# Definition A *vector space [[isomorphism]]* is a [[linear map]] $\Phi: V \rightarrow W$ between [[vector space|vector spaces]] which preserves their structure $ \Phi(\pmb{v}_1 + \pmb{v}_2) = \Phi(\pmb{v}_1) + \Phi(\pmb{v}_2), \quad \forall \pmb{v}_1, \pmb{v}_2 \in V $ The map must be [[Types of Maps|bijective]] to be an isomorphism. If $V$ and $W$ are isomorphic, we write $V \cong W$. Note that this implies $\dim V = \dim W$ (i.e., $V$ and $W$ have the same [[dimension]]) and $\Phi(\pmb{v}) = 0 \implies \pmb{v} = 0$ See [[Linear Map#Homomorphisms and Isomoprhisms|here]] for more discussion.