Discover Linear Algebra

Section 42.2 Terminology and notation

(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 $\funcdef{T}{V}{W}$ between vector spaces $V$ and $W\text{,}$ 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 a 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 a 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

$$\begin{equation*} T(\uvec{v}) = \stdmatrixOf{T} \uvec{v} \end{equation*}$$

for all $\uvec{v}$ in $\R^n$

zero transformation (between vector spaces $V,W$)

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 $W\text{;}$ also referred to as the trivial transformation from $V$ to $W$

zero operator (on a vector space $V$)

the zero linear transformation $\funcdef{\zerovec}{V}{V}\text{;}$ also referred to as the trivial operator on $V$

identity operator (on a vector space $V$)

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 a vector space $V$ for a 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 a basis $\basisfont{B}$ for a finite-dimensional vector space $V$)

the transformation $\funcdef{\coordmap{B}}{V}{\R^n}$ defined by $\coordmap{B}(\uvec{v}) = \rmatrixOf{\uvec{v}}{B}$ for each $\uvec{v}$ in $V$

$L(V,W)$

the space of all linear transformations $V \to W$ for vector spaces $V,W\text{,}$ 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 $W$

linear functional (on a vector space $V$)

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 a vector space $V$)

the space $L(V,\R^1)$ of linear functionals on $V$

$\vecdual{V}$

alternative notation for the dual space $L(V,\R^1)$ of vector space $V$