Let be an orthonormal basis of . Then is it true that every element of this vector space can be written thus,

.

If so, why?

( etc. is the dot product).

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- January 13th 2010, 03:04 AMSwlabrOrthonormal Basis
Let be an orthonormal basis of . Then is it true that every element of this vector space can be written thus,

.

If so, why?

( etc. is the dot product). - January 13th 2010, 04:54 AMDefunkt
Yes, this is correct for any finite-dimensional vector space V (with some inner product) over .

Assume is an orthonormal basis of .

Let for some scalars . But then,

Therefore: