Suppose u, v E V. Prove that (u, v) = 0 if and only if ||u|| ≤ ||u + av|| for all a € F.

Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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From Linear Algebra Done Right by Axler, how do I prove this for both F=C and F=R? I need to know both proofs and the difference. Appreciate your help!

Suppose \( u, v \in V \). Prove that \(\langle u, v \rangle = 0\) if and only if

\[
\| u \| \leq \| u + av \|
\]

for all \( a \in \mathbb{F} \).
Transcribed Image Text:Suppose \( u, v \in V \). Prove that \(\langle u, v \rangle = 0\) if and only if \[ \| u \| \leq \| u + av \| \] for all \( a \in \mathbb{F} \).
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