Is there correspondence between vector spaces and independent theories?

Question

I'm not a mathematician, but I have this idea.

A vector space is defined by a basis, that is a set of linearly independent vectors. On the other hand, an independent theory is defined by a set of independent axioms. A set of axioms is said to be independent if neither of them can be proved from the others.

Correspondence can be recognized in the following parallel sentences:

1. A vector space over a field is defined by a set of linearly independent basis vectors.
2. New vectors can be expressed as linear combinations of basis vectors.
3. A vectorial basis spans a vector space.

vs.

1. An independent theory over a formal language is defined by a set of logically independent axioms.
2. New statements (theorems) can be derived as combinations of axioms.
3. An axiomatic system "spans" a deductive theory.

If the correspondence holds, the following questions make sense:

• Is it possible to assign coordinates to the theorems of an independent theory?
• A vector space may have multiple bases. Is it possible to translate this statement into the language of axiomatic systems?

Disclaimer: I tried to be as exact as I could be. Feel free to correct me if I accidentally misused some concepts.