Question

Suppose the set {v1, v2, v3} is an orthonormal basis for V (||x+y|| = ||x-y||) , and that w is any
vector in V which can be represented as
w = λ1v1 + λ2v2 + λ3v3.
Show that λk = (w, vk) for k = 1, 2, 3. Using the above representation of w, or
otherwise, show that
||w||^2=(w,v1)^2 + (w,v2)^2 + (w,v3)^2

Answer 1

aka proof of dot product (repeated Pythagorus)

Answer 2

http://en.wikipedia.org/wiki/Dot_product#Proof_of_the_geometric_interpretation

Answer 3

w1 = v1
w2 = v2 - projection of v1 onto w1
w3 = v3 - projection of v1 onto w3 - projection of v2 onto w3

thats orthogonal, orthonmormal you now have to do this:

u1 = w1/||w1||
u2 = w2/||w2||
u3 = w3/||w3||

leme know if you dont know how to do the projections

Answer 4

In English, orthonormal are unit as well as perpendicular....

The first part of the link above is all u need (IMHO)

Answer 5

Thanks guys!!

Answer 6

ur welcome...