Isomorphisms

Def: Isomorphism

A linear transformation is an isomorphism if $\mathrm{\exists }S:W\to V$ which is also linear such that:
$T\circ S$=$I_W$ and $S\circ T$=$I_V$ in this case $V\cong W$ , V and W are isomorphic.

Thm: Isomorphism iff bijective

Follows right from the definition and the existence of a set theoretic inverse.

Thm: Isomorphic Then Same Dimensions

If V and W are fdvs then V$\cong$W iff dim(V)=dim(W)

Say T is linear s.t T: $V\to W$
Follows right from the bijection. Surjective means the dim(V) $\ge$ dim(W) and injective means that dim(W) $\ge$ dim(V).

Other way I suppose you map a basis to a basis and use linear extension