Re: Gram-Schmidt process

On Sun, 27 Nov 2005 17:32:07 -0500, quasi <quasi@xxxxxxxx> wrote:

>On Sun, 27 Nov 2005 17:15:55 EST, SusanP <susanp@xxxxxxxxxxx> wrote:
>
>>Is it true that if {w1, w2, ..., wn} is an orthogonal
>>set of nonzero vectors, then the vectors v1, v2,..., vn
>>derived from the Gram-Schmidt process satisfy vi=wi,
>>for i= 1,2, ..., n?
>>
>>If so, can anyone think of a way to prove it? (Possibly
>>without induction... I don't like it very much....)
>>
>>I know the orthogonal vectors w1, w2,..., wn are
>>linearly independent.
>
>First, prove it for n=1.
>
>Then try it for n=2.
>
>Then n=3.
>
>Continue for a while ..., n=4, n=5, ...
>
>By the time you get to 100 and realize that you're still short of your
>goal of arbitrary n, you will pay anything for induction.
>
>Induction is your friend -- a magic bullet for many problems.
>
>Is induction natural for this problem? Well, just answer this
>question:
>
>How is the Gram-Schmidt process defined?
>
>Inductively, right?
>
>quasi

On the other hand, if it fails for some n, for example n=2, then you
don't have to worry about induction.

But my point was that if the result is true, then an inductive proof
is natural here since the Gram-Schmidt process is defined inductively.
If you can't clinch the induction, maybe check a few small values of n
such as n=1, n=2 to see if the result is even true. If the result is
false, then induction won't help you. Induction may be a magic bullet
but there's a limit to the magic -- it can't prove things true if
they're false.

quasi
.

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