If S is orthonormal and $y \in span (S)$, then $y=\sum_{i=1}^{k}<y,v_i> v_i$.

Question

Corollary:If S is orthonormal and $y \in span (S)$, then $y=\sum_{i=1}^{k} \langle y,v_i \rangle v_i$.

What's the general idea of proving this?

attempt: since $y \in span(S)$, then it is a linear combination of $\sum_{i=1}^{k} c_i v_i$ where $c_i$ is a scalar $\in F$. How am I supposed to prove that $c_i=\langle y,v_i \rangle$ from the ground up? What fact to use?

Answer 1

$y= \sum\limits_{k=1}^{n} a_iv_i$for some scalars $a_1,a_2,...,a_n$. Take inner product with $v_j$ to get $\langle y, v_j \rangle =a_j$ in view of orthonormality.