Consider the parity check error correcting code on words of length n$.$ That is$,$ f : Bn $->$

Bn$+1$ is the code that attaches parity digit b to the word w depending on whether the number of $1$s

in W is even $($then b $=0)$ or odd $($then b $=1).$ Prove that the codewords $($image of the code$)$ are those

words in Bn$+1$ with even number of $1$s$.$ Find their generator matrix.