To show the code is not linear

Question

Consider the projective plane of order 2 or the Fano plane and its incidence matrix.https://en.wikipedia.org/wiki/Fano_plane. Any two rows of the incidence matrix thought of as binary words, have distance 4 from each other, since they have 3 ones,whose positions agree exactl once. If we denote these rows by $r_1,r_2,...,r_7$ then $d(r_i,r_j)=4$. Now consider the complements $s_1,...,s_7$ of the rows, where $s_i$ is obtained from $r_i$ by interchanging $0's$ and $1's$. So, $d(s_i,s_j)=4$ and $d(r_i,s_j)=3$. So the code $C={(0,1,r_1,...,r_7,s_1,...,s_7)}$ is a (7,16,3) code. We need to show that this code is non linear. Now I know what linear code is. but I am unable to that it is not a subspace of $F_2^7$. I was thinking that if I get a generator matrix and then taking two rows $r_1,r_2$ I will show the non-linearity. But I don't know how to get the generator matrix or if I am wrong please correct me, I know the structure of a generator matrix for linear code.So,please tell me how to show the non-linearity of that code?