Re: Gram-Schmidt process

On Sun, 27 Nov 2005 17:44:18 -0500, quasi <quasi@xxxxxxxx> wrote:

>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

One more comment about this.

If you don't see a proof immediately, consider it automatic that you
should then try to test it with some simple examples. It might lead to
a counterexample, but if not, it may give you some intuition about why
the result is true.

Try a small special case where you actually carry out the Gram-Schmidt
process, checking to see if relations v_i=w_i are forced.

quasi
.

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