In article
<2ce93f0b-13e3-45f8-a8d7-d7cdf5b2e2ad@[EMAIL PROTECTED]
>,
<SneakyElf@[EMAIL PROTECTED]
> wrote:
>Was going over my notes on Gram-Schmidt procedure and came across a
>problem..
>given P2 (R) consider and inner product given by
>
><p,q> = integral from 0 to 1 p(x)q(x) dx
>
>need to apply G-S procedure to the basis (1,x,x^2) to produce
>orthonormal basis of P2(R).
Okay... And after coming across the problem what happened?
Did you try running over the Gram-Schmidt process with that basis and
that inner product? It should be a matter of doing some computations
with some easy integrals. Why is this being a problem?
>Also, staying on the same topic if given a list of vectors, say in
>R^3, and if any two vectors of that set are orthogonal to each other,
>what does it mean in terms of applying G-S to that set find an
>orthonormal basis?
Read the procedure. If the set is already orthogonal, and does not
contain the zero vector, then the first part of the Gram-Schmidt
process (the lengthy part that involves taking inner products with
every previously defined vector and doing vector subtractions) will be
trivial, since all those inner products will already be zero; you will
just end up with the same list as you started. The second part of the
Gram-Schmidt process, which is the normalization, consists of dividing
each vector by its norm, and that still needs to be done even if your
original set was orthogonal. So you ->apply it<-.
That's what "applying" means.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidin-at-member-ams-org
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