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  1. Can you help me find an example for $\,u,v \in \Bbb C^2\,$ unit vectors that $\langle u,v\rangle\neq 0$ and $\|u+v\|=\sqrt 2$

  2. $u,v$ over $C^n$ and we know that $\|u\|=\|u+v\|=\|u-v\|$ need to prove $v=0$

after I opened every thing I got to this: $4Re(\langle u,v\rangle)=\langle u,u\rangle$ but I don't see how it helps me.

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    I changed $$ to $\langle u,v\rangle$ and $||x||$ to $\|x\|$.2012-11-28

1 Answers 1

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Responding to problem 2.
Lets break this down:

  1. $||u||=||u+v||$
  2. $||u+v||=||u-v||$

From 1 you will get: $\langle u,u\rangle=\langle u+v,u+v\rangle$
From 2: $\langle u+v,u+v\rangle=\langle u-v,u-v\rangle$ and from there you easily get to: $\langle u,v \rangle = - \overline{\langle u,v \rangle}$. Place that into the first equation and you will get your answer.

Regarding your first question: You can simply define $a=(a_1,a_2)$ and $b=(b_1,b_2)$ place into the standard complex inner product and solve.