If an orthogonal set is linearly independent how can we get the linear combination of these vectors to form another vector that is in the orthogonal set? I thought linear independence meant you cant form another vector that is in the set based off of the vectors in a set.
The only thing that can equal $c_1v_1+c_2v_2+...+c_nv_n=0$ is where scalars $c_1,c_2,...,c_n$ are $0$, meaning no multiples of those vectors can form another vector.
For example S=(v1,v2,v3) where v1=[1,2,3] v2=[1,1,-1] v3=[5,-4,1] is an orthogonal subset of R^3.
Vectors in S are non zero so S is linearly indepdent.
Let u=[3,2,1], now we can write u as a linear combination of the vectors in S.
Using the equation ci=(dot product of u and vi)/||vi||^2 we can obtain scalars for c where u=c1*v1+c2*v2+c3*v3
Again how can you find this vector u using the linear combination of the vectors in S if S is linearly independent? There should not be a vector u that is a linear combination of the vectors in S if S is linearly independent.