Orthonormal basis for subsets of C^3

[gysush]

We want to find a basis for W and W_perpendicular for W=span({(i,0,1)}) =Span({w1}) in C^3

a vector x =(a,b,c) in W_perp satisfies <w1,x> = 0 => ai + c = 0 => c=-ai
Thus a vector x in W_perp is x = (a,b,-ai)

So an orthonormal basis in W would be simply w1/norm(w1) ...but the norm(w1)=0 (i^2 + 1 = 0)

What am I missing here? Does a basis for W satisfy that it has zero length? Thus it is just the origin. Then would all of C^3 be W_perp?

 

[Dick]

The complex inner product norm <x,y> is defined by (x*)^T y. You are forgetting the complex conjugate.

 

[gysush]

Ahh...I forgot to remember that a norm for F=C requires we take the complex conjugate of the 2nd vector.

You beat me to to it. :-)