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Topic: Orthogonal complement
Replies: 15   Last Post: Mar 5, 2013 6:48 PM

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 David C. Ullrich Posts: 21,553 Registered: 12/6/04
Re: Orthogonal complement
Posted: Mar 3, 2013 12:11 PM

On Sun, 03 Mar 2013 17:43:57 +0200, Kaba <kaba@nowhere.com> wrote:

>Hi,
>
>Let V be a finite-dimensional vector space over K together with a
>reflexive bilinear form f : V^2 --> K, and let S subset V be a subspace
>of V. Let C(S) stand for the orthogonal complement of S.
>
>Claim
>-----
>
>The bilinear form f is non-degenerate if and only if V = S + C(S).
>
>Any ideas?

One of us is missing something. Say f = 0. Then C(S) = V, hence
V = S + C(S).

Maybe the "+" was supposed to mean _direct_ sum? It seems likely
that it's easy to show that f is non-degenerate if and only if
S intersect C(S) = 0 for all S.

And it seems likely to me that if f is non-degenerate then it does
follow that V = S + C(S). To give an inelegant proof, start
with a basis for S, extend that to a basis for V, then
apply Gram-Schmidt to convert to an orthonormal basis... ?