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).
>
> --
> 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?
>
> thanks in advance, any comments are appreaciated.
Unless you start with an orthogonal set of vectors, which is not the
case with your problem, the final set will depend on the order in which
you apply the G-S process. If you are given (1,x,x^2), take them in that
order.
It is often just as easy to separate the orthogonalization from the
normalization, first finding an orthogonal set then normalizing it.
In your problem, start with f1(x) = 1 and f2(x) = x + A and solve for A
so that integral_0^1 f1(x)*f2(x) dx = 0
Then do similarly for f3(x) = x^2 + B*x + C so that both
integral_0^1 f1(x)*f3(x) dx = 0 and integral_0^1 f2(x)*f3(x) dx = 0
At this point, your f1(x0, f2(x) and f3(x) "vectors" are orthogonal but
not necessarily unit vectors.
Finally, to unitize any "vector" f(x) divide it by the square root of
its inner product with itself, integral_0^1 f(x)^2 dx.
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