- #1
Hitman2-2
Homework Statement
Prove that if V is a vector space over [tex] \mathbb{C}^n [/tex] with the standard inner product, then
[tex]
|<x,y>| = ||x|| \cdot ||y||
[/tex]
implies one of the vectors x or y is a multiple of the other.
The Attempt at a Solution
Assume the identity holds and that y is not zero. Let
[tex]
a = \frac {<x,y>} {||y||^2}
[/tex]
and let z = x - ay. I've shown that y and z are orthogonal and want to show
[tex]
|a| = \frac {||x||} {||y||}
[/tex]
Well,
[tex]
|a| \cdot ||y|| = \frac {|<x,y>|} {||y||} = \frac {|\sum_{i=1}^n a_i \overline{b_i} |} {\sqrt{\sum_{i=1}^n |b_i|^2}}
= \sqrt{\frac {\left(\sum_{i=1}^n a_i \overline{b_i} \right) \left(\sum_{i=1}^n \overline{a_i} b_i \right)} {\sum_{i=1}^n |b_i|^2} }
[/tex]
but now I don't see how to simply this further to get this equal to the norm of x.