Symmetric bilinear form/pq/Dimension degeneracy space/Zero restriction/Exercise

Let ${}V$ be a finite-dimensional ${}\mathbb {R}$-vector space, equipped with a symmetric bilinear form of type ${}(p,q)$, and let ${}a$ denote the dimension of the degeneracy space of the form. Show that there exists a linear subspace ${}U\subseteq V$ such that the restriction of the form to ${}U$ is the zero form, and with

${}\dim _{\mathbb {R} }{\left(U\right)}={\min {\left(m,q\right)}}+a\,.$

Moreover, show that there does not exist a linear subspace of larger dimension such that the restriction is the zero form.

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