1.II.17B

Let $\phi$ be a symmetric bilinear form on a finite dimensional vector space $V$ over a field $k$ of characteristic $\neq 2$. Prove that the form $\phi$ may be diagonalized, and interpret the rank $r$ of $\phi$ in terms of the resulting diagonal form.

For $\phi$ a symmetric bilinear form on a real vector space $V$ of finite dimension $n$, define the signature $\sigma$ of $\phi$, proving that it is well-defined. A subspace $U$ of $V$ is called null if $\left.\phi\right|_{U} \equiv 0$; show that $V$ has a null subspace of dimension $n-\frac{1}{2}(r+|\sigma|)$, but no null subspace of higher dimension.

Consider now the quadratic form $q$ on $\mathbb{R}^{5}$ given by

$2\left(x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{4}+x_{4} x_{5}+x_{5} x_{1}\right)$

Write down the matrix $A$ for the corresponding symmetric bilinear form, and calculate $\operatorname{det} A$. Hence, or otherwise, find the rank and signature of $q$.

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