Every vector space of dimension $\kappa$ has $\kappa$ elements when $\kappa$ is at least $\aleph_1$

Question

I'm reading this proof from http://web.mat.bham.ac.uk/R.W.Kaye/logic/vspaces.

But I don't see why every vector space of dimension $\kappa$ has $\kappa$ elements.

Also, I know that vector spaces of finite dimensions are an isomorphism. But I'm not sure how they reached the conclusion $T_{VS}$ is $\kappa$-categorical.

Thanks!