Orthonormal Basis Question Follow Up?

Question

This is a follow up to the question posted here:

http://answers.yahoo.com/question/index;_ylt=ArIze...

My main question is, can the best answer posted there be proven to be correct? As I mentioned in my comment, the statement made by the Best Answer does not prove in and of itself that all < u_i , u_j > are 0, only that a certain collection of sums are.

For the case n = 4, there are six different inner products and four equations. I can get solutions to the equations that are nonzero, but then I start hitting problems when trying to construct unique Vectors of magnitude 1 to fit these values. This is expected, seeing as the statement can be proven by other means: but is there a way to algebraically verify stephenmdalton's claim with Vector properties?

As an addition to my old post, here is the proof I came up with:

Let W_k = U_k / || U_k ||, where U_k is the cross product of all vectors u_i for i = 1 to n excluding k. W_k is a unit vector, and a member of V, so with it we obtain the equation:

|| W_k || ^ 2 = Sum from i = 1 to n of < W_k , u_i >

Because W_k is proportional to U_k, which is orthogonal to all u_i for i not equal to k, W_k is also orthogonal to these Vectors. Therefore:

|| W_k || ^ 2 = < W_k , u_k > = || W_k || * || u_k || * cos(θ_k)

This simplifies to cos(θ_k) = 1, showing that W_k = some scalar λ * u_k, where λ can equal ±1. Since u_k is proportional to W_k, u_k is also orthogonal to all u_i for i not equal to k. This goes for all k from 1 to n, so all < u_i , u_j > = 0 for i not equal to j, making this an orthonormal set.

Different proofs or insights welcome as well. Please contribute.

Mise à jour:

@zeta: Valid contradiction. However, we can't assume that the inner product is equivalent to the dot product in space V. I don't know how to define it, but we do know it obeys the more abstract properties listed on this article:

http://en.wikipedia.org/wiki/Inner_product

Zeta · Accepted Answer

I just want to ask this, to see if I understand the question, or there is something I'm misunderstanding,

Let V = |R², v = (3,3). Let <u,v> be the dot product. And let {u_1,u_2} = {(1,0) , (0,1)} be an orthonormal basis for V.

Then || v ||² = <v, v> = v dot v = 3² + 3² = 18

And sum from i = 1 to n of < v , u_i > = <v, u_1> + <v, u_2> = (3,3) dot (1,0) + (3,3) dot (0,1) = 3 + 3 = 6

The two are not equal. Is there something wrong with my argument, or is there something wrong with the question?

EIDT: Right, and the dot product satisfies all of the axioms for an inner product. So inner product would be a generalization of a dot product. If the assertion of the question holds for an inner product, then it must also hold for a dot product.

In fact, in the article it explicity states, "Inner product spaces generalize Euclidean spaces (in which the inner product is the dot product, also known as the scalar product)"

Orthonormal Basis Question Follow Up?

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