4.II.10B
(i) Let be a finite-dimensional real vector space with an inner product. Let be a basis for . Prove by an explicit construction that there is an orthonormal basis for such that the span of is equal to the span of for every .
(ii) For any real number , consider the quadratic form
on . For which values of is nondegenerate? When is nondegenerate, compute its signature in terms of .
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