If G = [I_k|A] is a generator matrix for the [n, k]codec in standard form, then H = [-A^T|I_n - k] is a parity check matrix for C. Proof: We clearly have HG^T = -A^T + A^T = O. Thus C is contained in the kernel of the linear transformation x rightarrow Hx^T. As rank n - k, this linear transformation has kernel of dimension k, which is also the dimension of C. The result follows. Prior to the statement of Theorem 1.2.1, it was noted that the rows of the (n -k) times n parity check matrix satisfying (1.1) are independent. Why is that so