Skip to main content
Contents Index
Search Book
Search Results:
No results.
Read aloud Readability settings Prev Up Next \(\require{cancel}
\newcommand{\bigcdot}{\mathbin{\large\boldsymbol{\cdot}}}
\newcommand{\basisfont}[1]{\mathcal{#1}}
\newcommand{\iddots}{{\mkern3mu\raise1mu{.}\mkern3mu\raise6mu{.}\mkern3mu \raise12mu{.}}}
\DeclareMathOperator{\RREF}{RREF}
\DeclareMathOperator{\adj}{adj}
\DeclareMathOperator{\proj}{proj}
\DeclareMathOperator{\matrixring}{M}
\DeclareMathOperator{\poly}{P}
\DeclareMathOperator{\Span}{Span}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\nullity}{nullity}
\DeclareMathOperator{\nullsp}{null}
\DeclareMathOperator{\uppermatring}{U}
\DeclareMathOperator{\trace}{trace}
\DeclareMathOperator{\dist}{dist}
\DeclareMathOperator{\negop}{neg}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\im}{im}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\ci}{\mathrm{i}}
\newcommand{\cconj}[1]{\bar{#1}}
\newcommand{\lcconj}[1]{\overline{#1}}
\newcommand{\cmodulus}[1]{\left\lvert #1 \right\rvert}
\newcommand{\bbrac}[1]{\bigl(#1\bigr)}
\newcommand{\Bbrac}[1]{\Bigl(#1\Bigr)}
\newcommand{\irst}[1][1]{{#1}^{\mathrm{st}}}
\newcommand{\ond}[1][2]{{#1}^{\mathrm{nd}}}
\newcommand{\ird}[1][3]{{#1}^{\mathrm{rd}}}
\newcommand{\nth}[1][n]{{#1}^{\mathrm{th}}}
\newcommand{\leftrightlinesubstitute}{\scriptstyle \overline{\phantom{xxx}}}
\newcommand{\inv}[2][1]{{#2}^{-{#1}}}
\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}
\newcommand{\degree}[1]{{#1}^\circ}
\newcommand{\rad}{\;\mathrm{rad}}
\newcommand{\blank}{-}
\newenvironment{sysofeqns}[1]
{\left\{\begin{array}{#1}}
{\end{array}\right.}
\newcommand{\iso}{\simeq}
\newcommand{\absegment}[1]{\overline{#1}}
\newcommand{\abray}[1]{\overrightarrow{#1}}
\newcommand{\abctriangle}[1]{\triangle #1}
\newcommand{\abcdquad}[1]{\square\, #1}
\newenvironment{abmatrix}[1]
{\left[\begin{array}{#1}}
{\end{array}\right]}
\newenvironment{avmatrix}[1]
{\left\lvert\begin{array}{#1}}
{\end{array}\right\rvert}
\newcommand{\mtrxvbar}{\mathord{|}}
\newcommand{\utrans}[1]{{#1}^{\mathrm{T}}}
\newcommand{\rowredarrow}{\xrightarrow[\text{reduce}]{\text{row}}}
\newcommand{\bidentmattwo}{\begin{bmatrix} 1 \amp 0 \\ 0 \amp 1 \end{bmatrix}}
\newcommand{\bidentmatthree}{\begin{bmatrix} 1 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \end{bmatrix}}
\newcommand{\bidentmatfour}{\begin{bmatrix} 1 \amp 0 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \amp 0
\\ 0 \amp 0 \amp 0 \amp 1 \end{bmatrix}}
\newcommand{\uvec}[1]{\mathbf{#1}}
\newcommand{\zerovec}{\uvec{0}}
\newcommand{\bvec}[2]{#1\,\uvec{#2}}
\newcommand{\ivec}[1]{\bvec{#1}{i}}
\newcommand{\jvec}[1]{\bvec{#1}{j}}
\newcommand{\kvec}[1]{\bvec{#1}{k}}
\newcommand{\injkvec}[3]{\ivec{#1} - \jvec{#2} + \kvec{#3}}
\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}
\newcommand{\unorm}[1]{\norm{\uvec{#1}}}
\newcommand{\dotprod}[2]{#1 \bigcdot #2}
\newcommand{\udotprod}[2]{\dotprod{\uvec{#1}}{\uvec{#2}}}
\newcommand{\crossprod}[2]{#1 \times #2}
\newcommand{\ucrossprod}[2]{\crossprod{\uvec{#1}}{\uvec{#2}}}
\newcommand{\uproj}[2]{\proj_{\uvec{#2}} \uvec{#1}}
\newcommand{\adjoint}[1]{{#1}^\ast}
\newcommand{\matrixOfplain}[2]{{\left[#1\right]}_{#2}}
\newcommand{\rmatrixOfplain}[2]{{\left(#1\right)}_{#2}}
\newcommand{\rmatrixOf}[2]{\rmatrixOfplain{#1}{\basisfont{#2}}}
\newcommand{\matrixOf}[2]{\matrixOfplain{#1}{\basisfont{#2}}}
\newcommand{\invmatrixOfplain}[2]{\inv{\left[#1\right]}_{#2}}
\newcommand{\invrmatrixOfplain}[2]{\inv{\left(#1\right)}_{#2}}
\newcommand{\invmatrixOf}[2]{\invmatrixOfplain{#1}{\basisfont{#2}}}
\newcommand{\invrmatrixOf}[2]{\invrmatrixOfplain{#1}{\basisfont{#2}}}
\newcommand{\stdmatrixOf}[1]{\left[#1\right]}
\newcommand{\ucobmtrx}[2]{P_{\basisfont{#1} \to \basisfont{#2}}}
\newcommand{\uinvcobmtrx}[2]{\inv{P}_{\basisfont{#1} \to \basisfont{#2}}}
\newcommand{\uadjcobmtrx}[2]{\adjoint{P}_{\basisfont{#1} \to \basisfont{#2}}}
\newcommand{\coordmapplain}[1]{C_{#1}}
\newcommand{\coordmap}[1]{\coordmapplain{\basisfont{#1}}}
\newcommand{\invcoordmapplain}[1]{\inv{C}_{#1}}
\newcommand{\invcoordmap}[1]{\invcoordmapplain{\basisfont{#1}}}
\newcommand{\similar}{\sim}
\newcommand{\inprod}[2]{\left\langle\, #1,\, #2 \,\right\rangle}
\newcommand{\uvecinprod}[2]{\inprod{\uvec{#1}}{\uvec{#2}}}
\newcommand{\orthogcmp}[1]{{#1}^{\perp}}
\newcommand{\vecdual}[1]{{#1}^\ast}
\newcommand{\vecddual}[1]{{#1}^{\ast\ast}}
\newcommand{\change}[1]{\Delta #1}
\newcommand{\dd}[2]{\frac{d{#1}}{d#2}}
\newcommand{\ddx}[1][x]{\dd{}{#1}}
\newcommand{\ddt}[1][t]{\dd{}{#1}}
\newcommand{\dydx}{\dd{y}{x}}
\newcommand{\dxdt}{\dd{x}{t}}
\newcommand{\dydt}{\dd{y}{t}}
\newcommand{\intspace}{\;}
\newcommand{\integral}[4]{\int^{#2}_{#1} #3 \intspace d{#4}}
\newcommand{\funcdef}[3]{#1\colon #2\to #3}
\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 42.2 Terminology and notation
Where they appear in the following definitions, \(V\) and \(W\) represent abstract vector spaces.
(real) matrix transformation
a function
\(\funcdef{T_A}{\R^n}{\R^m}\) defined by multiplication by a real
\(m \times n\) matrix
\(A\text{,}\) so that
\(T_A(\uvec{x}) = A \uvec{x}\) for each
\(\uvec{x}\) in
\(\R^n\)
(complex) matrix transformation
a function
\(\funcdef{T_A}{\C^n}{\C^m}\) defined by multiplication by a complex
\(m \times n\) matrix
\(A\text{,}\) so that
\(T_A(\uvec{x}) = A \uvec{x}\) for each
\(\uvec{x}\) in
\(\C^n\)
linear transformation
a function \(\funcdef{T}{V}{W}\) so that both
\begin{align*}
T(\uvec{u} + \uvec{v}) \amp = T(\uvec{u}) + T(\uvec{v}) \text{,} \amp
T(k \uvec{v}) = k T(\uvec{v})
\end{align*}
are true for all vectors \(\uvec{u},\uvec{v}\) in \(V\) and all scalars \(k\)
vector space homomorphism
a synonym for
linear transformation
domain space
the vector space
\(V\) of “input” vectors for a linear transformation
\(\funcdef{T}{V}{W}\)
codomain space
the vector space
\(W\) which contains the “output” vectors for a linear transformation
\(\funcdef{T}{V}{W}\)
linearity properties
the defining properties of a linear transformation: that it
respects the vector addition and scalar multiplication operations of the domain and codomain spaces
image vector (of domain space vector \(\uvec{v}\) )
the output vector
\(\uvec{w} = T(\uvec{v})\) in the codomain space corresponding to the input vector
\(\uvec{v}\)
linear operator
a linear transformation
\(\funcdef{T}{V}{V}\) with the same codomain space as domain space
standard matrix (of linear transformation \(\funcdef{T}{\R^n}{\R^m}\) )
the \(m \times n\) matrix
\begin{equation*}
\stdmatrixOf{T} = \begin{bmatrix}
| \amp | \amp \amp | \\
T(\uvec{e}_1) \amp T(\uvec{e}_2) \amp \cdots \amp T(\uvec{e}_n) \\
| \amp | \amp \amp |
\end{bmatrix}\text{,}
\end{equation*}
where \(\uvec{e}_j\) is the \(\nth[j]\) standard basis vector of \(\R^n\text{,}\) so that
\begin{equation*}
T(\uvec{x}) = \stdmatrixOf{T} \uvec{x}
\end{equation*}
for all \(\uvec{x}\) in \(\R^n\)
zero transformation (between \(V,W\) )
the linear transformation
\(\funcdef{\zerovec}{V}{W}\) defined by
\(\zerovec(\uvec{v}) = \zerovec_W\) for all
\(\uvec{v}\) in
\(V\text{,}\) where
\(\zerovec_W\) is the zero vector in
\(W\)
trivial transformation
synonym for
zero transformation
zero operator (on \(V\) )
the zero transformation
\(\funcdef{\zerovec}{V}{V}\)
trivial operator
synonym for
zero operator
identity operator (on \(V\) )
the identity linear transformation
\(\funcdef{I_V}{V}{V}\) defined by
\(I_V(\uvec{v}) = \uvec{v}\) for each
\(\uvec{v}\) in
\(V\)
scalar operator (on \(V\) for scalar \(a\) )
the scalar multiplication transformation
\(\funcdef{m_a}{V}{V}\) defined by
\(m_a(\uvec{v}) = a \uvec{v}\) for each
\(\uvec{v}\) in
\(V\)
coordinate transformation (relative to basis \(\basisfont{B}\) for \(V\) )
the transformation
\(\funcdef{\coordmap{B}}{V}{\R^n}\) (for
\(V\) finite-dimensional with
\(\dim V = n\) ) defined by
\(\coordmap{B}(\uvec{v}) = \rmatrixOf{\uvec{v}}{B}\) for each
\(\uvec{v}\) in
\(V\)
linear functional (on \(V\) )
a linear transformation
\(\funcdef{f}{V}{\R^1}\) (for
\(V\) a real vector space) or
\(\funcdef{f}{V}{\C^1}\) (for
\(V\) a complex vector space)
dual space (of \(V\) )
the collection of all linear functionals on
\(V\)
\(L(V,W)\)
the space of all linear transformations \(V \to W\text{,}\) with addition and scalar multiplication defined by
\begin{align*}
(T_1 + T_2)(\uvec{v}) \amp = T_1(\uvec{v}) + T_2(\uvec{v}) \text{,} \amp
(k T)(\uvec{v}) \amp = k \, T(\uvec{v})
\end{align*}
for all \(\funcdef{T,T_1,T_2}{V}{W}\) and all \(\uvec{v}\) in \(V\)
\(\Hom(V,W)\)
alternative notation for
\(L(V,W)\text{,}\) sometimes referred to as the
hom-space of
\(V\) into
\(W\)
\(L(V,\R^1)\)
\(\vecdual{V}\)
alternative notation for the dual space of
\(V\)
