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, isomorphism and isometry
A linear map is a vector space homomorphism, if it is a bijection then it is invertible, but the word isomorphism should be used sparingly, to avoid confusion with linear isometries which ought to be called "isometries"
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]
