Linear map
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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]
Homomorphism and isomorphism
A linear map is a vector space homomorphism, if it is a bijection then it is a vector space isomorphism.
