Orthogonal Sets are Linearly Independent

Theorem

1. Let $V$ be an Inner Product Space
2. An Orthogonal Set $S\subset V$ of non-zero vectors is linearly independent

Proof

1. With ${\alpha }_1,\dots ,{\alpha }_n$ be distinct non-zero vectors
2. In $S$, Let $\beta =c_1{\alpha }_1+\cdots +c_n{\alpha }_n$
3. Then, $⟨\beta |{\alpha }_k⟩$
4. $=⟨{\sum }_{i=1}^n{c}_i{\alpha }_i|{\alpha }_k⟩$
5. $={\sum }_{i=1}^n⟨{\alpha }_i|{\alpha }_k⟩$
6. $={\sum }_{i=1}^k{c}_i\ast 0+c_k⟨{\alpha }_k|{\alpha }_k⟩+{\sum }_{i=k+1}^n{c}_i\ast 0$
7. $=c_k\ast ||{\alpha }_k||$
8. Thus, $c_k=\frac{⟨\beta |{\alpha }_k⟩}{||{\alpha }_k|{|}^2},\forall 1\le k\le n$
9. This is valid as ${\alpha }_k\ne 0$
10. If $\beta =0$, then $c_k=0$. Therefore, $({\alpha }_1,\dots ,{\alpha }_n)$ is linearly independent