Finding expansion coefficients of a generic vector?

[RJLiberator]

Homework Statement

Find the expansion coefficients of a generic vector (λ, μ) ∈ ℂ^2 in the orthonormal basis:
{ i/sqrt(2) (1, 1), 1/sqrt(2) (1, -1) }

Homework Equations

λ_i = |v_i> |v>

The Attempt at a Solution

I don't think this should be difficult.
But, clearly, as I am posting here, I do not understand the terminology used yet.

So the question says "generic" vector, so I am assuming we choose some values say
λ = (x, y)
and then we simply say
λ_x = |x> |isqrt(2) (1,1)>

Which would be x*i/sqrt(2)

What am I doing wrong here? It can't be that straightforward?

 

[Mark44]

Homework Statement

Find the expansion coefficients of a generic vector (λ, μ) ∈ ℂ^2 in the orthonormal basis:
{ i/sqrt(2) (1, 1), 1/sqrt(2) (1, -1) }

In $\mathbb{R}^2$, the "usual" basis is {<1, 0>, <0, 1>}. I don't know what the "usual" basis is in $\mathbb{C}^2$, but I'm fairly sure that the generic vector (λ, μ) is in terms of the usual basis in $\mathbb{C}^2$ (whatever it is). IOW, (λ, μ) = λc₁ + μc₂, with the c_i being the basis vectors.

I believe what this problem is asking you to do is to find the coefficients of (λ, μ) in terms of the vectors in your orthonormal basis. That is, find the coefficients (a₁, a₂) such that (λ, μ) = a_a * i/sqrt(2) (1, 1) + a₂ * 1/sqrt(2) (1, -1).

Homework Equations

λ_i = |v_i> |v>

The Attempt at a Solution

I don't think this should be difficult.
But, clearly, as I am posting here, I do not understand the terminology used yet.

So the question says "generic" vector, so I am assuming we choose some values say
λ = (x, y)
and then we simply say
λ_x = |x> |isqrt(2) (1,1)>

Which would be x*i/sqrt(2)

What am I doing wrong here? It can't be that straightforward?

 

[RJLiberator]

Ugh, I just can't get it for the life of me.
Any blatant examples around? With one example, I'm sure it is as easy as clockwork.

λ_i = |v_i> |v> is the formula that should be used.

I'm guessing |v> is the spot for the orthonormal basis given.
But that still leaves me two unknowns, the one I'm trying to find, and the |v_i>.

 

[RJLiberator]

Update:

Any help on this problem?
Best we have:
(λ, μ) = a_a*i/sqrt(2) (1, 1) + a_2*1/sqrt(2) (1, -1).

I'm nearly positive that I have to use the formula that
λ_i = |v_i> |v>

In some respect.

I cannot put the pieces of the puzzle together.

 

[RJLiberator]

Aha! I think I may have solved it. OR perhaps not...

Question: Find the expansion coefficients of a generic vector \binom{λ}{μ}∈ℂ^2 in the orthonormal basis ((\binom{i/\sqrt{2}}{i/\sqrt{2}},\binom{1/\sqrt{2}}{-1/\sqrt{2}}).

So I set it up as such:

\binom{λ}{μ} = a_1\binom{i/\sqrt{2}}{i/\sqrt{2}}+a_2\binom{1/\sqrt{2}}{-1/\sqrt{2}}

where a_1, a_2 ∈ ℂ.

I did this due to what was in my notes and the hints giving in this thread.

Now, with a bit of algebra, it is clear to see that

λ = \frac{ia_1+a_2}{\sqrt{2}}
and
μ = \frac{ia_1-a_2}{\sqrt{2}}

And here we have the expansion coefficients of a generic vector.

1. Is this correct?
2. Is it correct to state that a_1 and a_2 exist in the complex numbers, or must they be real numbers as scalars?

 

[Mark44]

Aha! I think I may have solved it. OR perhaps not...

Question: Find the expansion coefficients of a generic vector \binom{λ}{μ}∈ℂ^2 in the orthonormal basis ((\binom{i/\sqrt{2}}{i/\sqrt{2}},\binom{1/\sqrt{2}}{-1/\sqrt{2}}).

So I set it up as such:

\binom{λ}{μ} = a_1\binom{i/\sqrt{2}}{i/\sqrt{2}}+a_2\binom{1/\sqrt{2}}{-1/\sqrt{2}}

where a_1, a_2 ∈ ℂ.

I did this due to what was in my notes and the hints giving in this thread.

Now, with a bit of algebra, it is clear to see that

λ = \frac{ia_1+a_2}{\sqrt{2}}
and
μ = \frac{ia_1-a_2}{\sqrt{2}}

You haven't really done anything here. The expansion coefficients are the multipliers of the vectors in your base so that you get $\binom{λ}{μ}$. So all you've done is write out the question in mathematical form. What you need to do, and what I said in post #2, is find formulas for $a_1$ and $a_2$, for given values of λ and μ.

And here we have the expansion coefficients of a generic vector.

Nope

1. Is this correct?
2. Is it correct to state that a_1 and a_2 exist in the complex numbers, or must they be real numbers as scalars?

The vector space is $\mathbb{C}^2$ and the field (of scalars) is $\mathbb{C}$.

 

[RJLiberator]

What you need to do, and what I said in post #2, is find formulas for a1 and a2, for given values of λ and μ.

So, what you are saying is that I have to find a_1 and a_2 such that the equations that I found yield λ and υ respectively?

 

[RJLiberator]

Then from the equations
a_2 = λ*sqrt(2)-μ*sqrt(2)
a_1 = -μ*i*sqrt(2)

Now we see what the expansion coefficients are for any vectors. So, we can think of vector λ=2, μ=1, and can find coefficient expansions.EDIT: I'm not sure if this works, when I choose λ = 2 and μ = 1, and plug it into

\binom{λ}{μ} = a_1\binom{i/\sqrt{2}}{i/\sqrt{2}}+a_2\binom{1/\sqrt{2}}{-1/\sqrt{2}}

With a_1 and a_2 the above coefficients.
I get whacky answers.

It seems as if the first part works out well, If we are looking for λ = 2 then certainly -2*i*sqrt(2) * i/(sqrt(2)) = 2.
However, I assume you have to add the second part of the equation to it, which is -sqrt(2).

:/

 

[RJLiberator]

Any last minute help on this problem gentlemen?
I will turn this in in a few hours, you all have been a wonderful help. (Props to you Mark44)

Some questions I could use help on:

1) Is my method here mostly correct? Is there an algebraic mistake, or is my method not right?

2) If it is my method, do you have any suggestions on a new method?
This made sense to me when Mark44 told me to find formulas for a_1 and a_2. I thought I had done that in my previous post, but they don't seem to be working in the big scheme of things. This could be an algebraic mistake OR it could be not understanding how to use everything properly.

 

[Mark44]

So, what you are saying is that I have to find a_1 and a_2 such that the equations that I found yield λ and υ respectively?

Yes, that's what you need to do.

Any last minute help on this problem gentlemen?
I will turn this in in a few hours, you all have been a wonderful help. (Props to you Mark44)

Some questions I could use help on:

1) Is my method here mostly correct? Is there an algebraic mistake, or is my method not right?

2) If it is my method, do you have any suggestions on a new method?

The method is correct, and you're doing well to make up an example to verify that you're getting the right values. If you're not getting the right values, it's likely due to a mistake in the algebra.

This made sense to me when Mark44 told me to find formulas for a_1 and a_2. I thought I had done that in my previous post, but they don't seem to be working in the big scheme of things. This could be an algebraic mistake OR it could be not understanding how to use everything properly.

 

[RJLiberator]

Indeed, it was a mistake in the algebra.

It turned out to be very close to the right result, I appreciate you taking the time to walk me through some steps. I know understand the problem, for the most part.