For a field F, char(F) > 3, and an affine map A : F^n -> F^m we form a contradiction by picking some b not in A({0,1}^n) and saying Ax = b. This can be written as a system of 0-1 unsatisfiable linear equations over F expressible in the language of algebraic proof systems like Res(lin), NS, PC over F. One may think of these instances as the Subset Sum principle in positive characteristic. I'll suggest some hardness criterions for such instances and prove lower bounds for linear decision trees, tree-like Res(lin) and some restricted dag-like Res(lin) refutations. I'll also discuss work in progress on lower bounds for general dag-like Res(lin) refutations and PC refutations of these instances.