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

Coordinate vector

Let $V$ be a vector space, and let $\mathcal {B}=\{\vec {v}_1, \ldots ,\vec {v}_n\}$ be a basis for $V$ . If $\vec {v}=a_1\vec {v}_1+\ldots +a_n\vec {v}_n$ , then the vector in $\RR ^n$ whose components are the coefficients $a_1, \ldots ,a_n$ is said to be the coordinate vector for $\vec {v}$ with respect to $\mathcal {B}$ . We denote the coordinate vector by $[\vec {v}]_{\mathcal {B}}$ and write: $[\vec {v}]_{\mathcal {B}}=\begin {bmatrix}a_1\\\vdots \\a_n\end {bmatrix}$

Finite-dimensional vector space

A vector space is said to be finite-dimensional if it is spanned by finitely many vectors.

Isomorphism

Let $V$ and $W$ be vector spaces. If there exists an invertible linear transformation $T:V\rightarrow W$ we say that $V$ and $W$ are isomorphic and write $V\cong W$ . The invertible linear transformation $T$ is called an isomorphism.

Matrix of a linear transformation

Let $V$ and $W$ be finite-dimensional vector spaces with ordered bases $\mathcal {B}=\{\vec {v}_1,\vec {v}_2,\ldots ,\vec {v}_n\}$ and $\mathcal {C}$ , respectively. Suppose $T:V\rightarrow W$ is a linear transformation. $\text {Let}\quad A=\begin {bmatrix} | & |& &|\\ [T(\vec {v}_1)]_{\mathcal {C}} & [T(\vec {v}_2)]_{\mathcal {C}}&\dots &[T(\vec {v}_n)]_{\mathcal {C}}\\ |&| & &| \end {bmatrix}$ Then $A[\vec {v}]_{\mathcal {B}}=[T(\vec {v})]_{\mathcal {C}}$ for all vectors $\vec {v}$ in $V$ .

Matrix $A$ is called the matrix of $T$ with respect to ordered bases $\mathcal {B}$ and $\mathcal {C}$ .

One-to-one linear transformation

A linear transformation $T:V\rightarrow W$ is one-to-one if $T(\vec {v}_1)=T(\vec {v}_2)\quad \text {implies that}\quad \vec {v}_1=\vec {v}_2$

Onto linear transformation

A linear transformation $T:V\rightarrow W$ is onto if for every element $\vec {w}$ of $W$ , there exists an element $\vec {v}$ of $V$ such that $T(\vec {v})=\vec {w}$ .

Subspace

A nonempty subset $U$ of a vector space $V$ is called a subspace of $V$ , provided that $U$ is itself a vector space when given the same addition and scalar multiplication as $V$ .

Vector Space

Let $V$ be a nonempty set. Suppose that elements of $V$ can be added together and multiplied by scalars. The set $V$ , together with operations of addition and scalar multiplication, is called a vector space provided that

$V$ is closed under addition
$V$ is closed under scalar multiplication

and the following properties hold for $\vec {u}$ , $\vec {v}$ and $\vec {w}$ in $V$ and scalars $k$ and $p$ :

(a): Commutative Property of Addition: $\vec {u}+\vec {v}=\vec {v}+\vec {u}$
(b): Associative Property of Addition: $(\vec {u}+\vec {v})+\vec {w}=\vec {u}+(\vec {v}+\vec {w})$
(c): Existence of Additive Identity: $\vec {u}+\vec {0}=\vec {u}$
(d): Existence of Additive Inverse: $\vec {u}+(-\vec {u})=\vec {0}$
(e): Distributive Property over Vector Addition: $k(\vec {u}+\vec {v})=k\vec {u}+k\vec {v}$
(f): Distributive Property over Scalar Addition: $(k+p)\vec {u}=k\vec {u}+p\vec {u}$
(g): Associative Property for Scalar Multiplication: $k(p\vec {u})=(kp)\vec {u}$
(h): Multiplication by $1$ : $1\vec {u}=\vec {u}$

We will refer to elements of $V$ as vectors.

Note that definitions of span, linear independence, basis, and dimension are analogous to those for subspaces of $\RR ^n$ .

2024-09-06 02:10:02