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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
\(V\)
