Re: Vector space or ot's orthogonal

In article
<1151650411.812593.312750@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"eugene" <jane1806@xxxxxxxxxx> wrote:

The World Wide Wade wrote:
In article
<1151593115.134097.66840@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"eugene" <jane1806@xxxxxxxxxx> wrote:

Suppose that you have a vector subspace F in R^n. Prove that either F
or its orthogonal complement contains a non-zero vector all of whose
components
are non-negative.

This is true when n = 1. Suppose true for n - 1, where n > 1. Let
M + N = R^n, where N is the orthogonal complement (OC) of M in
R^n. Set m = dim M => n-m = dim N.

Let M', N' be the intersections of M and N with R^(n-1) resp. (by
R^(n-1) I mean R^(n-1) x {0}). If M is contained in R^(n-1)

Thanks a lot. But i have a question: what if M' and N' are trivial, i
mean , if

M' = N' = {0}.

This happens iff n = 2 and M, N are perpendicular lines through
the origin. I see no problem.

then
N' is the OC of M in R^(n-1), and we are done by the induction
hypothesis. Same thing for N. So assume neither M nor N is
contained in R^(n-1). Then by dimension counting, dim M' = m-1,
dim N' = n-m-1.

If either M' or N' contain a vector of the desired form, we're
done. Otherwise we look at the OC of M' + N' in R^(n-1), which by
dimension counting must be [w] for some nonzero unit vector w in
R^(n-1). (Here [ ] denotes linear span.) By the induction
hypothesis, w has the desired form.

Choose u, v in M, N orthogonal to M', N' resp. Now R^n = M' + N'
+ [w] + [e_n], and these are orthogonal sums. So both u, v lie in
[w] + [e_n], and hence one of them has a nonzero component in the
w-direction. Suppose it's u. Then u = aw + be_n, and WLOG a > 0.
If b >= 0, we are done: u has the desired form. If b < 0, then w
+ (-a/b)e_n is orthogonal to M, hence is in N, and has the
desired form.
.

Relevant Pages

  • Re: Vector space or ots orthogonal
    ... or its orthogonal complement contains a non-zero vector all of whose ... N' be the intersections of M and N with R^resp. ... If either M' or N' contain a vector of the desired form, ... Choose u, v in M, N orthogonal to M', N' resp. ...
    (sci.math)
  • Re: Vector space or ots orthogonal
    ... or its orthogonal complement contains a non-zero vector all of whose ... Then by dimension counting, dim M' = m-1, ... If either M' or N' contain a vector of the desired form, ...
    (sci.math)
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