Let $W$ and $V$ be vector spaces over the same ground field $F$. A function $T: V \to W$ is a linear transformation if for all $v,w \in V$ and $\alpha,\beta \in F$, \(T(\alpha v+\beta w) = \alpha T(v) + \beta T(w).\)
This is the correct notion for a homomorphism of vector spaces. If a linear transformation is bijective, then it is an isomorphism of vector spaces.
Wikidata ID: Q207643