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]

Which is eqivalent to the following:

• [math]T(x+y)=T(x)+T(y)[/math]
• [math]T(\lambda x)=\lambda T(x)[/math]

Or indeed:

• [math]T(x+\lambda y)=T(x)+\lambda T(y)[/math]