Suppose $S = {[v_1, . . . , v_n]}$ is an orthonormal set and $v = α_1v_1 + · · · + α_nv_n$. Show that $\langle v, v\rangle =$ $α_1\overline{α_1} + · · · + α_n\overline{α_n} $.
Here I tried subbing in $v$, as a linear combination of vectors in $S$, into the inner product, but I became tangled up because I know that if the set is orthonormal, the inner product of any two elements in the set is zero, so my proof diminished. Any ideas on how to prove this?
The number $\langle v,v\rangle$ is the sum$$\sum_{k,k\in\{1,2,\ldots,n\}}\alpha_j\overline{\alpha_k}\langle v_j,v_k\rangle.\tag1$$But$$\langle v_j,v_k\rangle=\begin{cases}1&\text{ if }j=k\\0&\text{ if }j\neq k\end{cases}$$and therefore $(1)$ is equal to$$\sum_{j=1}^n\alpha_j\overline{\alpha_j}.$$