show that $V^* = A(W) \oplus A(U)$.
\item Let $T:V \to V$ be a linear operator and suppose that $T(W) \subseteq W$ for some subspace $W$ (i.e. $W$ is a $T$-invariant subspace).  
          Show that $T^t(A(W)) \subseteq A(W)$ (i.e. $A(W)$ is $T^t$-invariant).
\end{enumerate}

\end{description}

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