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 \(V\)