Invariants between two isomorphic vector spaces

Question

I have a general question about isomorphisms between vector spaces.

From a general point of view, there are common properties (invariants) between two isomorphic structures (e.g., properties about compactness or connectedness between two isomorphic topological spaces, commutativity between two isomorphic groups). Can you give me some examples of invariants between two isomorphic vector spaces ?

Thank you for your help !

Answer 1

The number one is the dimension, of course, since two vector spaces over the same field are isomorphic if and only if they have the same dimension.

Answer 2

One of the fundamental theorems of linear algebra is that there is only one invariant you really need to care about: the dimension. Two vector spaces are isomorphic if and only if they have the same dimension. The reason is that any map that bijectively takes a basis of one vector space to a basis of another vector space must extend linearly (in a unique way) to a vector space isomorphism.

Another example is the cardinality of the underlying set (of course, the isomorphism is a bijection).

This observation is actually useful in field theory. Suppose you have a field $K$ which you know is finite. Then the prime subfield of $K$ is $\mathbb{F}_{p}$ for some prime $p.$ Now $K$ must have finite dimension as a vector space over $\mathbb{F}_{p}$ (since $K$ is finite), so, again as a vector space, $K=(\mathbb{F}_{p})^{\oplus r}$ for some $r.$ It follows immediately that $\lvert{K}\rvert=p^{r}$ for the same $r.$ Moreover, if two finite fields are of the same cardinality, then that cardinality must be $p^{r}$ for some prime $p$ and some integer $r,$ and it follows that those two fields are isomorphic as vector spaces over the prime subfield $\mathbb{F}_{p}.$ (With a bit more work, one can show that any two finite fields of the same cardinality are isomorphic as fields, and that a finite field of cardinality $p^{r}$ does exist for every prime $p$ and every integer $r>0.$)