What are the isomorphisms in the category of vector spaces, $Vec_K$?

Here is an interesting question from "The Rising Sea" book of Algebraic Geometry: http://math.stanford.edu/~vakil/216blog/FOAGapr2915public.pdf

1.2.3 Example. Consider the category $$Vec_K$$ of vector spaces over a field $$K$$. The objects are $$K$$-vector spaces, and the morphisms are linear transformations. What are the isomorphisms?

I would say that the isomorphisms are the invertible linear transformations, is that correct?

• Yes, that's right. There's not much more to say about this, except to remark that it suffices to require that the linear transformation is invertible as a map of sets; this kind of thing isn't always true in other concrete categories. Dec 28, 2020 at 6:54

Answer: If $$C$$ is any category and $$U,V$$ objects in $$C$$, we define $$U$$ to be isomorphic to $$V$$ (written $$U \cong V$$) iff there are morphisms (in $$C$$) $$f:U \rightarrow V$$ and $$g: V\rightarrow U$$ such that $$f\circ g=Id_V$$ and $$g \circ f=Id_U$$. In your case the maps $$f,g$$ are invertible linear maps/linear transformations between vector spaces.