I was given the following question:
Let $V$ be an inner product space and let $u,v\in V$ be two nonzero vectors. Prove or disprove:
- If $\langle u,v\rangle=0$, then $u,v$ are linearly independent.
- If $u,v$ are independent, then $\langle u,v\rangle=0$.
I know that $u,v$ are arthogonal if $\langle u,v\rangle = 0$. So, since $\langle u,v\rangle = 0$, and $u,v$ are non zero vectors can I claim linear independence between the vectors directly? And if so, how do I explain it?
This just seems wrong... I don't see how linear independence leads to this vectors having inner product of zero, meaning they are orthogonal. Any help or direction would be very helpful.