I want to solve a linear matrix equation in mod 2, which takes the form AX = B but with additional restrictions described below. Here, columns of A are linearly independent to begin with. The RHS, B on the other hand, has some redundancy as some columns are all 0. X is unknown, but can be solved using known GF(2) solvers.
Clearly X is not a one-to-one transformation as it maps distinct columns in A to the same vector (the all 0 vector). So X is not invertible in this system.