Chapter 5.3, Problem 64E

Guided Proof Prove that if $w$ is orthogonal to each vector in $S=\left\{v_1,v_2,\dots ,v_n\right\}$, then $w$ is orthogonal to every linear combination of vector in $S$.

Getting Started: To prove that $w$ is orthogonal to every linear combination of vectors in $S$, you need to show that their inner product is $0$.

(i)	Write $v$ as a linear combination of vectors, with arbitrary scalars $c_1,\dots ,c_n$ in $S$.
(ii)	Form the inner product of $w$ and $v$.
(iii)	Use the properties of inner products to rewrite the inner product $\langle w,v\rangle$ as a linear combination of the inner products $\langle w,v_i\rangle$, $i=1,\dots ,n$.
(iv)	Use the fact that $w$ is orthogonal to each vector in $S$ to lead to the conclusion that $w$ is orthogonal to $v$.