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

Notations

Some authors use $$L$$ for a linear map.

Because linear maps can often (always if $U$ and $V$ are finite dimensional) be represented as a matrix sometimes the notation $$Tv$$ is used instead of $$T(v)$$

Homomorphism and isomorphism

A linear map is a vector space homomorphism, if it is a bijection then it is a vector space isomorphism.

Categories

The set of all linear maps from $(U,F)$ to $(V,F)$ is often denoted by $$\mathcal{L}(U,V)$$ or $$\text{Hom}(U,V)$$

See also

Example comparing bilinear to linear maps