(i) Assume that $B = \{w_1,\dots,w_k\}$ is an orthogonal basis for $E$. Let $v \in E^{\perp}$ such that $v\neq O_{V}$. Prove that $v=w+c_1w_1+c_2w_2+\dots +c_kw_k$ for some nonzero $w\notin E$ and some $c_1,\dots , c_k \in \mathbb{R}$
(ii) (converse of (i)) Let $B = \{w_1,\dots , w_k\}$ be an orthogonal basis for E. Let $w\notin E$. Prove that $v=w-proj^{(w)}_{w_1}-\dots -proj^{(w)}_{w_k} \in E^{\perp}$
this is how i approached it:
suppose you have an element $v=w+c_1w_1+c_2w_2+\dots +c_kw_k$. For v to belong to $E^\perp$, then v has to be orthogonal to $w_i, \forall i$. hence,
$\langle v,w_i\rangle = \langle w+c_1w_1+c_2w_2+\dots +c_kw_k,w_i\rangle = \langle w,w_i\rangle+\langle c_iw_i,w_i\rangle =\langle w_i,c_iw_i+w\rangle$ Therefore, for $v$ to be in $E^\perp$, $\langle w,w_i\rangle =-c_i\langle w_i,w_i\rangle$. Also, we require w not to be in $E$
I don't know how to move next.