Let $(V, \gamma)$ be a quadratic space, where $V$ is an $n$-dimensional $\mathbb{Z}/(7)$-vector space and $r = r(\gamma)$ is the rank of the bilinear form.
I want to show: either, $(V, \gamma)$ is isometric to $(V, \gamma')$, where $\gamma'$ is given by the associated matrix
$$ \begin{pmatrix} E_r & 0 \\ 0 & 0 \\ \end{pmatrix}$$
or, it's isometric to $(V, \gamma'')$, where $\gamma''$ is given by the associated matrix
$$ \begin{pmatrix} E_{r-1} & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}$$
where $E_m$ is the identity matrix of size $m$.
Thanks in advance. I thought that this might be solved via induction over $r$, but I don't really know how to do it.