Section 44.2 Terminology and notation
- composite function (for functions f:X→Y and g:Y→Z)
the function g∘f:X→Z defined by (g∘f)(x)=g(f(x))
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 ST in place of composite function notation S∘T for all linear transformations.- injective function
a function for which outputs are always distinct; i.e. for which f(x1)≠f(x2) in the codomain whenever x1≠x2 in the domain
- one-to-one
synonym for injective
- invertible transformation
a linear transformation T:V→W for which there exists linear T−1:imT→V so that both T−1T and TT−1 are the identity transformation (on V and imT, respectively)
- inverse transformation (of an invertible transformation T:V→W)
the transformation T−1:imT→V as in the definition of invertible transformation
Remark 44.2.1.
Technically, for functions f and g the composite g∘f is only defined in the case that the domain of g is the same as the codomain for f. But as long as the domain for g contains the image of f as a subset, then the definition (g∘f)(x)=g(f(x)) still makes sense.
In particular, a linear transformation T:V→W can always be “redefined” by restricting the codomain to T:V→imT. This way, in the case that T is invertible, the composition T−1T makes sense even though T−1 is only defined on imT rather than on all of the codomain space W for T.
- surjective function
a function for which every codomain element is an image element; i.e. for which f:X→Y with imf=Y
- onto
synonym for surjective
- isomorphism
a surjective, invertible transformation T:V→W
- isomorphic spaces
vector spaces V and W for which there exists an isomorphism V→W
- V≃W
notation to mean that spaces V and W are isomorphic