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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

synonym for injective

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 \(\funcdef{T}{V}{W}\))

the transformation \(\funcdef{\inv{T}}{\im T}{V}\) as in the definition of invertible transformation

Remark 44.2.1.

Technically, for functions \(f\) and \(g\) the composite \(g \circ f\) is only defined in the case that the domain of \(g\) is the same as the codomain for \(f\text{.}\) But as long as the domain for \(g\) contains the image of \(f\) as a subset, then the definition \((g \circ f)(x) = g\bigl(f(x)\bigr)\) still makes sense.

In particular, a linear transformation \(\funcdef{T}{V}{W}\) can always be “redefined” by restricting the codomain to \(\funcdef{T}{V}{\im T}\text{.}\) This way, in the case that \(T\) is invertible, the composition \(\inv{T} T\) makes sense even though \(\inv{T}\) is only defined on \(\im T\) rather than on all of the codomain space \(W\) for \(T\text{.}\)

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

synonym for surjective

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