Linear map
Definition
Given two vector spaces [ilmath](U,F)[/ilmath] and [ilmath](V,F)[/ilmath] (it is important that they are over the same field) we say that a map, [math]T:(U,F)\rightarrow(V,F)[/math] or simply [math]T:U\rightarrow V[/math] (because mathematicians are lazy), is a linear map if:
[math]\forall \lambda,\mu\in F[/math] and [math]\forall x,y\in U[/math] we have [math]T(\lambda x+\mu y) = \lambda T(x) + \mu T(y)[/math]
Notations
Some authors use [math]L[/math] for a linear map.
Because linear maps can often (always if [ilmath]U[/ilmath] and [ilmath]V[/ilmath] are finite dimensional) be represented as a matrix sometimes the notation [math]Tv[/math] is used instead of [math]T(v)[/math]
Homomorphism and isomorphism
A linear map is a vector space homomorphism, if it is a bijection then it is a vector space isomorphism.
Categories
The set of all linear maps from [ilmath](U,F)[/ilmath] to [ilmath](V,F)[/ilmath] is often denoted by [math]\mathcal{L}(U,V)[/math] or [math]\text{Hom}(U,V)[/math]
