For a set of orthonormal vectors \((v_1,v_2,...,v_m)\) in an m-dimensional vector space , and set of constants \(r_1,r_2,...r_m \in \mathbb{R}\) , if we consider the equation \(r_1v_1+r_2v_2+...+r_mv_m=0\) with the objective of demonstrating that the vectors in \((v_1,v_2,...,v_m)\) are linearly independent, then we must show that \(r_1=r_2=...=r_m=0\) . If we take the inner product on both sides of the equation , we have that
$$ <r_1v_1+r_2v_2+…+r_mv_m, v_1> = <0,v_1> $$
Using the properties of inner products, we have that
$$ r_1<v_1,v_1> +r_2<v_2,v_1> … +r_m<v_m,v_1> = 0 \implies r_1=0 $$
Repeating the same process but isolating \(v_2,v_3,...\) on the right hand side using inner products, we will deduce that \(r_2=r_3=...=r_m=0\) . Thus, since
$$ r_1v_1+r_2v_2+…+r_mv_m \iff r_1=r_2=…=r_m=0 $$
, we conclude that \((v_1,v_2,...,v_m)\) is linearly independent.