Re: Gram-Schmidt process

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
.

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