Let $V$ be a finite dimensional inner product space over $k$ with basis $\{v_1,\dots,v_n\}$ and inner product $\langle \cdot,\cdot\rangle$. For any $\alpha_1,\dots,\alpha_n\in k$, there exists a unique $v\in V$ such that $\langle v,v_i\rangle = \alpha_i$ for each $i$.
I read this and didn't think much of it at first, but I am having difficulty proving existence. Showing uniqueness is easy. For existence, I tried using the matrix associated to the inner product, say $Q$, and writing $$ v^tQv_i = \alpha_i $$ and computing $Qv_i$ to see how I might choose $v$, but this doesn't seem to give me anything. How does existence work in this claim?