By a formal system I mean a system for inferring theorems from axioms according to a set of rules. Essentially I wonder, whether a linear algebra could be viewed as some particular set of axioms from which by rules of inference I would generate all its theorems. Or is linear algebra just a field of mathematics dealing with linear equations and everything around them so that there could be more formal systems built around it? In my university, we don't learn to do formal proofs in linear algebra so I wonder whether linear algebra could be formalized.
Note: I have just done a course in mathematical logic (model theory - first order logic, proof theory - Hilbert's calculus) and I've just started learning linear algebra. If there is anything unclear on my question, please let me know.