Verification of an Inner Product

I should be able to show that it's positive even with the 3 in there, but I'm having a lot of trouble deriving that.I don't know what you mean by "positive coefficient, 3". You are not proving anything unless you actually prove it. If you don't know how to prove it, then maybe you should assume it as given, and continue to use it as a hypothesis for other results that you are trying to prove. But I don't know if you can use it as given, because it is not clear what you are doing in other places.In summary, the given exercise is to show that the expression ##\langle {\bf u}, {\bf v} \rangle = 3u_1v_1
  • #1
Parmenides
37
0
My analysis instructor has posed an exercise to me in the following format:

"For [tex]{\bf{u}} = (u_1,u_2), {\bf{v}} = (v_1,v_2) \in R^2[/tex] define [tex]\left\langle{\bf{u}},{\bf{v}}\right\rangle = 3u_1u_2 - u_1v_2 - u_2v_1 + \frac{1}{2}u_2v_2..[/tex]
Show that this is an inner product on ##R^2##."

Not sure what the dots at the end represent, but I'm just posting it exactly as it appears. The textbook we're using (Advanced Calculus, Edwards) gives the criteria for an inner product as
1)[tex]\left\langle{\bf{x}},{\bf{x}}\right\rangle > 0, {\bf{x}} \neq 0 [/tex]
2) [tex]\left\langle{\bf{x}},{\bf{y}}\right\rangle = \left\langle{\bf{y}},{\bf{x}}\right\rangle[/tex]
3) [tex]\left\langle{a}{\bf{x}} + b{\bf{y}},{\bf{z}}\right\rangle = a\left\langle{\bf{x}},{\bf{z}}\right\rangle + b\left\langle{\bf{y}},{\bf{z}}\right\rangle[/tex]

I assume proving these would be sufficient, since this course doesn't deal with complex space. Thus, my attempt at a solution is:
1) [tex]\left\langle{\bf{u}},{\bf{u}}\right\rangle = 3u_1u_2 - u_1u_2 - u_2u_1 + \frac{1}{2}{u_2}^2 = u_1u_2 + \frac{1}{2}{u_2}^2[/tex]
Which is non-zero if ##{\bf{u}}## is non-zero. Next,
2) [tex]\left\langle{\bf{v}},{\bf{u}}\right\rangle = 3v_1v_2 - v_1u_2 - v_2u_1 + \frac{1}{2}v_2u_2 = \left\langle{\bf{u}},{\bf{v}}\right\rangle[/tex]
Proving property 3 seems to be giving me trouble. I believe I should begin by letting: [tex]{\bf{w}} = (w_1,w_2)[/tex]
Then, [tex]\left\langle{a}{\bf{u}} + b{\bf{v}},{\bf{w}}\right\rangle = 3(au_1 + bv_1)(au_2 + bv_2) - (au_1 + bv_1)w_2 - (au_2 + bv_2)w_1 + \frac{1}{2}(au_2 + bv_2)w_2[/tex]
The first term seems to pose a problem in rewriting in the desired form because I'm getting a quadratic-like form with the constants. Other than that, that's the only way I see how to show this. Any thoughts? Much appreciated.
 
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  • #2
Looking back, I think I've messed up proving property 2 and am now lost on that one, too.
 
  • #3
I didn't even look at the third property because the firs and the second don't add up.
In the second as you sadi yourself what you wrote doesn't make sense.
but even more problematic is the fact that the first condition isn't met, and that there is no way to make it work.
As you wrote yourself condition is that an inner product of a vector with itself shouldn't only be non zero, it should also be definitely positive for each non-zero vector.

This term is interlinked with the way we define metrics, and as you perhaps know, due to this condition we can say that an inner product defines a norm over the space considered.

As you may know a norm is a non-negative operator and therefore your supposed innner product isn't an inner product at all.
 
  • #4
Parmenides said:
1) [tex]\left\langle{\bf{u}},{\bf{u}}\right\rangle = 3u_1u_2 - u_1u_2 - u_2u_1 + \frac{1}{2}{u_2}^2 = u_1u_2 + \frac{1}{2}{u_2}^2[/tex]
Which is non-zero if ##{\bf{u}}## is non-zero.
That's not true. ##\langle(1,0), (1,0)\rangle = 0##

Even if it were, you actually need to show that the inner product of a non-zero vector with itself is positive as simba_lk noted, but you have, for example, ##\langle(-1,1), (-1,1)\rangle = -1/2##.
 
  • #5
Correction; my instructor made a typo. The ##u_2## in the term ##3u_1u_2## should be a ##v_1## in which case I think all three properties come together easily:
[tex]\left\langle{\bf{u}},{\bf{v}}\right\rangle = 3u_1v_1 - u_1v_2 - u_2v_1 + \frac{1}{2}u_2v_2[/tex]
So that:
1) [tex]\left\langle{\bf{u}},{\bf{u}}\right\rangle = 3{u_1}^2 - 2u_1u_2 + \frac{1}{2}{u_2}^2[/tex]
2) [tex]\left\langle{\bf{v}},{\bf{u}}\right\rangle = 3v_1u_1 - v_1u_2 - v_2u_1 + \frac{1}{2}v_2u_2 = \left\langle{\bf{u}},{\bf{v}}\right\rangle[/tex]
3) Let ##{\bf{w}} = (w_1,w_2)## so that:
[tex]\left\langle{a}{\bf{u}} + b{\bf{v}},{\bf{w}}\right\rangle = 3(au_1 + bv_1)w_1 -(au_1 + bv_1)w_2 - (au_2 + bv_2)w_1 + \frac{1}{2}(au_2 + bv_2)w_2 = (3au_1w_1 - au_1w_2 - au_2w_1 + \frac{1}{2}au_2w_2) + (3bv_1w_1 - bv_1w_2 - bv_2w_1 + \frac{1}{2}bv_2w2) = a\left\langle{\bf{u}},{\bf{w}}\right\rangle + b\left\langle{\bf{v}},{\bf{w}}\right\rangle[/tex]
 
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  • #6
Did you prove that the expression you got in #1 is positive?
 
  • #7
I rested on the theorem of quadratic forms that it is positive-definite due to the positive coefficient, 3. The proof of *that* escapes me, however.
 

1. What is an inner product?

An inner product is a mathematical operation that takes two vectors and returns a scalar value. It is often denoted by ⟨u, v⟩ and is a generalization of the dot product. The result of an inner product is a measure of the similarity between the two vectors.

2. How is the inner product verified?

The inner product can be verified by checking if it satisfies the four properties of linearity, symmetry, positive definiteness, and conjugate symmetry. These properties ensure that the inner product operation produces a valid result and follows certain rules.

3. What is the importance of verifying an inner product?

Verifying an inner product is important because it ensures that the operation is valid and follows established rules. This allows for the use of inner products in various mathematical and scientific applications, such as in vector spaces, optimization problems, and quantum mechanics.

4. Can an inner product be verified for any type of vector?

Yes, an inner product can be verified for any type of vector as long as it satisfies the four properties mentioned earlier. This includes real-valued vectors, complex-valued vectors, and even abstract vectors in higher dimensions.

5. What are some examples of inner product verification?

Some examples of inner product verification include verifying the dot product of two vectors in Euclidean space, verifying the inner product of two wave functions in quantum mechanics, and verifying the inner product of two polynomials in abstract vector spaces.

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