Simple Inner Product Clarification

[RJLiberator]

Homework Statement

I'm having trouble understanding the definition of a complex inner product.

Let λ ∈ ℂ
So if we have <λv|w> what does it equal to?
Does it equal λ*<v|w> where * is the complex conjugate?Are all these correct:
<λv|w> = λ*<v|w>
<v|λw> = λ<v|w>
<v|w> = (<w|v>)*

<v|w> = Σvw
<λv|w> = Σλ*vw
<v|λw> = Σλvw

λ(μ+α) = Σλ*α + Σλ*μ

Is there any other tough ones that you can present to me?

Homework Equations

All info is above
* = complex conjugate

The Attempt at a Solution

All info is above

 

[Ray Vickson]

Homework Statement

I'm having trouble understanding the definition of a complex inner product.

Let λ ∈ ℂ
So if we have <λv|w> what does it equal to?
Does it equal λ*<v|w> where * is the complex conjugate?Are all these correct:
<λv|w> = λ*<v|w>
<v|λw> = λ<v|w>
<v|w> = (<w|v>)*

<v|w> = Σvw
<λv|w> = Σλ*vw
<v|λw> = Σλvw

λ(μ+α) = Σλ*α + Σλ*μ

Is there any other tough ones that you can present to me?

Homework Equations

All info is above
* = complex conjugate

The Attempt at a Solution

All info is above

The first three are correct; the next three are incorrect; the last one makes no sense.

 

[RJLiberator]

The first three are correct; the next three are incorrect; the last one makes no sense.

I'm sorry for not writing out in full, let me try to explain.

There is a sum notation for inner products, correct?

 

[Ray Vickson]

I'm sorry for not writing out in full, let me try to explain.

There is a sum notation for inner products, correct?

Yes, there is. But what you wrote is incorrect.

Go back and review your course notes and/or the textbook material to see why, or failing that, just go back to previous discussions on similar topics in this forum, where all those issues were already thoroughly dealt with and you claimed at the time that you understood the material.

 

[RJLiberator]

I'm trying to find verification since my notes on summation representations are quite sloppy and the one area I am not 100% on is the summation representations. I was lazy in my initial thread here, let me try to present:

If we have vectors (v_1,...v_n) and (w_1,...,w_n) ∈ ℂ then the inner product between the two will result in the summation from i=1 to n of v_1*w_1 where * denotes complex conjugate

I have this ^^ written in notes. I do not believe this to be true. I don't see why the complex conjugate would exist there.

However, since <λv|w> = λ*<v|w> is true, we can see that in similar fashion, the sum from i=1 to n of this inner product would be λ_i*v_iw_i.

 

[Fredrik]

What you think of as a "summation representation" is just the option to write the vectors as linear combinations of basis vectors.

An n-tuple of vectors ##(e_1,\dots,e_n)## is called an ordered basis if the set ##\{e_1,\dots,e_n\}## is a basis. An ordered basis ##(e_1,\dots,e_n)## is said to be orthonormal if ##\{e_1,\dots,e_n\}## is an orthonormal set. If E is an orthonormal ordered basis and x is a vector, we can define the component n-tuple of x with respect to E as the unique n-tuple ##(x_1,\dots,x_n)\in\mathbb C^n## such that ##x=\sum_{i=1}^n x_i e_i##.

Using these definitions and your notation for inner products, we see that
$$\langle x|y\rangle =\bigg\langle\sum_{i=1}^n x_i e_i\bigg| \sum_{j=1}^n y_j e_j\bigg\rangle =\sum_{i=1}^n \sum_{j=1}^n x_i^* y_j\langle e_i|e_j\rangle =\sum_{i=1}^n x_i^* y_i.$$

 

[RJLiberator]

Ah, Fredrik! Beautiful explanation.

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