Supose $v_1,...,v_n$ is a basis of a inner product space $V$. Prove that there exists a unique vector $u$ of $V$ such that $\langle u,v_i\rangle=c_i$ for all $i=1,...,n$ (each $c_i$ is constant).
I'm not sure how to prove this. I tried proving it by contradiction. Suppose there's $w$ in $V$ such that $\langle w,v_i\rangle=\langle u,v_i\rangle=c_i$ then $\langle w-u,v_i\rangle=0$. But this doesn't get me very far, as it does not imply $w=u$? How do I prove this? Also, how to prove existence in here?