c8ec07…: “Ahead of the gray ring”

$\newenvironment {prompt}{}{} \newcommand {\DeclareMathOperator }[2]{\@OldDeclareMathOperator {##1}{##2}\immediate \write \myfile {\unexpanded {\DeclareMathOperator }{\unexpanded {##1}}{\unexpanded {##2}}}} \newcommand {\amatrix }[1]{\left [\begin {array}{@{}*{##1}{c}|c@{}} } \newcommand {\RR }[0]{\bf R} \newcommand {\adj }[0]{\mathop {\rm adj}} \newcommand {\degrees }[0]{$^{\circ }$} \newcommand {\Col }[0]{\mathop {\rm Col}} \newcommand {\diag }[0]{\mathop {\rm diag}} \newcommand {\Inn }[0]{\mathop {\rm Inn}} \newcommand {\Nul }[0]{\mathop {\rm Nul}} \newcommand {\rank }[0]{\mathop {\rm rank}} \newcommand {\rk }[0]{\mathop {\rm rk}} \newcommand {\Row }[0]{\mathop {\rm Row}} \newcommand {\proj }[0]{\mathop {\rm proj}} \newcommand {\Span }[0]{\mathop {\rm Span}} \newcommand {\spec }[0]{\mathop {\rm spec}} \newcommand {\tr }[0]{\mathop {\rm tr}} \newcommand {\vol }[0]{\mathop {\rm vol}} \newcommand {\reserved@a }[2]{} \newcommand {\reserved@a }[2]{} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument } \newcommand {\reserved@a }[1]{} \newcommand {\reserved@a }[2]{} \newcommand {\AppendGraphicsExtensions }[0]{\@ifundefined {Gin@extensions}{\let \Gin@extensions \@empty }{}\@ifstar {\grfext@Append \grfext@Check }{\grfext@Append \grfext@@Add }} \newcommand {\PrependGraphicsExtensions }[0]{\@ifundefined {Gin@extensions}{\let \Gin@extensions \@empty }{}\@ifstar {\grfext@Prepend \grfext@Check }{\grfext@Prepend \grfext@@Add }} \newcommand {\RemoveGraphicsExtensions }[1]{\@ifundefined {Gin@extensions}{\def \Gin@extensions {}}{\edef \grfext@tmp {\zap@space ##1 \@empty }\@for \grfext@ext :=\grfext@tmp \do {\def \grfext@next {\let \grfext@tmp \Gin@extensions \@expandtwoargs \@removeelement \grfext@ext \Gin@extensions \Gin@extensions \ifx \grfext@tmp \Gin@extensions \let \grfext@next \relax \fi \grfext@next }\grfext@next }}\grfext@Print \RemoveGraphicsExtensions } \newcommand {\epstopdfsetup }[0]{\setkeys {ETE}} \newcommand {\epstopdfcall }[1]{\ifETE@InsideSetfile \expandafter \@firstoftwo \else \expandafter \@secondoftwo \fi {‘##1}{\Gin@base \Gin@ext }} \newcommand {\epstopdfDeclareGraphicsRule }[4]{\ifx \\##4\\\@PackageError {epstopdf-base}{Conversion command is missing}\@ehc \else \begingroup \@ifundefined {Gin@rule@##1}{}{\@PackageInfo {epstopdf-base}{Redefining graphics rule for ‘##1’}}\endgroup \@namedef {Gin@rule@##1}####1{{##2}{##3}{\epstopdfcall {##4}}}\fi } \newcommand {\mdf@patchamsthm }[0]{\let \mdf@deferred@thm@head \deferred@thm@head \pretocmd {\deferred@thm@head }{\@inlabelfalse }{\mdf@PackageInfo {mdframed detected package amsthm changed the theorem header of amsthm\MessageBreak }}{\mdf@PackageError {mdframed detected package amsthm changed the theorem header of amsthm failed\MessageBreak }}} \newcommand {\GetTitleStringSetup }[0]{\setkeys {gettitlestring}} \newcommand {\GetTitleString }[0]{\ifGTS@expand \expandafter \GetTitleStringExpand \else \expandafter \GetTitleStringNonExpand \fi } \newcommand {\GetTitleStringExpand }[1]{\def \GetTitleStringResult {##1}\begingroup \GTS@DisablePredefinedCmds \GTS@DisableHook \edef \x {\endgroup \noexpand \def \noexpand \GetTitleStringResult {\GetTitleStringResult }}\x } \newcommand {\GetTitleStringNonExpand }[1]{\def \GetTitleStringResult {##1}\global \let \GTS@GlobalString \GetTitleStringResult \begingroup \GTS@RemoveLeft \GTS@RemoveRight \endgroup \let \GetTitleStringResult \GTS@GlobalString } \newcommand {\GTS@DisableHook }[0]{} \newcommand {\label@hook }[0]{} \newcommand {\@currentlabelname }[0]{} \newcommand {\reserved@a }[0]{} \newcommand {\@safe@activestrue }[0]{} \newcommand {\@safe@activesfalse }[0]{} \newcommand {\nameref }[0]{\@ifstar \T@nameref \T@nameref } \newcommand {\Sectionformat }[2]{##1} \newcommand {\vnameref }[1]{\unskip ~\nameref {##1}\@vpageref [\unskip ]{##1}} \newcommand {\ref }[0]{\@ifstar \@refstar \T@ref } \newcommand {\pageref }[0]{\@ifstar \@pagerefstar \T@pageref } \newcommand {\nameref }[0]{\@ifstar \@namerefstar \T@nameref } \newcommand {\reserved@a }[2]{} \newcommand {\reserved@a }[2]{} \newcommand {\reserved@a }[0]{\AtBeginDocument } \newcommand {\reserved@a }[1]{} \newcommand {\reserved@a }[2]{}$

True/False: The set of vectors, $\left \{ \begin {bmatrix} 2\\1\\3\end {bmatrix}, \begin {bmatrix} -1\\1\\-5\end {bmatrix}, \begin {bmatrix} 0\\-4\\3\end {bmatrix} \right \}$ is orthogonal.

True False

Two vectors are orthogonal iff their dot product is zero.

True/False: The set of vectors, $\left \{ \begin {bmatrix} 1\\1\\-2\end {bmatrix}, \begin {bmatrix} 2\\0\\1\end {bmatrix}, \begin {bmatrix} 1\\-5\\-2\end {bmatrix} \right \}$ is orthogonal.

True False

Two vectors are orthogonal if and only if their dot product is zero.

True/False: An orthogonal basis for a subspace $W$ of $\RR ^n$ is a basis for $W$ that is also an orthogonal set.

True False

If $\{\vec {u}_1,\vec {u}_2,\vec {u}_3\}$ is a basis for $\RR ^3$ , then for any $\vec {w}$ in $\RR ^3$ , $\vec {w}$ can be written as a linear combination of $\vec {u}_1,\vec {u}_2,\vec {u}_3$ . Given the vectors $\vec {w},\vec {u}_1,\vec {u}_2$ and $\vec {u}_3$ below, compute the coefficients in the linear combination.

$\vec {w} =\begin {bmatrix} 1\\2\\3\end {bmatrix}, \hspace {10pt} \vec {u}_1 =\begin {bmatrix} 3\\-3\\0\end {bmatrix}, \hspace {10pt} \vec {u_2} =\begin {bmatrix} 2\\2\\-1\end {bmatrix}, \hspace {10pt} \vec {u}_3 =\begin {bmatrix} 1\\1\\4\end {bmatrix}$
$\vec {w} =- \frac {\answer {1}}{\answer {6}}\vec {u}_1 + \frac {\answer {1}}{\answer {3}}\vec {u}_2+ \frac {\answer {5}}{\answer {6}}\vec {u}_3$ . (Simplify fractions as much as possible.)

$c_j = \frac {\vec {w}\cdot \vec {u}_j}{\vec {u}_j\cdot \vec {u}_j}$

Let $\vec {y} = \begin {bmatrix} 7\\2\end {bmatrix}$ and $\vec {u} = \begin {bmatrix} -1\\6\end {bmatrix}$ . Find the orthogonal projection of $\vec {y}$ onto $\vec {u}$ .

First, find $\vec {y}\cdot \vec {u} = \answer {5}$ , then find $\vec {u}\cdot \vec {u} = \answer {37}$ .

Then the orthogonal projection of $\vec {y}$ onto $\vec {u}$ is ${\proj }_L\vec {y} = \begin {bmatrix} -\answer {5}/\answer {37} \\\answer {30}/\answer {37} \end {bmatrix}$ .

Let $L$ be the line through $\vec {u}$ and $\vec {0}$ . The orthogonal projection of $\vec {y}$ onto $\vec {u}$ (or the orthogonal projection of $\vec {y}$ onto $L$ ) is ${\proj }_L\vec {y} = \left (\frac {\vec {y}\cdot \vec {u}}{\vec {u}\cdot \vec {u}}\right )\vec {u}$

True/False: Not every orthogonal set in $\RR ^n$ is linearly independent.

True False

True/False: Every orthogonal set of non-zero vectors in $\RR ^n$ is linearly independent.

True False

What is the definition of an orthonormal set?

An orthogonal set that is linearly independent An orthogonal basis A basis consisting only of unit vectors An orthogonal basis consisting only of unit vectors An orthogonal set consisting only of unit vectors