I'm confronted with this question:
Let $V$ be an inner product space and $B=\{u_{1}, ..., u_{n}\}$ a basis of $V$.
Suppose there exists $\lambda_{1},...,\lambda_{n} \in F (=R \text{ or } C)$ such that: $||\sum_{k=1}^{n}\lambda_{k}u_{k}||^2=\sum_{k=1}^{n}|\lambda_{k}|^2$
Prove or disprove: $B$ is an orthonormal basis.
Not sure where to begin.
I tried finding a counter example in the case of $V=R^2$ using the identity: $||u+v||^2=||u||^2+2Re\langle u, v\rangle+||v||^2$ which gave me this result: $||\lambda_{1}u_{1}+\lambda_{2}u_{2}||^2=|\lambda_{1}|||u_{1}||^2+2Re\langle\lambda_{1}u_{1},\lambda_{2}u_{2}\rangle+|\lambda_{2}|||u_{2}||^2=|\lambda_{1}|^2+|\lambda_{2}|^2$
But I can't see how this helps me.
edit:
Here's a little follow up question if anyone is interested: suppose the equality above is true for all $\lambda_{1},...,\lambda_{n}$, does that imply the basis is orthonormal?