We introduce two-sorted theories in the style of [CN10] for the complexity classes \oplusL and DET, whose complete problems include determinants over Z2 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 Z2 and Z can be proved in the corresponding theory, but leaves open the interesting question of whether the theorems themselves can be proved.

• 03B30 - Foundations of classical theories (including reverse mathematics)
• 03F20 - Complexity of proofs
• 15A15 - Determinants, permanents, traces, other special matrix functions
• 15B33 - Matrices over special rings (quaternions, finite fields, etc.)
• 68Q15 - Complexity classes (hierarchies, relations among complexity classes, etc.)