Linear Algebra | Part IB, 2004

Let QQ be a quadratic form on a real vector space VV of dimension nn. Prove that there is a basis e1,,en\mathbf{e}_{1}, \ldots, \mathbf{e}_{n} with respect to which QQ is given by the formula

Q(i=1nxiei)=x12++xp2xp+12xp+q2Q\left(\sum_{i=1}^{n} x_{i} \mathbf{e}_{i}\right)=x_{1}^{2}+\ldots+x_{p}^{2}-x_{p+1}^{2}-\ldots-x_{p+q}^{2}

Prove that the numbers pp and qq are uniquely determined by the form QQ. By means of an example, show that the subspaces e1,,ep\left\langle\mathbf{e}_{1}, \ldots, \mathbf{e}_{p}\right\rangle and ep+1,,ep+q\left\langle\mathbf{e}_{p+1}, \ldots, \mathbf{e}_{p+q}\right\rangle need not be uniquely determined by QQ.

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