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I'm not sure how to approach this question. I have proved $det(G(v_1,...v_k)≥0$ and have proven the triangle inequality using the Cauchy-Schwarz inequality.

George
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1 Answers1

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Well, I think the most clean way to proceed here is by induction. Fon $n=1$ we have: $$det(G(v_1))=Vol_1(P(v_1))^2=<v_1,v_1>$$ Now let us suppose the formula valid for $n-1$ and induce the formula for $n$. First of all we notice that both $det(G(v_1,...,v_N))$ and $Vol_n(P(v_n))$ do not depend from a specific choice of the base (that's because the first one is a determinant and the second one is originated by the norm). Now instead of calculating directly $det(G(v_1,...,v_n))$ we calculate the determinant after a change of basis $$det(G(v_1,...,v_n))=det(S^{-1}G(v_1',...,v_n')S)$$ where $v_n'$ is orthogonal to all $v_i'$ with $i<n$. In this coordinates the calculation should be a little more convenient: $$det(G(v_1,...,v_n))=<v_n',v_n'>det(S^{-1}G(v_1',...,v_{n-1}')S)$$ now using the inductive hypotesis $$<v_n',v_n'>det(G(v_1',...,v_{n-1}'))=<v_n',v_n'>Vol_{n-1}(P(v_1',..,v_{n-1}'))^2$$ And finally using the recursive definition of the Volume and using the fact that the volume is invariant for change of basis we obtain $$Vol_{n}(P(v_1',..,v_{n}'))^2=Vol_{n}(P(v_1,..,v_{n}))^2$$

Dac0
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  • What is S? And for part (ii) what would I need to answer it? (what theorems) – George Nov 24 '15 at 23:32
  • S is the change of basis matrix? – George Nov 24 '15 at 23:39
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    S is just a matrix for the change of basis. Note that $det(S^{-1}G(v_1',...,v_n')S)=det(S^{-1})det(G(v_1',...,v_n'))det(S)=det(G(v_1',...,v_n'))$ – Dac0 Nov 24 '15 at 23:39
  • The last equality holds because $det(S)det(S^{-1}) = det(SS^{-1}) = det(I) = 1$? – George Nov 24 '15 at 23:41
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    It's a way of viewing that, but simply $det(S^{-1})=det(S)^{-1}$ and since $det(S)$ is a real number as $det(G)$ you can switch them and obtain: $det(S)^{-1}det(S)det(G(v_1...v_n))=det(G(v_1...v_n))$ – Dac0 Nov 24 '15 at 23:45
  • Ok thanks. For part (ii) I can see I need to prove the square root of the determinant of the Gram matrix is equal to the modulus of det (A) but I'm not sure how to start – George Nov 24 '15 at 23:47
  • To thank me you should put an up arrow to the answer – Dac0 Nov 24 '15 at 23:48
  • How did u take <vn prime, vn prime out>? – George Nov 29 '15 at 00:06