I see the terms linear transformation and (vector space) homomorphism used more or less interchangeably, and the set (space) of linear transformations from V to W referred to as Hom(V, W) or equivalently as L(V, W).
From category theory, a homomorphism is a mapping between objects that preserves structure, so a vector space homomorphism is a linear transformation as this is necessary to preserve the structure of vector addition and scalar multiplication.
But to say a linear transformation is a homomorphism seems to be an invalid assumption. For example, with real / complex inner product spaces only orthogonal / unitary linear transformations preserve the inner product structure and therefore deserve to be considered as homomorphisms.
So, should linear transformations always be referred to as such (not homomorphisms), and the set of linear transformations from V to W as L(V, W), not Hom(V, W) ?
What then about a vector space isomorphism ?