Let $C*$ be the length 8 binary code obtained by adding a parity check symbol to each word in $C$. (so a word $c_1, c_2, c_3, c_4, c_5, c_6, c_7$ is extended to  the word $c_1, c_2, c_3, c_4, c_5, c_6, c_7, c_8$ for which $\sum_{i=0}^8 c_i =0$. Show that $C*$ is self dual (I.e show that $C*=C*^{\bot}$. Where $C*^{\bot}$ is the orthogonal complement of $C*$I am given the generator matrix for $C$:$$G= \begin{bmatrix}    1 & 1 & 0 & 1 & 0 & 0 & 0 \\    0 & 1 & 1 & 0 & 1 & 0 & 0  \\    0 & 0 & 1 & 1 & 0 & 1 & 0 \\0 & 0 & 0 & 1 & 1 & 0 & 1 \end{bmatrix}$$So I think $C*$ is given by:$$G*= \begin{bmatrix}    1 & 1 & 0 & 1 & 0 & 0 & 0 & 1\\    0 & 1 & 1 & 0 & 1 & 0 & 0  & 1\\    0 & 0 & 1 & 1 & 0 & 1 & 0 & 1\\0 & 0 & 0 & 1 & 1 & 0 & 1 &1 \end{bmatrix}$$I can show  $C*$ and $C*^{\bot}$ have the same dimensions but don't know what to do from here. 
Showing that a binary linear code $C$ is self-dual
matrices;coding theory;error correcting codes