In Theorem 3.37, prove that if B = A, then PGx = 0 for every x in

Question:

In Theorem 3.37, prove that if B = A, then PGx = 0 for every x in Z^k₂.

(Throughout this proof we denote by ai the ith column of a matrix A.) With P and G as in the statement of the theorem, assume first that they are standard parity check and generator matrices for the same binary code. Therefore, for every x in Z^k₂,PGx 0. In terms of block multiplication,