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My attempt at a proof:

Let [itex]T:V\rightarrow W[/itex] be an isomorphism and [itex]\{v_{1},...,v_{n}\}[/itex] be a basis for [itex]V[/itex]. Since [itex]T[/itex] is an isomorphism, [itex]\forall v\in V,\exists T(v)\in W[/itex]. Therefore, if [itex]\{v_{1},...,v_{n}\}[/itex] spans [itex]V[/itex], [itex]\{T(v_{1}),...,T(v_{n})\}[/itex] spans [itex]W[/itex]. Since [itex]dim V=dimW[/itex], [itex]\{v_{1},...,v_{n}\}[/itex] is linearly independent, and is a basis for [itex]W[/itex].

I'm pretty sure I went wrong somewhere/claimed something I needed to prove. Any ideas where?