Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Vector Space Isomorphism

Definition

Let $\struct {V, +, \circ}$ and $\struct {W, +', \circ'}$ be $K$-vector spaces.

Then $\phi: V \to W$ is a vector space isomorphism if and only if:

$(1): \quad \phi$ is a bijection

$(2): \quad \forall \mathbf x, \mathbf y \in V: \map \phi {\mathbf x + \mathbf y} = \map \phi {\mathbf x} +' \map \phi {\mathbf y}$

$(3): \quad \forall \mathbf x \in V: \forall \lambda \in K: \map \phi {\lambda \mathbf x} = \lambda \map \phi {\mathbf x}$