Parity check matrix intuition

Question

I am currently learning algebraic coding theory. There is something I would like to understand, so according to my book the way we generate this parity check matrix is that we pick some matrix A and based on this matrix we generate the parity check matrix H as follows H = (A | I). They state that the identity matrix provides as a method for parity check, for example in the example below they state this as well. I don't understand why does the identity in the above or the example below provides a method for parity check.

Answer 1

Think of the vector $v$ being formatted as $[m|v^{\prime}]^{T}$, where $m$ is the message you want to send and $v^{\prime}$ consists of your "check" bits, this vector will be $Am$. You are working in characteristic $2$, so when you compute $[A|I]v = Am + v^{\prime}$, you will get a $0$ in positions where they match and a $1$ otherwise. This lets you know if you get the proper message which was encoded by the sender, or if some errors have arisen.