Essential Vocabulary

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\npnoround }[0]{\nprounddigits {-1}} \newcommand {\npnoroundexp }[0]{\nproundexpdigits {-1}} \newcommand {\npunitcommand }[1]{\ensuremath {\mathrm {#1}}} \newcommand {\tdplotsinandcos }[3]{\pgfmathsetmacro {#1}{sin(#3)}\pgfmathsetmacro {#2}{cos(#3)}} \newcommand {\tdplotmult }[3]{\pgfmathsetmacro {#1}{#2*#3}} \newcommand {\tdplotdiv }[3]{\pgfmathsetmacro {#1}{#2/#3}} \newcommand {\tdplotcheckdiff }[5]{\par \par \pgfmathparse { abs(#2 -#1)<#3 } \par \ifthenelse {\equal {\pgfmathresult }{1}}{#4}{#5} } \newcommand {\tdplotsetmaincoords }[2]{\pgfmathsetmacro {\tdplotmaintheta }{#1} \pgfmathsetmacro {\tdplotmainphi }{#2} \tdplotcalctransformmainscreen \tikzset {tdplot_main_coords/.style={x={(\raarot cm,\rbarot cm)},y={(\rabrot cm,\rbbrot cm)},z={(\racrot cm,\rbcrot cm)}}}} \newcommand {\tdplotcalctransformmainscreen }[0]{\tdplotsinandcos {\sintheta }{\costheta }{\tdplotmaintheta }\tdplotsinandcos {\sinphi }{\cosphi }{\tdplotmainphi }\tdplotmult {\stsp }{\sintheta }{\sinphi }\tdplotmult {\stcp }{\sintheta }{\cosphi }\tdplotmult {\ctsp }{\costheta }{\sinphi }\tdplotmult {\ctcp }{\costheta }{\cosphi }\pgfmathsetmacro {\raarot }{\cosphi }\pgfmathsetmacro {\rabrot }{\sinphi }\pgfmathsetmacro {\racrot }{0}\pgfmathsetmacro {\rbarot }{-\ctsp }\pgfmathsetmacro {\rbbrot }{\ctcp }\pgfmathsetmacro {\rbcrot }{\sintheta }\pgfmathsetmacro {\rcarot }{\stsp }\pgfmathsetmacro {\rcbrot }{-\stcp }\pgfmathsetmacro {\rccrot }{\costheta }} \newcommand {\tdplotcalctransformrotmain }[0]{\tdplotsinandcos {\sinalpha }{\cosalpha }{\tdplotalpha } \tdplotsinandcos {\sinbeta }{\cosbeta }{\tdplotbeta } \tdplotsinandcos {\singamma }{\cosgamma }{\tdplotgamma } \tdplotmult {\sasb }{\sinalpha }{\sinbeta } \tdplotmult {\sbsg }{\sinbeta }{\singamma } \tdplotmult {\sasg }{\sinalpha }{\singamma } \tdplotmult {\sasbsg }{\sasb }{\singamma } \tdplotmult {\sacb }{\sinalpha }{\cosbeta } \tdplotmult {\sacg }{\sinalpha }{\cosgamma } \tdplotmult {\sbcg }{\sinbeta }{\cosgamma } \tdplotmult {\sacbsg }{\sacb }{\singamma } \tdplotmult {\sacbcg }{\sacb }{\cosgamma } \tdplotmult {\casb }{\cosalpha }{\sinbeta } \tdplotmult {\cacb }{\cosalpha }{\cosbeta } \tdplotmult {\cacg }{\cosalpha }{\cosgamma } \tdplotmult {\casg }{\cosalpha }{\singamma } \tdplotmult {\cacbsg }{\cacb }{\singamma } \tdplotmult {\cacbcg }{\cacb }{\cosgamma } \pgfmathsetmacro {\raaeul }{\cacbcg -\sasg } \pgfmathsetmacro {\rabeul }{-\cacbsg -\sacg } \pgfmathsetmacro {\raceul }{\casb } \pgfmathsetmacro {\rbaeul }{\sacbcg + \casg } \pgfmathsetmacro {\rbbeul }{-\sacbsg + \cacg } \pgfmathsetmacro {\rbceul }{\sasb } \pgfmathsetmacro {\rcaeul }{-\sbcg } \pgfmathsetmacro {\rcbeul }{\sbsg } \pgfmathsetmacro {\rcceul }{\cosbeta } } \newcommand {\tdplotcalctransformmainrot }[0]{\tdplotsinandcos {\sinalpha }{\cosalpha }{\tdplotalpha } \tdplotsinandcos {\sinbeta }{\cosbeta }{\tdplotbeta } \tdplotsinandcos {\singamma }{\cosgamma }{\tdplotgamma } \tdplotmult {\sasb 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{\tdplotgamma }{#3} \tdplotcalctransformrotmain \par \tdplotmult {\raaeaa }{\raarot }{\raaeul } \tdplotmult {\rabeba }{\rabrot }{\rbaeul } \tdplotmult {\raceca }{\racrot }{\rcaeul } \tdplotmult {\raaeab }{\raarot }{\rabeul } \tdplotmult {\rabebb }{\rabrot }{\rbbeul } \tdplotmult {\racecb }{\racrot }{\rcbeul } \tdplotmult {\raaeac }{\raarot }{\raceul } \tdplotmult {\rabebc }{\rabrot }{\rbceul } \tdplotmult {\racecc }{\racrot }{\rcceul } \tdplotmult {\rbaeaa }{\rbarot }{\raaeul } \tdplotmult {\rbbeba }{\rbbrot }{\rbaeul } \tdplotmult {\rbceca }{\rbcrot }{\rcaeul } \tdplotmult {\rbaeab }{\rbarot }{\rabeul } \tdplotmult {\rbbebb }{\rbbrot }{\rbbeul } \tdplotmult {\rbcecb }{\rbcrot }{\rcbeul } \tdplotmult {\rbaeac }{\rbarot }{\raceul } \tdplotmult {\rbbebc }{\rbbrot }{\rbceul } \tdplotmult {\rbcecc }{\rbcrot }{\rcceul } \pgfmathsetmacro {\raarc }{\raaeaa + \rabeba + \raceca } \pgfmathsetmacro {\rabrc }{\raaeab + \rabebb + \racecb } \pgfmathsetmacro {\racrc }{\raaeac + \rabebc + \racecc } 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{fill,stroke} }{} } } \newcommand {\tdplotshowargcolorguide }[4]{ \par \pgfmathsetmacro {\tdplotx }{#1} \pgfmathsetmacro {\tdploty }{#2} \pgfmathsetmacro {\tdplothuestep }{5} \pgfmathsetmacro {\tdplotxsize }{#3} \pgfmathsetmacro {\tdplotysize }{#4} \par \pgfmathsetmacro {\tdplotyscale }{\tdplotysize /360} \par \foreach \tdplotphi in {0,\tdplothuestep ,...,360} { \pgfmathdivide {\tdplotphi }{360} \definecolor {tdplotfillcolor}{hsb}{\pgfmathresult ,1,1} \color {tdplotfillcolor} \par \pgfmathsetmacro {\tdplotstarty }{\tdploty + \tdplotphi * \tdplotyscale } \pgfmathsetmacro {\tdplotstopy }{\tdplotstarty + \tdplothuestep * \tdplotyscale } \pgfmathsetmacro {\tdplotstartx }{\tdplotx } \pgfmathsetmacro {\tdplotstopx }{\tdplotx + \tdplotxsize } \filldraw [tdplot_screen_coords] (\tdplotstartx ,\tdplotstarty ) rectangle (\tdplotstopx ,\tdplotstopy ); } \par \pgfmathsetmacro {\tdplotstopy }{\tdploty + (360+\tdplothuestep )*\tdplotyscale } \pgfmathsetmacro {\tdplotstopx }{\tdplotx + \tdplotxsize } 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Essential Vocabulary

Here is a link to a list of these terms on Quizlet

Basis

A set $\mathcal {S}$ of vectors is called a basis of $\RR ^n$ (or a basis of a subspace $V$ of $\RR ^n$ ) provided that

(a): $\mbox {span}(\mathcal {S})=\RR ^n$ (or $V$ )
(b): $\mathcal {S}$ is linearly independent.

Closed under addition

A set $V$ is said to be closed under addition if for each element $\vec {u} \in V$ and $\vec {v} \in V$ the sum $\vec {u}+\vec {v}$ is also in $V$ .

Closed under scalar multiplication

A set $V$ is said to be closed under scalar multiplication if for each element $\vec {v} \in V$ and for each scalar $k \in \RR$ the product $k\vec {v}$ is also in $V$ .

Column space of a matrix

Let $A$ be an $m\times n$ matrix. The column space of $A$ , denoted by $\mbox {col}(A)$ , is the subspace of $\RR ^m$ spanned by the columns of $A$ .

Coordinate vector with respect to a basis

Let $\mathcal {B} = \{\vec {v}_1, \vec {v}_2,\ldots ,\vec {v}_k\}$ be an basis. Then the coordinate vector $\vec {v}$ is the column vector $\begin {bmatrix}c_1\\ c_2\\ \vdots \\c_k\end {bmatrix}$ such that $\vec {v} = c_1\vec {v}_1+c_2\vec {v}_2+\ldots +c_p\vec {v}_k$ .

Dimension

Let $V$ be a subspace of $\RR ^n$ . The dimension of $V$ is the number, $m$ , of elements in any basis of $V$ . We write $\mbox {dim}(V)=m$

Null space of a matrix

Let $A$ be an $m\times n$ matrix. The null space of $A$ , denoted by $\mbox {null}(A)$ , is the set of all vectors $\vec {x}$ in $\RR ^n$ such that $A\vec {x}=\vec {0}$ . It is a subspace of $\RR ^n$ .

Nullity of a matrix

Let $A$ be a matrix. The dimension of the null space of $A$ is called the nullity of $A$ . $\mbox {dim}\Big (\mbox {null}(A)\Big )=\mbox {nullity}(A)$

Ordered basis

A basis in which the elements appear in a specific fixed order. Establishing an order is necessary because a coordinate vector with respect to a given basis relies on the order in which the basis elements appear.

Rank of a matrix

Let $A$ be a matrix. The dimension of the row space of $A$ is called the rank of $A$ . $\mbox {dim}\Big (\mbox {row}(A)\Big )=\mbox {rank}(A)$

Rank-Nullity theorem

Let $A$ be an $m\times n$ matrix. Then $\mbox {rank}(A)+\mbox {nullity}(A)=n$

Row space of a matrix

Let $A$ be an $m\times n$ matrix. The row space of $A$ , denoted by $\mbox {row}(A)$ , is the subspace of $\RR ^n$ spanned by the rows of $A$ .

Subspace

Suppose that $V$ is a nonempty subset of $\RR ^n$ that is closed under addition and closed under scalar multiplication. Then $V$ is a subspace of $\RR ^n$ .

2024-09-06 02:11:46