Orthonormal Vector Property

This problem comes from a second-semester course in linear algebra. We are currently covering Gram Scmidt - but outside of the application problems, I'm a bit lost.

Problem: Suppose V is an inner product space and {v_1, ... , v_k} is an orthonormal set of vectors in V. Show that for all v in V, we have:

$\displaystyle \sum_{i\ =\ 0}^k |<v,v_j>|^2 \leq ||v||^2$.

Furthermore, show that the equality holds if and only if v is in the span of {v_1, ... , v_k}.

Intuition: I can't picture the problem in anything larger than 2-space. If in two space, it seems plausible that a right triangle could be formed using the orthonormal set and v_j (which would be the hypotenuse). I'm not sure if this stems from a theorem, or if I should just be playing with the definitions in the summation/inequality.

Re: Orthonormal Vector Property

Quote:

Originally Posted by

**jsndacruz** This problem comes from a second-semester course in linear algebra. We are currently covering Gram Scmidt - but outside of the application problems, I'm a bit lost.

Problem: Suppose V is an inner product space and {v_1, ... , v_k} is an orthonormal set of vectors in V. Show that for all v in V, we have:

$\displaystyle \sum_{i\ =\ 0}^k |<v,v_j>|^2 \leq ||v||^2$.

Furthermore, show that the equality holds if and only if v is in the span of {v_1, ... , v_k}.

Intuition: I can't picture the problem in anything larger than 2-space. If in two space, it seems plausible that a right triangle could be formed using the orthonormal set and v_j (which would be the hypotenuse). I'm not sure if this stems from a theorem, or if I should just be playing with the definitions in the summation/inequality.

Write $\displaystyle v=v'+v''$ where $\displaystyle v'\in {\rm{Span}}(v_1, .. v_k)$ and $\displaystyle v''$ is in the orthogonal complement of $\displaystyle {\rm{Span}}(v_1, .. v_k)$

CB