Linear Map - Algebraic Classifications of Linear Transformations

Algebraic Classifications of Linear Transformations

No classification of linear maps could hope to be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on the vector space.

Let V and W denote vector spaces over a field, F. Let T: V → W be a linear map.

• T is said to be injective or a monomorphism if any of the following equivalent conditions are true:
  • T is one-to-one as a map of sets.
  • kerT = {0_V}
  • T is monic or left-cancellable, which is to say, for any vector space U and any pair of linear maps R: U → V and S: U → V, the equation TR = TS implies R = S.
  • T is left-invertible, which is to say there exists a linear map S: W → V such that ST is the identity map on V.

• T is said to be surjective or an epimorphism if any of the following equivalent conditions are true:
  • T is onto as a map of sets.
  • coker T = {0_W}
  • T is epic or right-cancellable, which is to say, for any vector space U and any pair of linear maps R: W → U and S: W → U, the equation RT = ST implies R = S.
  • T is right-invertible, which is to say there exists a linear map S: W → V such that TS is the identity map on W.

• T is said to be an isomorphism if it is both left- and right-invertible. This is equivalent to T being both one-to-one and onto (a bijection of sets) or also to T being both epic and monic, and so being a bimorphism.

• If T: V → V is an endomorphism, then:
  • If, for some positive integer n, the n-th iterate of T, Tn, is identically zero, then T is said to be nilpotent.
  • If T2 = T, then T is said to be idempotent
  • If T = kI, where k is some scalar, then T is said to be a scaling transformation or scalar multiplication map; see scalar matrix.