Skip to main content Contents Prev Up Next \(\usepackage{amsmath,mathabx}
\usepackage{mathpazo}
\normalfont
\def\bp{}
\def\ep{}
\def\R{{\mathbb R}}
\def\Pos{{\mathbb P}}
\def\Q{{\mathbb Q}}
\def\Ir{{\mathbb I}}
\def\N{{\mathbb N}}
\def\Z{{\mathbb Z}}
\def\T{{\mathbb T}}
\def\H{{\mathbb H}}
\def\mb{{\mathcal m}}
\def\Mb{{\mathcal M}}
\def\F{{\mathcal F}}
\def\P{{\mathcal P}}
\def\Qp{{\mathcal Q}}
\def\a{{\alpha}}
\def\d{{\delta}}
\def\eps{{\epsilon}}
\def\e{\varepsilon}
\def\exp{\text{exp }}
\def\g{{\gamma}}
\def\G{{\Gamma}}
\def\k{{\kappa}}
\def\p{{\psi}}
\def\t{{\tau}}
\def\l{{\ell}}
\def\la{{\lambda}}
\def\La{{\Lambda}}
\def\m{{\mu}}
\def\n{{\nu}}
\def\O{{\Omega}}
\def\r{{\rho}}
\def\th{{\theta}}
\def\s{{\sigma}}
\def\ze{{\zeta}}
\def\ph{{\phi}}
\def\Ph{{\Phi}}
\def\Ps{{\Psi}}
\def\U{{\boldsymbol{U}}}
\def\A{{\boldsymbol{A}}}
\def\B{{\boldsymbol{B}}}
\def\C{{\boldsymbol{C}}}
\def\D{{\boldsymbol{D}}}
\def\E{{\boldsymbol{E}}}
\def\J{{\boldsymbol{J}}}
\def\I{{\boldsymbol{I}}}
\def\L{{\boldsymbol{L}}}
\def\Lam{{\boldsymbol{\Lambda}}}
\def\M{{\boldsymbol{M}}}
\def\P{{\boldsymbol{P}}}
\def\X{{\boldsymbol{X}}}
\def\lmatrix{\left(\begin{array}}
\def\rmatrix{\end{array}\right)}
\def\ldmatrix{\left|\begin{array}}
\def\rdmatrix{\end{array}\right|}
\def\bcm{}
\def\pmkn{{(-1)}^{k+n}}
\def\np{{}}
\def\noi{{}}
\def\lmatrix{\left(\begin{array}}
\def\rmatrix{\end{array}\right)}
\def\ldmatrix{\left|\begin{array}}
\def\rdmatrix{\end{array}\right|}
\def\ninN{{n{\in}\N}}
\def\pcp2{\textcolor{forestgreen}\textbf{(CP)}}
\def\pw2{\textcolor{magenta}\textbf{(CW)}}
\newcommand{\chf}[1]{\chi_{{ }_{#1}}(x)}
\def\ok{\textcolor{goodgreen}\textbf{(OK) }}
\def\toinf{{\rightarrow \infty }}
\def\tozero{{\rightarrow 0 }}
\def\supp{{supp}}
\def\Lsp{{\boldsymbol L}}
\def\LtR{{\Lsp^2(\R)}}
\def\Ra{{\,\,\Rightarrow\,\,}}
\def\ra{{\,\rightarrow\,}}
\def\Lfa{{\,\,\Leftarrow\,\,}}
\def\LfRa{{\,\,\Leftrightarrow\,\,}}
\def\lra{{\,\,\longrightarrow\,\,}}
\def\Lra{{\,\,\Longrightarrow\,\,}}
\def\LLa{{\,\,\Longleftarrow\,\,}}
\def\LRa{{\,\,\Longleftrightarrow\,\,}}
\def\fa{{\,\,\forall\,\,}}
\def\ex{{\,\,\exists\,\,}}
\def\st{{\,\,\ni\,\,}}
\def\contr{{(\Rightarrow\,!\,\Leftarrow)}}
\def\ntoinf{{\underset{n\rightarrow\infty}{\lim}\,}}
\def\mtoinf{{\underset{m\rightarrow\infty}{\lim}\,\,}}
\def\and{{\,\,\wedge\,\,}}
\def\oor{{\,\,\vee\,\,}}
\def\blt{$\bullet$}
\def\tcb{\textcolor{blue}}
\newcommand{\ve}[1]{\overrightarrow{\boldsymbol#1}}
\newcommand{\vec}[1]{\overset{\rightharpoonup}{\boldsymbol#1}}
\def\b{\vec{b}}
\def\c{\vec{c}}
\def\d{\vec{d}}
\def\x{\vec{x}}
\def\y{\vec{y}}
\def\z{\vec{z}}
\def\u{\vec{u}}
\def\v{\vec{v}}
\def\w{\vec{w}}
def\z{\vec{z}}
\newcommand{\ip}[2]{{\left\langle #1,#2\right\rangle}}
\newcommand{\dpr}[2]{{#1\,{\boldsymbol \cdot}\,#2}}
\newcommand{\row}[2]{{#1}_{#2\ast}}
\newcommand{\col}[2]{{#1}_{\ast#2}}
\newcommand{\wh}{\widehat}
\newcommand{\wt}{\widetilde}
\def\lmatrix{\left(\begin{array}}
\def\rmatrix{\end{array}\right)}
\def\0matrix{\mathcal{O}}
\newcommand{\unit}[1]{\wh{\boldsymbol{#1}}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 3.14 The LU-Factorization: In-Class Practice
Worksheet 3.14.1 Worksheet
Exercise Group.
Remark 3.13.14 gives us an easy way to compute
\(\E_{(a)21}\E_{(b)31}\E_{(c)32}\in\R^{3 \times 3}.\) Here we will investigate the extents and limitations of that shortcut. Compute each of the following (break up the work amongst your group members). Which products follow the rule described in
Remark 3.13.14 ? Do you see a pattern?
1. \(\E_{(a)21}\E_{(b)31}\E_{(c)32}\) 2. \(\E_{(b)31}\E_{(a)21}\E_{(c)32}\) 3. \(\E_{(b)31}\E_{(c)32}\E_{(a)21}\) 4. \(\E_{(a)21}\E_{(c)32}\E_{(b)31}\) 5. \(\E_{(a)21}\E_{(c)32}\E_{(b)31}\) 6. \(\E_{(c)32}\E_{(a)21}\E_{(b)31}\)
Exercise Group. For each matrix
\(\A\) in the
\(\L\U\) -factorization of
\(\A\) by 1) using
Theorem 3.13.9 and 2) then
Algorithm 3.13.13 , then factor
\(\A\) as
\(\A=\L\D\widetilde{\U}.\)
7. \(\A=\left(\begin{array}{rrr}5\amp 1\amp -2\\-30\amp 0\amp 10\\15\amp -21\amp 11 \end{array} \right)\) 8. \(\A=\left(\begin{array}{rrr}2\amp 2\amp -3\\8\amp 3\amp -15\\-6\amp -31\amp 2\end{array} \right).\) 9. \(\A=\left(\begin{array}{rrr}4\amp 2\amp -6\\-4\amp 5\amp 20\\8\amp -52\amp -103\end{array} \right).\) 10. \(\A=\left(\begin{array}{rrr}4\amp 3\amp -4\\12\amp 7\amp -5\\20\amp 19\amp -34\end{array} \right).\)
11.
