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I really need help with this one:

1. The problem statement, all variables and given/known data

I have a 3 dimensional state space H and its subspace H_{1}which is spanned with

|Psi> = a x_{1}+ b x_{2}+ c x_{3}

and

|Psi'> = d x_{1}+ e x_{2}+ f x_{3}

Those two "rays" are linearly independent and x_{1}, x_{2}, and x_{3}is an (orthonormal) basis for H.

Now I need to find coefficients g, h and i so that

|psi_othogonal> = g x_{1}+ h x_{2}+ i x_{3}

is not a trivial element of the subspace H_{1}_orthogonal.

2. Relevant equations

not sure

3. The attempt at a solution

I think that I need to find another element in H which is orthogonal both to |Psi> and |Psi'>

In R^{3}I would normally use cross product to find the third base vector but how does this transforms to the complex valued coefficients?

I think it may be:

g = (bf~ - ce~)

h = (cd~ - af~)

i = (ae~ - bd~)

where f~ means complex conjugate of f.

but is this right?

Also how would I go about prooving this? What pops in my mind is to build scalar products and see if they give 0...

Thanks alot

/Nathan

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# Homework Help: State space and its subspace : finding a basis

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