Linear map/Definition

Given two vector spaces $(U,F)$ and $(V,F)$ (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:

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