Linear combination of vectors in orthogonal set 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.
Edit:
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.
u=c1*v1+c2*v2+c3*v3
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.
 A: The definition of linear independence says you can't make 0 out of a linear combination. It says nothing about not being able to make any other vector out of linear combinations.
(1,0) and (0,1) are independent since you cannot write (0,0) = c(1,0) + d(0,1) without c=d=0. But you can write every other vector as a nontrivial linear combination of these. (2,3) = 2(1,0)+3(0,1) for example. Spend some time making sense of the definitions with some concrete examples like this one and it will make sense eventually.
If you call your orthogonal set $\{v_1, v_2, \dots, v_n\}$, you can trivially write any vector in your set as a linear combination (take all coefficients $0$ except the coefficient of $v_k$ which is $1$).
$v_k = 0\cdot v_1+0\cdot v_2+\dots+0\cdot v_{k-1}+1\cdot v_k+0\cdot v_{k+1}+\dots+0\cdot v_n$
This is true of any set, whether it is orthogonal or not.
Moreover, any vector in the span of $\{v_1, v_2, \dots, v_n\}$ can be written as a linear combination of these vectors. This is again true of any set, whether orthogonal or not.
