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

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Definition

Let $\left({V, +, \circ }\right)$ and $\left({W, +', \circ'}\right)$ be $K$-vector spaces.

Then $\phi: V \to W$ is a vector space isomorphism iff:

$(1): \quad \phi$ is a bijection
$(2): \quad \forall \mathbf x, \mathbf y \in V: \phi \left({\mathbf x + \mathbf y}\right) = \phi \left({\mathbf x}\right) +' \phi \left({\mathbf y}\right)$
$(3): \quad \forall \mathbf x \in V: \forall \lambda \in K: \phi \left({\lambda \mathbf x}\right) = \lambda \phi \left({\mathbf x}\right)$