  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.
