Have you set up your augmented system? (Hint: Setup the augmented system [A | b] and solve for [I | x])
I have an orthogonal basis consisting of the following vectors in C^4 -
[0,1,0,-1] , [1,0,0,0] , [0,0,i,0]
Now, I have the vector [1, -i, 0, i] in standard coordinates. I wish to express this vector with respect to my orthogonal basis.
So, I solve c1 [0,1,0,-1] + c2 [1,0,0,0] + c3 [0,0,0,0] + c4[0,0,i,0] = [1, -i, 0, i].
[c1,c2,c3,c4] will be the vector wrt my othogonal basis.
Is this correct?
IF (1,-i,0,i) is in the span of your three vectors, then you only need to solve:
c1(0,1,0,-1) + c2(1,0,0,0) + c3(0,0,i,0) = (1,-i,0,i) for c1,c2 and c3 (adding a 0-vector to a basis is a waste of time).
inspection of the first coordinate shows c2 = 1. inspection of the fourth coordinate shows c1 = -i.
finally, since (-i)(0,1,0,-1) + (1)(1,0,0,0) = (0,-i,0,i) + (1,0,0,0) = (1,-i,0,i), we conclude c3 = 0, so our coordinates in this basis are [-i,1,0].
note you only specified THREE basis vectors, so we only have THREE coordinates (that is, if our basis only has 3 elements, the span of this basis is a 3-dimensional subspace of C4).