Formal Theories for Linear Algebra

We introduce two-sorted theories in the style of [CN10] for the complexity classes circle plus L and DET, whose complete problems include determinants over Z(2) and Z, respectively. We then describe interpretations of Soltys' linear algebra theory LAP over arbitrary integral domains, into each of our new theories. The result shows equivalences of standard theorems of linear algebra over Z(2) and Z can be proved in the corresponding theory, but leaves open the interesting question of whether the theorems themselves can be proved.