#LyX 1.4.3 created this file. For more info see http://www.lyx.org/ \lyxformat 245 \begin_document \begin_header \textclass article \begin_preamble %% % \setlength{\topmargin}{-.8in} \setlength{\headheight}{0in} \setlength{\headsep}{0in} \setlength{\footskip}{0in} \setlength{\evensidemargin}{-.5in} \setlength{\oddsidemargin}{-.5in} \setlength{\textheight}{9.3in} \setlength{\textwidth}{7in} \setlength{\parindent}{0in} % \newcommand{\rank}{\operatorname{rank}} \newcommand{\dimn}{\operatorname{dim}} \newcommand{\spann}{\operatorname{span}} \newcommand{\vc}[1]{\mathbf{#1}} \end_preamble \language english \inputencoding auto \fontscheme default \graphics default \paperfontsize 12 \papersize default \use_geometry false \use_amsmath 2 \cite_engine basic \use_bibtopic false \paperorientation portrait \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \papercolumns 1 \papersides 1 \paperpagestyle empty \tracking_changes false \output_changes true \end_header \begin_body \begin_layout Standard \series bold Exam\InsetSpace ~ 3 \hfill Math\InsetSpace ~ 314 \hfill Sample \newline Name: \series default \begin_inset ERT status collapsed \begin_layout Standard \backslash rule{2.4in}{.01in} \end_layout \end_inset \hfill \series bold Score: \series default \begin_inset ERT status collapsed \begin_layout Standard \backslash rule{1in}{.01in} \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash hspace{.25in} \end_layout \end_inset \shape italic Instructions: \begin_inset ERT status collapsed \begin_layout Standard \backslash /{} \end_layout \end_inset \shape default Show your work in the spaces provided below for full credit. Use the reverse side for additional space, \shape italic but clearly so indicate. \begin_inset ERT status collapsed \begin_layout Standard \backslash /{} \end_layout \end_inset \shape default You must clearly identify answers and show supporting work to receive any credit. Exact answers (e.g., \begin_inset Formula $\pi$ \end_inset ) are preferred to inexact (e.g., \begin_inset Formula $3.14$ \end_inset ). Point values of problems are given in parentheses. Notes or text in \emph on any \emph default form not allowed. Calculator is allowed. \newline \begin_inset ERT status collapsed \begin_layout Standard \backslash rule[.3in]{7in}{.01in} \end_layout \end_inset (27) \series bold 1. \series default Let \begin_inset Formula $\mathbf{v}_{1}=\left(-1,-1,1,1\right)$ \end_inset , \begin_inset Formula $\mathbf{v}_{2}=\left(1,1,1,1\right)$ \end_inset , and \begin_inset Formula $\mathbf{v}_{3}=\left(-1,1,-1,1\right)$ \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{0.1in} \end_layout \end_inset (a) Show that \begin_inset Formula $\mathbf{v}_{1}$ \end_inset , \begin_inset Formula $\mathbf{v}_{2}$ \end_inset , \begin_inset Formula $\mathbf{v}_{3}$ \end_inset is an orthogonal set of vectors and convert it to an orthonormal set \begin_inset Formula $\mathbf{u}_{1}$ \end_inset , \begin_inset Formula $\mathbf{u}_{2}$ \end_inset , \begin_inset Formula $\mathbf{u}_{3}$ \end_inset by normalizing each vector. \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{0.1in} \end_layout \end_inset (b) Use the orthogonal coordinates theorem to determine if \begin_inset Formula $\left(4,5,0,1\right)$ \end_inset belongs to \begin_inset Formula $\spann\left\{ \mathbf{u}_{1},\mathbf{u}_{2},\mathbf{u}_{3}\right\} $ \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.1in} \end_layout \end_inset (c) Set up and solve the normal equations for the system \begin_inset Formula $A\mathbf{x}=\mathbf{b}$ \end_inset , where \begin_inset Formula $A=\left[\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\right]$ \end_inset and \begin_inset Formula $\mathbf{b}=\left(-1,0,0,1\right)$ \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.2in} \end_layout \end_inset (15) \series bold 2. \series default Let \begin_inset Formula $A=\left[\begin{array}{cc} -4 & 6\\ -3 & 5\end{array}\right].$ \end_inset Then the eigenvalues of \begin_inset Formula $A$ \end_inset are \begin_inset Formula $-1,2$ \end_inset and two eigenvectors are \begin_inset Formula $(1,1)$ \end_inset and \begin_inset Formula $(2,1).$ \end_inset Assume this information and use it to derive a formula for the powers of \begin_inset Formula $A$ \end_inset in terms of the eigenvalues of \begin_inset Formula $A.$ \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.2in} \end_layout \end_inset (20) \series bold 3. \series default (a) Find all eigenvalues and eigenvectors of the matrix \begin_inset Formula $A=\left[\begin{array}{ccc} 2 & 1 & 1\\ 0 & 3 & 1\\ 0 & 0 & 2\end{array}\right].$ \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.1in} \end_layout \end_inset (b) Is the matrix \begin_inset Formula $A$ \end_inset diagonalizable? If not, give reasons, otherwise compute the matrix \begin_inset Formula $P$ \end_inset that diagonalizes \begin_inset Formula $A$ \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{0.2in} \end_layout \end_inset (18) \series bold 4. \series default Fill in the blanks or answer T/F: \end_layout \begin_layout Standard (a) Every real matrix is diagonalizable (T/F) \begin_inset ERT status collapsed \begin_layout Standard \backslash rule{1in}{.01in} \end_layout \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.1in} \end_layout \end_inset (b) Every orthonormal set of vectors is linearly independent (T/F) \begin_inset ERT status collapsed \begin_layout Standard \backslash rule{1in}{.01in} \end_layout \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.1in} \end_layout \end_inset (c) Eigenvalues of a matrix cannot be zero (T/F) \begin_inset ERT status collapsed \begin_layout Standard \backslash rule{1in}{.01in} \end_layout \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.1in} \end_layout \end_inset (d) The CBS inequality says that for vectors \begin_inset Formula $\mathbf{u}$ \end_inset and \begin_inset Formula $\mathbf{v}$ \end_inset , we have that \begin_inset ERT status collapsed \begin_layout Standard \backslash rule{2in}{.01in} \end_layout \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.1in} \end_layout \end_inset (e) Every eigenvalue of a Hermitian matrix is real (T/F) \begin_inset ERT status collapsed \begin_layout Standard \backslash rule{1in}{.01in} \end_layout \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.1in} \end_layout \end_inset (f) The projection of \begin_inset Formula $\mathbf{u}=(1,2,0,1)$ \end_inset along the vector \begin_inset Formula $\mathbf{v}=(1,1,1,0)$ \end_inset is \begin_inset Formula $\operatorname{proj}_{\mathbf{v}}\left(\mathbf{u}\right)=$ \end_inset \begin_inset ERT status collapsed \begin_layout Standard \backslash rule{1.3in}{.01in} \end_layout \end_inset and the orthogonal projection of \begin_inset Formula $\mathbf{u}$ \end_inset to \begin_inset Formula $\mathbf{v}$ \end_inset is \begin_inset Formula $\operatorname{orth}_{\mathbf{v}}\left(\mathbf{u}\right)=$ \end_inset \begin_inset ERT status collapsed \begin_layout Standard \backslash rule{1.3in}{.01in} \end_layout \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.1in} \end_layout \end_inset (g) The component of \begin_inset Formula $\mathbf{u}=(-3,2,1,0)$ \end_inset along the vector \begin_inset Formula $\mathbf{v}=(0,2,-1,1)$ \end_inset is \begin_inset Formula $\operatorname{comp}_{\mathbf{v}}\left(\mathbf{u}\right)=$ \end_inset \begin_inset ERT status collapsed \begin_layout Standard \backslash rule{1.3in}{.01in} \end_layout \end_inset and the cosine of an angle between \begin_inset Formula $\mathbf{u}$ \end_inset and \begin_inset Formula $\mathbf{v}$ \end_inset is \begin_inset ERT status collapsed \begin_layout Standard \backslash rule{1.3in}{.01in} \end_layout \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.2in} \end_layout \end_inset (20) \series bold 5. \series default Give brief answers to the following questions. \end_layout \begin_layout Standard (a) The matrix \begin_inset Formula $A=\left[\begin{array}{cc} 0 & i\\ -i & 0\end{array}\right]$ \end_inset has eigenvectors \begin_inset Formula $(i,1)$ \end_inset and \begin_inset Formula $(1,i).$ \end_inset Assume this and unitarily diagonalize \begin_inset Formula $A,$ \end_inset that is, produce the appropriate matrices \begin_inset Formula $U$ \end_inset and \begin_inset Formula $D$ \end_inset such that \begin_inset Formula $U^{*}AU=D$ \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.1in} \end_layout \end_inset (b) Suppose that the diagonalizable matrix \begin_inset Formula $A$ \end_inset is the transition matrix of a discrete dynamical system and that the spectral radius (largest eigenvalue in absolute value) of \begin_inset Formula $A$ \end_inset is \begin_inset Formula $1.$ \end_inset What can you say about the states \begin_inset Formula $\mathbf{x}^{(k)}=A^{k}\mathbf{x}^{(0)}?$ \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.1in} \end_layout \end_inset (c) Define what an eigenvector and eigenvalue of the matrix \begin_inset Formula $A$ \end_inset is. Use this to show that if \begin_inset Formula $\lambda$ \end_inset is an eigenvalue of \begin_inset Formula $A$ \end_inset , then \begin_inset Formula $\lambda^{2}$ \end_inset is an eigenvalue of \begin_inset Formula $A^{2}$ \end_inset . \end_layout \end_body \end_document