4.II.10B

Linear Algebra | Part IB, 2005

(i) Let VV be a finite-dimensional real vector space with an inner product. Let e1,,ene_{1}, \ldots, e_{n} be a basis for VV. Prove by an explicit construction that there is an orthonormal basis f1,,fnf_{1}, \ldots, f_{n} for VV such that the span of e1,,eie_{1}, \ldots, e_{i} is equal to the span of f1,,fif_{1}, \ldots, f_{i} for every 1in1 \leqslant i \leqslant n.

(ii) For any real number aa, consider the quadratic form

qa(x,y,z)=xy+yz+zx+ax2q_{a}(x, y, z)=x y+y z+z x+a x^{2}

on R3\mathbf{R}^{3}. For which values of aa is qaq_{a} nondegenerate? When qaq_{a} is nondegenerate, compute its signature in terms of aa.

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