# Two finite dimensional vector spaces are isomorphic if and only if they have the same dimension (Proof review)

I would like to know if this HALF proof is ok:

Theorem: Two finite dimensional vector spaces are isomorphic if and only if they have the same dimension.

Using these two corollaries:

Corollary 1: If $$V$$ and $$W$$ are ﬁnite-dimensional vector spaces such that $$dim( V) > dim (W)$$, then no linear map from $$V$$ to $$W$$ is injective.

Corollary 2: If $$V$$ and $$W$$ are ﬁnite-dimensional vector spaces such that $$dim (V) < dim (W)$$, then no linear map from V to W is surjective.

HALF Proof: Contraposition

Let us assume $$V$$ and $$W$$ are isomorphic. That means there is an invertible linear map $$T$$ between them. Because $$T$$ is invertible, in particular it is injective. So there exists a linear map injective from $$V$$ to $$W$$ which means that $$dim(V) \leq dim(W)$$. Then apply the same reasoning to Corollary 2. So we get $$dim(V) = dim(W)$$

Thanks as always!

Maybe you can explicitly mention that $$\dim(V)\geq\dim(W)$$ after applying corollary 2, before making your final conclusion.