Orthonormal basis for an inner product space (R^3)

[Idealistic]

consider the inner product space defined by:

(v, w) = v[sup:3h3rlghc]T[/sup:3h3rlghc]Aw

A = [2 1 2]
.....[1 1 1]
.....[2 1 3]

The question asks for an orthonormal basis of R[sup:3h3rlghc]3[/sup:3h3rlghc] for this inner product space using the gram-schmidt method.

the formula is:

w1 = v1

w2 = v2 - w1(v2[sup:3h3rlghc]T[/sup:3h3rlghc]Aw1)/(w1[sup:3h3rlghc]T[/sup:3h3rlghc]Aw1)

w3 = v3 - w1(v3[sup:3h3rlghc]T[/sup:3h3rlghc]Aw1)/(w1[sup:3h3rlghc]T[/sup:3h3rlghc]Aw1) - w2(v3[sup:3h3rlghc]T[/sup:3h3rlghc]Aw2)/(w2[sup:3h3rlghc]T[/sup:3h3rlghc]Aw2)

Correct?

can I use the standard basis vectors as my {v1, v2, v3}? (i.e. (1,0,0), (0,1,0), (0,0,1)}?

when I use this formula and the standard basis vectors I get:

w1 = v1 = (1,0,0)

w2 = (2/root5)(-1/2, 1, 0) *root5/2 was it's normal magnitude (sqrt(a^2 + b^2 + c^2))

w3 = (1/root2)(-1,0,1) *root2 was the magnitude

if I were to do w1[sup:3h3rlghc]T[/sup:3h3rlghc]Aw1 I don't get 1 as my answer, I get 2; isn't one of the properties of an orthonormal basis: (w1, w1) = 1

or do i multiply w1 by 1/2 so that when I do w1[sup:3h3rlghc]T[/sup:3h3rlghc]Aw1 I get 1

?