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Section 44.2 Terminology and notation
composite function (for functions \(\funcdef{f}{X}{Y}\) and \(\funcdef{g}{Y}{Z}\) )
the function
\(\funcdef{g \circ f}{X}{Z}\) defined by
\((g \circ f)(x) = g\bigl(f(x)\bigr)\)
Notation. As in
Discovery set 44.1 , in analogy with the relationship between standard matrices of matrix transformations and the standard matrix of their composition, we will use multiplication notation
\(S T\) in place of composite function notation
\(S \circ T\) for
all linear transformations.
injective function
a function for which outputs are always distinct; i.e. for which
\(f(x_1) \neq f(x_2)\) in the codomain whenever
\(x_1 \neq x_2\) in the domain
one-to-one
invertible transformation
a linear transformation
\(\funcdef{T}{V}{W}\) for which there exists linear
\(\funcdef{\inv{T}}{\im T}{V}\) so that both
\(\inv{T} T\) and
\(T \inv{T}\) are the identity transformation (on
\(V\) and
\(\im T\text{,}\) respectively)
inverse transformation (of an invertible transformation \(\funcdef{T}{V}{W}\) )
surjective function
a function for which every codomain element is an image element; i.e. for which
\(\funcdef{f}{X}{Y}\) with
\(\im f = Y\)
onto
isomorphism
a surjective, invertible transformation
\(\funcdef{T}{V}{W}\)
isomorphic spaces
vector spaces
\(V\) and
\(W\) for which there exists an isomorphism
\(V \to W\)
\(V \iso W\)
notation to mean that spaces
\(V\) and
\(W\) are isomorphic
