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Thread: Inner product proof

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    Inner product proof

    I could use some help on this proof...

    Suppose $\displaystyle u,v\in V$. Prove that $\displaystyle <u,v>=0$ if and only if $\displaystyle ||u||\leq ||u+av||$ for all $\displaystyle a\in F$.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by zebra2147 View Post
    I could use some help on this proof...

    Suppose $\displaystyle u,v\in V$. Prove that $\displaystyle <u,v>=0$ if and only if $\displaystyle ||u||\leq ||u+av||$ for all $\displaystyle a\in F$.
    What do you think? And, I assume that the norm is the natural one induced by the inner product.
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    Well, I'm guess that we are going to use the fact that since $\displaystyle <u,v>=0$ then $\displaystyle u,v$ are orthogonal. Then, we can evaluate the norm of $\displaystyle u+v$. That is, $\displaystyle ||u+v||$. Then, we can use Pythagorean Theorem to show that $\displaystyle ||u||^2\leq ||u+v||^2$. Then, the proof the other way would probably be similar???
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