# 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, $T:(U,F)\rightarrow(V,F)$ or simply $T:U\rightarrow V$ (because mathematicians are lazy), is a linear map if:

• $\forall \lambda,\mu\in F$ and $\forall x,y\in U$ we have $T(\lambda x+\mu y) = \lambda T(x) + \mu T(y)$

Which is eqivalent to the following:

• $T(x+y)=T(x)+T(y)$
• $T(\lambda x)=\lambda T(x)$

Or indeed:

• $T(x+\lambda y)=T(x)+\lambda T(y)$[1]

## References

1. Linear Algebra via Exterior Products - Sergei Winitzki