CE_Apr6.tex

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\usepackage{systeme, array}
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\newcommand{\adj}{\textrm{adj}\,}
\newcommand{\rank}{\textrm{rank}\,}
\newcommand{\Ker}{\textrm{Ker}\,}
\newcommand{\Image}{\textrm{Im}\,}
\newcommand{\Proj}{\textrm{Proj}\,}
\newcommand{\Span}{\textrm{Span}\,}

\begin{document}

\begin{center}
	\Large
	\rm{Math 221}
	\\
	\rm{Class Exercises:  Apr. 6}
	\\
\end{center}
\vspace{0.2in}
\fboxsep0.5cm

\begin{enumerate}
\item{
	The transformations $T:\mathbb{R}^2\to\mathbb{R}^3$ and $S:\mathbb{R}^3\to\mathbb{R}^2$ 
	are defined as follows:
	\begin{displaymath}
	T\left(\left[ \begin{array}{r} x \\ y \end{array} \right] \right) =  \left[ \begin{array}{c} x \\ 0 \\ x+y \end{array} \right] \quad \quad \quad 
	S\left(\left[ \begin{array}{r} x \\ y \\ z  \end{array} \right] \right) =  \left[ \begin{array}{c} z-x \\ y \end{array} \right] \quad
	\end{displaymath}
	\begin{enumerate}
		\item {Is $T$ injective (one-to-one)?}
		\item {Is $S\circ T$ an isomorphism?}
		\item {Find $[S]_{\alpha}^{\beta}$ where $\alpha$ and $\beta$ are the standard ordered basis for $\mathbb{R}^3$ and $\mathbb{R}^2$ respectively.}
	\end{enumerate}
}
\item{Let $T:\mathbb{M}_{2\times 2} \to \mathbb{M}_{2\times 2}$ be defined by $T(B)=B^T$.
	\begin{enumerate}
		\item {Show that $T$ is linear.}
		\item {Find $[T]_{\alpha}^{\alpha}$ if
			\begin{displaymath}
			\alpha = \left\{\left[ \begin{array}{rr} 1&0\\0& 0 \end{array} \right],
			\left[ \begin{array}{rr} 0&1\\0& 0 \end{array} \right],
			\left[ \begin{array}{rr} 0&0\\1& 0 \end{array} \right],
			\left[ \begin{array}{rr} 0&0\\0& 1 \end{array} \right] \right\}
			\end{displaymath}
		}
	\end{enumerate}
}

\vspace{0.1in}


\item{Decide if each of the following statements is true or false and explain why:
\begin{enumerate}
	\item {If $V\perp W$, then $V^{\perp}\perp W^{\perp}$.}
	\item {If $V\perp W$ and $W\perp Z$, then $V\perp Z$.} 
\end{enumerate}
}

\item{The equation $x+2y-3z=0$ defines a plane in $\mathbb{R}^3$.
\begin{enumerate}
	\item {Find a matrix that has this plane as its null space.  Is the matrix unique?}
	\item {Find a matrix that has this plane as its row space.  Is the matrix unique?}
\end{enumerate}
}

\vspace{0.1in}

\item{
	\begin{enumerate}
		\item {Find the matrix $P_1$ that projects orthogonally onto $\Span\left\{\left[ \begin{array}{c} 2 \\ 1   \end{array}\right]\right\}$.}
		\item{Find the matrix $P_2$ that projects orthogonally onto $\Span\left\{\left[ \begin{array}{c} -1 \\ 4   \end{array}\right]\right\}$.}
		\item{Is the product $P_1P_2$ also an orthogonal projection?}
	\end{enumerate}

}

\vspace{0.2in}

\item{Let $U = \Span\left\{\left[ \begin{array}{c} 0 \\ 1 \\ 0   \end{array}\right]\right\}$ and
	$W = \Span\left\{
	\left[ \begin{array}{c} 2 \\ 1 \\ 1  \end{array}\right],
	\left[ \begin{array}{c} 0 \\ 1 \\ 1  \end{array}\right]
	\right\}$
	
	\vspace{0.2in}
	
	Find a matrix $P$ so that $Px$ is the projection of $x$ onto $U$ along W.  
	\begin{enumerate}
		\item {Check that $P^2=P$.}
		\item {Check that $Px$ is in $U$ for any $x\in\mathbb{R}^3$.}
		\item {Check that $Pw=0$ for any $w\in W$.}
\end{enumerate}
}


\end{enumerate}

\vspace{0.1in}
\end{document}