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 3.6 Solving General \(\A\x=\b:\) Homework
Exercises Exercises
Exercise Group.
Find the solution set to \(\A\x=\b\) for each of the following, citing any theorems you use.
For any system having exactly one solution, write the linear combination of the columns of \(\A\) which equals \(\b.\)
For any system having more than one solution, choose different values of the parameters in the element form of the solution set to find three distinct linear combinations of the columns of \(\A\) which are equal to \(\b.\)
1. \(\A=\left(\begin{array}{rrr}-1\amp 3\amp 0\\6\amp 2\amp 7 \end{array} \right),\quad\b=\left(\begin{array}{r}1\\3 \end{array} \right)\)
2. \(\A=\left(\begin{array}{rrrr}1\amp -2\amp 4\amp -2\\1\amp 1\amp -3\amp 5\\0\amp 3\amp -7\amp 7 \end{array} \right),\quad\b=\left(\begin{array}{r}-4\\5\\9 \end{array} \right)\)
3. \(\A=\left(\begin{array}{rrr}1\amp 10\amp -1\\2\amp 3\amp 5\\-4\amp 11\amp 7 \end{array} \right),\quad\b=\left(\begin{array}{r}-11\\-30\\-76 \end{array} \right)\)
4. \(\A=\left(\begin{array}{rrrr}1\amp 10\amp -1\amp -3\\2\amp 20\amp 4\amp -3\\1\amp 10\amp 11\amp -6 \end{array} \right),\quad\b=\left(\begin{array}{r}5\\7\\16 \end{array} \right)\)
5. \(\A=\left(\begin{array}{rr}1\amp -2\\1\amp 5\\-4\amp 6 \end{array} \right),\quad\b=\left(\begin{array}{r}7\\-7\\19 \end{array} \right)\)
6. \(\A=\left(\begin{array}{rrrr}1\amp 10\amp -1\amp -3\\2\amp 20\amp 4\amp -3\\1\amp 10\amp 11\amp -6 \end{array} \right),\quad\b=\left(\begin{array}{r}5\\7\\17 \end{array} \right)\)
7. \(\A=\left(\begin{array}{rrr}1\amp 4\amp -3\\2\amp -3\amp 5\\3\amp -5\amp 8\\1\amp 0\amp 3 \end{array} \right),\quad\b=\left(\begin{array}{r}12\\-9\\-15\\5 \end{array} \right)\)
8. \(\A=\left(\begin{array}{r}3\\1\\5\\15 \end{array} \right),\quad\b=\left(\begin{array}{r}2\\1\\5\\15 \end{array} \right)\)
9. \(\A=\left(\begin{array}{rrrr}2\amp 15\amp 6\amp 10 \end{array} \right),\quad\b=(30)=30\)
10. \(\A=\left(\begin{array}{rrrr}3\amp 1\amp 5\amp 15 \end{array} \right),\quad\b=(30)=30\)
11. \(\A=\left(\begin{array}{rrr}9\amp 3\amp 1\\4\amp 2\amp 1\\5\amp 1\amp 1 \end{array} \right),\quad\b=\left(\begin{array}{r}3\\5\\1 \end{array} \right)\)
12. \(\A=\left(\begin{array}{rrr}1\amp 2\amp -1\\1\amp 3\amp 1\\3\amp 8\amp 3 \end{array} \right),\quad\b=\left(\begin{array}{r}4\\-3\\-2 \end{array} \right)\)
13. \(\A=\left(\begin{array}{rrr}1\amp 2\amp -1\\1\amp 3\amp 1\\3\amp 8\amp 3 \end{array} \right),\quad\b=\left(\begin{array}{r}4\\-3\\-1 \end{array} \right)\)
14. 15.
