Let $V$ be a vector space over $\Bbb R$.
Is it true that $\langle u,w\rangle= \langle v,w\rangle$ implies $u = v$? Prove it or give a counter-example.
The above is exactly what my assignment demands. It does not state "for every $w$" belonging to vector space $V$.
I wonder what difference does it make to not state that "for every $v$" in $V$.
I was able to find a counter-example, let $u = (1,-1,0)$, $w = (0,-1,1)$ and $v = (1,1,1)$.
$\langle(1,-1,0), (1,1,1)\rangle = 0$
$\langle (0,-1,1), (1,1,1)\rangle = 0$
but $u \neq w$
Is that correct?
As you've noted, $\langle u, v\rangle = \langle w, v\rangle$ is not sufficient to conclude $u = w$. In fact, what you do get is that $$\langle u - w, v\rangle = 0.$$ That is, $u - w$ is orthogonal to $v$. Now, if you are given that the above is true for all $v \in V$, then you can put $v = u - w$ to get $$\langle u - w, u - w\rangle = 0.$$ By the positive-definiteness of inner product, this implies that $u - w = 0$ or $u = w$, as desired.