WTS: $V\simeq$ W $\iff$ $dimV=dimW$
My proof: Assume $T: V \rightarrow W$ is an isomorphism, therefore the kernel for the linear map is the zero vector therefore $dim(ker(T))=0$ and since it is bijective, it is surjective therefore $imT=W$ hence $dimT=dimW$, by the rank-nullity theorem, dimV=dimW.
Assume $dimV=dimW$ since they are both finite dimensional, V=W hence defining a linear map $T:V \rightarrow V$ by $T(v) =v$ is an isomorphism.
Is my proof correct? May someone tell me how to improve it, please?