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.
Best Answer
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.