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%TCIDATA{<META NAME="Title" CONTENT="Exercises">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:11:04 1997}
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\begin{document}


\section{Exercises\label{Exercises}}

\begin{enumerate}
\item[1.]  \label{q1}The vectors $u=\left[ 
\begin{array}{ccc}
1 & 1 & 0
\end{array}
\right] $ and $v=\left[ 
\begin{array}{ccc}
1 & 1 & 1
\end{array}
\right] $ span a plane in $\Bbb{R}^{3}.$ Find the projection matrix $P$ onto
the plane, and find a nonzero vector $b$ that is projected to zero.\hfill 
\hyperref{\TCIIcon{BITMAPSETAnswer}{0.2214in}{0.205in}{0in}}{}{}{a1}

\item[2.]  \label{q2}For the following matrix, find the characteristic
polynomial, minimum polynomial, eigenvalues, and eigenvectors. Discuss the
relationships among these, and explain the multiplicity of the eigenvalue.
\medskip 
\[
\left[ 
\begin{array}{cccc}
2 & 0 & 0 & 0 \\ 
1 & 2 & 0 & 0 \\ 
0 & 0 & 2 & 0 \\ 
0 & 0 & -3 & 2
\end{array}
\right] 
\]

\hfill \hyperref{\TCIIcon{BITMAPSETAnswer}{0.2214in}{0.205in}{0in}}{}{}{a2}

\item[3.]  \label{q3}Which of the following statements are correct for the
matrix $A=\left[ 
\begin{array}{ccc}
1 & 1 & 1 \\ 
1 & 0 & 2
\end{array}
\right] $? The set of all solutions $x=\left[ 
\begin{array}{c}
x_{1} \\ 
x_{2} \\ 
x_{3}
\end{array}
\right] $ of the equation $Ax=\left[ 
\begin{array}{c}
0 \\ 
0
\end{array}
\right] $ is the column space of $A$; the row space of $A$; a nullspace of $A
$; a plane; a line; a point.\hfill \hyperref{\TCIIcon{BITMAPSETAnswer}{%
0.2214in}{0.205in}{0in}}{}{}{a3}

\item[4.]  \label{q4}The matrices that rotate the $xy$-plane are $A\left(
\theta \right) =\left[ 
\begin{array}{rr}
\cos \theta & -\sin \theta \\ 
\sin \theta & \cos \theta
\end{array}
\right] $. Verify that $A\left( \theta \right) A\left( \varphi \right)
=A\left( \theta +\varphi \right) $ and $A\left( -\theta \right) =A\left(
\theta \right) ^{-1}$, using matrix products and trigonometric
identities.\medskip \hfill \hyperref{\TCIIcon{BITMAPSETAnswer}{0.2214in}{%
0.205in}{0in}}{}{}{a4}
\end{enumerate}

\subsection{Solutions\label{Solutions}}

\begin{enumerate}
\item[1.]  \label{a1}The projection 
\index{Projection matrix@Projection matrix}\label{Projection matrix}matrix $%
P $ onto the plane in $\Bbb{R}^{3}$ spanned by the vectors $u=\left[
1,1,0\right] $ and $v=\left[ 1,1,1\right] $ is the product $P=A\left(
A^{T}A\right) ^{-1}A^{T}$, where $u$ and $v$ are the columns of $A$.\hfill 
\hyperref{\TCIIcon{BITMAPSETProbSolvHint}{0.2006in}{0.243in}{0in}}{}{}{q1}

\begin{itemize}
\item  $P=\left[ 
\begin{array}{cc}
1 & 1 \\ 
1 & 1 \\ 
0 & 1
\end{array}
\right] \left( \left[ 
\begin{array}{cc}
1 & 1 \\ 
1 & 1 \\ 
0 & 1
\end{array}
\right] ^{T}\left[ 
\begin{array}{cc}
1 & 1 \\ 
1 & 1 \\ 
0 & 1
\end{array}
\right] \right) ^{-1}\left[ 
\begin{array}{cc}
1 & 1 \\ 
1 & 1 \\ 
0 & 1
\end{array}
\right] ^{T}=\,\left[ 
\begin{array}{ccc}
\frac{1}{2} & \frac{1}{2} & 0 \\ 
\frac{1}{2} & \stackrel{}{\frac{1}{2}} & 0 \\ 
0 & 0 & 1
\end{array}
\right] $
\end{itemize}

Note that $Pw$ is a linear combination of $u$ and $v$ for any vector $%
w=\left( x,y,z\right) $ in $\Bbb{R}^{3}$, so $P$ maps $\Bbb{R}^{3}$ onto the
plane spanned by $u$ and $v$.

\begin{itemize}
\item  $\,\left[ 
\begin{array}{ccc}
\frac{1}{2} & \frac{1}{2}\vspace{4pt} & 0 \\ 
\frac{1}{2} & \frac{1}{2}\vspace{4pt} & 0 \\ 
0 & 0 & \stackrel{}{1}
\end{array}
\right] \left[ 
\begin{array}{c}
x \\ 
y \\ 
z
\end{array}
\right] =\,\left[ 
\begin{array}{c}
\frac{1}{2}x+\frac{1}{2}y\vspace{4pt} \\ 
\frac{1}{2}x+\frac{1}{2}y\vspace{4pt} \\ 
z
\end{array}
\right] =\left( \dfrac{x+y}{2}-z\right) \,\left[ 
\begin{array}{c}
1 \\ 
1 \\ 
0
\end{array}
\right] +z\left[ 
\begin{array}{c}
1 \\ 
1 \\ 
1
\end{array}
\right] $
\end{itemize}

To find a nonzero vector $b$ that is projected to zero, leave the insertion
point in the matrix $P$, and from the \textsf{Matrices} submenu choose 
\textsf{Nullspace Basis}.

\begin{itemize}
\item  $\,\left[ 
\begin{array}{ccc}
\frac{1}{2} & \frac{1}{2} & 0 \\ 
\frac{1}{2} & \stackrel{}{\frac{1}{2}} & 0 \\ 
0 & 0 & 1
\end{array}
\right] $, nullspace basis:$\left\{ \left[ 
\begin{array}{r}
-1 \\ 
1 \\ 
0
\end{array}
\right] \right\} $
\end{itemize}

\item[2.]  \label{a2}The matrix $\left[ 
\begin{array}{rrrr}
2 & 0 & 0 & 0 \\ 
1 & 2 & 0 & 0 \\ 
0 & 0 & 2 & 0 \\ 
0 & \;0 & -3 & \;2
\end{array}
\right] $ has characteristic polynomial $\left( X-2\right) ^{4}$, minimum
polynomial $4-4X+X^{2}=\,\left( X-2\right) ^{2}$, and eigenvalue and
eigenvectors \hfill \hyperref{\TCIIcon{BITMAPSETProbSolvHint}{0.2006in}{%
0.243in}{0in}}{}{}{q2} 
\[
\left\{ 2,4,\left[ 
\begin{array}{c}
0 \\ 
0 \\ 
0 \\ 
1
\end{array}
\right] ,\left[ 
\begin{array}{c}
0 \\ 
1 \\ 
0 \\ 
0
\end{array}
\right] \right\} 
\]
The minimal polynomial is a factor of the characteristic polynomial. The
eigenvalue $2$ occurs with multiplicity $4$ as a root of the characteristic
polynomial $\left( X-2\right) ^{4}$. The eigenvalue $2$ has two linearly
independent eigenvectors. Note that

\begin{itemize}
\item  $\left[ 
\begin{array}{cccc}
2 & 0 & 0 & 0 \\ 
1 & 2 & 0 & 0 \\ 
0 & 0 & 2 & 0 \\ 
0 & 0 & -3 & 2
\end{array}
\right] \left[ 
\begin{array}{c}
0 \\ 
0 \\ 
0 \\ 
1
\end{array}
\right] =\,\left[ 
\begin{array}{c}
0 \\ 
0 \\ 
0 \\ 
2
\end{array}
\right] =2\left[ 
\begin{array}{c}
0 \\ 
0 \\ 
0 \\ 
1
\end{array}
\right] $
\end{itemize}

$\;$

\begin{itemize}
\item  $\left[ 
\begin{array}{cccc}
2 & 0 & 0 & 0 \\ 
1 & 2 & 0 & 0 \\ 
0 & 0 & 2 & 0 \\ 
0 & 0 & -3 & 2
\end{array}
\right] \left[ 
\begin{array}{c}
0 \\ 
1 \\ 
0 \\ 
0
\end{array}
\right] =\,\left[ 
\begin{array}{c}
0 \\ 
2 \\ 
0 \\ 
0
\end{array}
\right] =2\left[ 
\begin{array}{c}
0 \\ 
1 \\ 
0 \\ 
0
\end{array}
\right] $
\end{itemize}

\item[3.]  \label{a3}The solutions of this equation are in $\Bbb{R}^{3}$,
and the column space of $A$ is a subset of $\Bbb{R}^{2}$, so these solutions
cannot be the column space of $A$. They do form the nullspace of $A$ by the
definition of nullspace; consequently, this set is a subspace of $\Bbb{R}%
^{3} $. The product of $A$ with the first row of $A$ is $\left[ 
\begin{array}{ccc}
1 & 1 & 1 \\ 
1 & 0 & 2
\end{array}
\right] \left[ 
\begin{array}{c}
1 \\ 
1 \\ 
1
\end{array}
\right] =\,\left[ 
\begin{array}{c}
3 \\ 
3
\end{array}
\right] $, which is not $\left[ 
\begin{array}{c}
0 \\ 
0
\end{array}
\right] $, so the solution set is not the row space of $A$.\hfill \hyperref{%
\TCIIcon{BITMAPSETProbSolvHint}{0.2006in}{0.243in}{0in}}{}{}{q3}

To determine whether this subspace of $\Bbb{R}^{3}$ is a point, line, or
plane, solve the system of equations: from the \textsf{Solve} submenu,
choose \textsf{Exact} 
\[
\left[ 
\begin{array}{ccc}
1 & 1 & 1 \\ 
1 & 0 & 2
\end{array}
\right] \left[ 
\begin{array}{c}
x_{1} \\ 
x_{2} \\ 
x_{3}
\end{array}
\right] =\left[ 
\begin{array}{c}
0 \\ 
0
\end{array}
\right] \text{, Solution is: }\left[ 
\begin{array}{r}
-2t_{1} \\ 
t_{1} \\ 
t_{1}
\end{array}
\right] 
\]

The subspace is a line. It is the line that passes through the origin and
the point $\left[ 
\begin{array}{ccc}
-2 & 1 & 1
\end{array}
\right] $.

\item[4.]  \label{a4}Apply the following operations in turn.\hfill \hyperref{%
\TCIIcon{BITMAPSETProbSolvHint}{0.2006in}{0.243in}{0in}}{}{}{q4}

\begin{itemize}
\item  \textsf{New Definition}: $A\left( \theta \right) =\left[ 
\begin{array}{rr}
\cos \theta & -\sin \theta \\ 
\sin \theta & \cos \theta
\end{array}
\right] $

\item  \textsf{Evaluate, Evaluate}: 
\begin{eqnarray*}
A\left( \theta \right) A\left( \varphi \right) &=&\,\left[ 
\begin{array}{rr}
\cos \theta & -\sin \theta \\ 
\sin \theta & \cos \theta
\end{array}
\right] \left[ 
\begin{array}{rr}
\cos \varphi & -\sin \varphi \\ 
\sin \varphi & \cos \varphi
\end{array}
\right] \\
&& \\
&=&\left[ 
\begin{array}{rr}
\cos \theta \cos \varphi -\sin \theta \sin \varphi & -\cos \theta \sin
\varphi -\sin \theta \cos \varphi \\ 
\sin \theta \cos \varphi +\cos \theta \sin \varphi & \cos \theta \cos
\varphi -\sin \theta \sin \varphi
\end{array}
\right]
\end{eqnarray*}
$\,$

\item  Leave the insertion point in the matrix, and from the \textsf{Combine}
submenu choose \textsf{Trig Functions}. 
\[
\begin{array}{c}
\left[ 
\begin{array}{rr}
\cos \theta \cos \varphi -\sin \theta \sin \varphi & -\cos \theta \sin
\varphi -\sin \theta \cos \varphi \\ 
\sin \theta \cos \varphi +\cos \theta \sin \varphi & \cos \theta \cos
\varphi -\sin \theta \sin \varphi
\end{array}
\right] = \\ 
\\ 
\left[ 
\begin{array}{rr}
\cos \left( \theta +\varphi \right) & -\sin \left( \theta +\varphi \right)
\\ 
\sin \left( \theta +\varphi \right) & \cos \left( \theta +\varphi \right)
\end{array}
\right]
\end{array}
\]
\end{itemize}

This result proves the first identity. For the second part, carry out the
following steps.

\begin{itemize}
\item  \textsf{Evaluate} 
\[
A\left( \theta \right) A\left( -\theta \right) =\,\left[ 
\begin{array}{rr}
\cos \theta & -\sin \theta \\ 
\sin \theta & \cos \theta
\end{array}
\right] \left[ 
\begin{array}{rr}
\cos \theta & \sin \theta \\ 
-\sin \theta & \cos \theta
\end{array}
\right] 
\]

\item  \textsf{Evaluate} 
\[
\begin{array}{l}
\left[ 
\begin{array}{rr}
\cos \theta & -\sin \theta \\ 
\sin \theta & \cos \theta
\end{array}
\right] \left[ 
\begin{array}{rr}
\cos \theta & \sin \theta \\ 
-\sin \theta & \cos \theta
\end{array}
\right] = \\ 
\\ 
\qquad \qquad \,\left[ 
\begin{array}{cc}
\cos ^{2}\theta +\sin ^{2}\theta & 0 \\ 
0 & \cos ^{2}\theta +\sin ^{2}\theta
\end{array}
\right]
\end{array}
\]

\item  Leave the insertion point in the matrix and choose \textsf{Simplify},
or from the \textsf{Combine} submenu choose \textsf{Trig Functions}. 
\[
\left[ 
\begin{array}{cc}
\cos ^{2}\theta +\sin ^{2}\theta & 0 \\ 
0 & \cos ^{2}\theta +\sin ^{2}\theta
\end{array}
\right] =\left[ 
\begin{array}{cc}
1 & 0 \\ 
0 & 1
\end{array}
\right] 
\]
\medskip
\end{itemize}
\end{enumerate}

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Working With Matrices}{}{}{DM8-1.tex#Matrix Algebra}

\item  \hyperref{Standard Operations}{}{}{DM8-2.tex#Standard matrix
operations}

\item  \hyperref{Row Operations and Echelon Forms}{}{}{DM8-3.tex#Matrices,
row operations}

\item  \hyperref{Systems of Linear Equations}{}{}{DM8-4.tex#Systems of
equations}

\item  \hyperref{Matrix Equations}{}{}{DM8-4.tex#Matrix equations}

\item  \hyperref{Matrix Operators}{}{}{DM8-5.tex#Trace}

\item  \hyperref{Polynomials and Vectors Associated with a Matrix}{}{}{%
DM8-6.tex#Characteristic, polynomial}

\item  \hyperref{Vector Spaces Associated with a Matrix}{}{}{DM8-7.tex#Row
space}

\item  \hyperref{Special Forms of Matrices}{}{}{DM8-8.tex#Smith normal form}
\end{itemize}

\rule{0.67in}{0.01in}

\hyperref{\FRAME{itbpF}{0.5275in}{0.3399in}{0.0519in}{}{}{index.wmf}{\special%
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DMIndex.tex#Math Index}\quad {\small Index entries: \hyperref{Matrices}{}{}{%
DMIndex.tex#Matrices}, \hyperref{Vector}{}{}{DMIndex.tex#Vector}, \hyperref{%
Vector spaces}{}{}{DMIndex.tex#Vector spaces}}

\end{document}
