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

1. Sep 12, 2011

### max_jammer

Hello.

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 H1 which is spanned with

|Psi> = a x1 + b x2 + c x3
and
|Psi'> = d x1 + e x2 + f x3

Those two "rays" are linearly independent and x1, x2, and x3 is an (orthonormal) basis for H.

Now I need to find coefficients g, h and i so that
|psi_othogonal> = g x1 + h x2 + i x3
is not a trivial element of the subspace H1_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 R3 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

2. Sep 12, 2011

### vela

Staff Emeritus
Try calculating the scalar products and see what you get. If they're 0, you have your answer.

3. Sep 14, 2011

### max_jammer

Thanks for the replay.

I tried to calculate the scalar product but it doesn't work; it all boils down to where I put conjugate. if I put conjugate on

g = (~bf - ~ce)
h = (~cd - ~af)
i = (~ae - ~bd)

then the scalar product of this vector and |psi> is zero BUT
it is not zero with |psi'>...

Or am I missing something?

~z w is generally not equal to z ~w ?

any help is appreciated...

4. Sep 14, 2011

### vela

Staff Emeritus
Try going the other way. Set the inner products to 0 and to get two equations involving g, h, and i, which you can solve up to a multiplicative constant.

5. Sep 14, 2011

### max_jammer

I did that before and I got expressions for g and h with i as a parameter. (1 equations and 3 unknowns - no surprise there) But I have no idea what i might be. If I were dealing with real numbers I would just set i = 1 and normalize the (g h i) vector, but these are complex numbers and I don't think I can do the same thing.

anyway I got:

g = i (b~ f~ - c~ e~) / (a~ e~ - b~ d~)
and
h = (-d~ g - f~ i) / e~

what am I missing? either my complex algebra is too rusty or there is some fundamental physical relations I didn't think of.

Please help me; this has been bugging me for a week now.

6. Sep 14, 2011

### vela

Staff Emeritus
Why not? You don't actually have to set i to 1. You could just normalize the vector, which will allow you to solve for |i|. You still have the freedom to set an arbitrary complex phase, which you can take to be 0.