#LyX 1.4.2 created this file. For more info see http://www.lyx.org/ \lyxformat 245 \begin_document \begin_header \textclass article \begin_preamble \setlength{\topmargin}{-0.7in} \setlength{\headheight}{0in} \setlength{\headsep}{0in} \setlength{\footskip}{0in} \setlength{\evensidemargin}{-.5in} \setlength{\oddsidemargin}{-.5in} \setlength{\textheight}{9.3in} \setlength{\textwidth}{7.5in} \setlength{\parindent}{0in} \newcommand{\D}{\displaystyle} \newcommand{\I}{\mathrm{i}} \newcommand{\rnk}{\operatorname{rank}} \newcommand{\E}{\mathrm{e}} \end_preamble \language english \inputencoding auto \fontscheme default \graphics default \paperfontsize 11 \spacing single \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 ~ 1 \hfill Math\InsetSpace ~ 314 \hfill Fall\InsetSpace ~ 2006 \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. The only electronic equipment allowed is a calculator. \newline \begin_inset ERT status collapsed \begin_layout Standard \backslash rule[.2in]{7.5in}{.01in} \end_layout \end_inset \end_layout \begin_layout Standard (24) \series bold 1. \series default Consider the linear system given by the following: \begin_inset Formula \begin{eqnarray*} x_{1}+x_{2}+x_{3}-x_{4} & = & 2\\ 2x_{1}+x_{2}-2x_{4} & = & 1\\ 2x_{1}+2x_{2}+2x_{3}-2x_{4} & = & 4\end{eqnarray*} \end_inset \end_layout \begin_layout Standard (a) (12) Use Gauss-Jordan elimination to find the general solution to this system. Clearly specify the elementary row operations you use. \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{2.5in} \end_layout \end_inset (b) (4) If we write the system as \begin_inset Formula $A\mathbf{x}=\mathbf{b}$ \end_inset , what are the coefficient matrix \begin_inset Formula $A$ \end_inset and right-hand-side vector \begin_inset Formula $\mathbf{b}$ \end_inset ? What are the rank and nullity of \begin_inset Formula $A$ \end_inset ? \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{1.4in} \end_layout \end_inset (c) (3) Express the reduced row echelon form \begin_inset Formula $R$ \end_inset of the augmented matrix \begin_inset Formula $\widetilde{A}$ \end_inset of this system as product of elementary matrices and \begin_inset Formula $\widetilde{A}$ \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{1.1in} \end_layout \end_inset (d) (5) Apply the row operations used in part (a) in the same order as in (a) to a general right hand side vector \begin_inset Formula $\mathbf{b}=(b_{1},b_{2},b_{3})$ \end_inset . What is the resulting vector? \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash clearpage \end_layout \begin_layout Standard \end_layout \end_inset (16) \series bold 2. \series default Let \begin_inset Formula $A=\left[\begin{array}{rrr} 1 & -2 & 1\\ 0 & 2 & 0\\ -1 & 0 & 1\end{array}\right]$ \end_inset . Find the inverse of \begin_inset Formula $A$ \end_inset and use it to solve \begin_inset Formula $A\mathbf{x}=\mathbf{b}$ \end_inset with \begin_inset Formula $\mathbf{b}=(2,-4,8)$ \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{4.8in} \end_layout \end_inset (12) \series bold 3. \series default Solve the following systems for the (complex) variable \begin_inset Formula $z$ \end_inset . Express your answere in standard form ( \begin_inset Formula $z=x+\I y$ \end_inset ) where possible. \end_layout \begin_layout Standard (a) \begin_inset Formula $z=\E^{\I\pi}+2\I$ \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{1.5in} \end_layout \end_inset \end_layout \begin_layout Standard (b) \begin_inset Formula $\left(2+\I\right)z=1$ \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{1.5in} \end_layout \end_inset (c) \begin_inset Formula $z^{3}=1$ \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{2.5in} \end_layout \end_inset (20) \series bold 4. \series default Carry out these calculations or indicate they are impossible. You are given that \begin_inset Formula $\mathbf{x}=\left[\begin{array}{r} 2\\ 1\end{array}\right]$ \end_inset , \begin_inset Formula $\mathbf{y}=\left[\begin{array}{rr} 3 & 4\end{array}\right]$ \end_inset , \begin_inset Formula $C=\left[\begin{array}{rr} 2 & 1+\I\\ 0 & 1\end{array}\right],$ \end_inset and \begin_inset Formula $D=\left[\begin{array}{rrr} 1 & 0 & 2\\ 0 & 1 & 0\end{array}\right]$ \end_inset . \end_layout \begin_layout Standard (a) \begin_inset Formula $\mathbf{y}C\mathbf{x}$ \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{1.3in} \end_layout \end_inset (b) \begin_inset Formula $\mathbf{xy}$ \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{1.3in} \end_layout \end_inset (c) \begin_inset Formula $\mathbf{x}+2\mathbf{x}^{T}$ \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{1.3in} \end_layout \end_inset (d) \begin_inset Formula $D^{*}$ \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{1.3in} \end_layout \end_inset (e) \begin_inset Formula $C^{-1}$ \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{1.3in} \end_layout \end_inset (f) \begin_inset Formula $CD$ \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash clearpage \end_layout \begin_layout Standard \end_layout \end_inset (18) \series bold 5. \series default Fill in the blanks or answer True/False. Parts (a) and (b) are worth 3 points and remaining parts are worth 2 points. \end_layout \begin_layout Standard (a) If \begin_inset Formula $A$ \end_inset is a \begin_inset Formula $2\times2$ \end_inset nonzero matrix and the system \begin_inset Formula $A\mathbf{x}=\mathbf{b}$ \end_inset has infinitely many solutions for some \begin_inset Formula $\mathbf{b}$ \end_inset then \end_layout \begin_layout Standard \begin_inset Formula $\rnk A=$ \end_inset \begin_inset ERT status collapsed \begin_layout Standard \backslash rule{1.0in}{.01in} \end_layout \end_inset and \begin_inset Formula $A$ \end_inset \begin_inset ERT status collapsed \begin_layout Standard \backslash rule{1.0in}{.01in} \end_layout \end_inset invertible. (Fill in \begin_inset Quotes eld \end_inset is \begin_inset Quotes erd \end_inset or \begin_inset Quotes eld \end_inset is not \begin_inset Quotes erd \end_inset .) \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.4in} \end_layout \end_inset (b) \begin_inset Formula $T((x,y))=(x+y,2x,4y-x)$ \end_inset is a matrix multiplication function \begin_inset Formula $T_{A}((x,y)),$ \end_inset where \begin_inset Formula $A=$ \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.5in} \end_layout \end_inset (c) If a Markov chain has transition matrix \begin_inset Formula $\left[\begin{array}{rr} \frac{1}{2} & \frac{1}{4}\\ \frac{1}{2} & \frac{3}{4}\end{array}\right],$ \end_inset and initial state \begin_inset Formula $x^{(0)}=\left(1,0\right)$ \end_inset , then \end_layout \begin_layout Standard \begin_inset Formula $x^{(2)}=$ \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.5in} \end_layout \end_inset (d) As a matrix--vector product, \begin_inset Formula $x_{1}\left[\begin{array}{c} 1\\ 2\\ 4\end{array}\right]+x_{2}\left[\begin{array}{c} 1\\ 2\\ 6\end{array}\right]+x_{3}\left[\begin{array}{c} 1\\ 5\\ 8\end{array}\right]=$ \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.3in} \end_layout \end_inset (e) If \begin_inset Formula $3\times3$ \end_inset matrix \begin_inset Formula $A$ \end_inset is invertible, then the reduced row echelon form of \begin_inset Formula $A$ \end_inset is: \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.3in} \end_layout \end_inset (f) Any homogeneous (right-hand-side vector \begin_inset Formula $\mathbf{0}$ \end_inset ) linear system is consistent (T/F): \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.3in} \end_layout \end_inset (g) If \begin_inset Formula $A,B$ \end_inset are \begin_inset Formula $2\times2$ \end_inset matrices, then \begin_inset Formula $\left(AB\right)^{2}=A^{2}B^{2}$ \end_inset (T/F): \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.3in} \end_layout \end_inset (h) Every diagonal matrix is symmetric (T/F): \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{.3in} \end_layout \end_inset (10) \series bold 6. \series default Let \begin_inset Formula $A=\left[\begin{array}{rrr} -1 & 0 & -1\\ 0 & 1 & 2\end{array}\right]$ \end_inset and \begin_inset Formula $B=\left[\begin{array}{rrr} 1 & 2 & -1\\ 4 & 1 & 3\end{array}\right]$ \end_inset . \end_layout \begin_layout Standard (a) Verify the commutative law of matrix addition for these two matrices. \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash vspace{1.5in} \end_layout \end_inset (c) (Honors only) Give a proof of this law. \end_layout \end_body \end_document