What is an inner product and how can it be verified for polynomials?

[bernoli123]

[-1]int[1]P(x)Q(x)dx P,Q\inS
verify that this is an inner product.

 

[henry_m]

What are the properties that define an inner product? You must show that your form satisfies all those properties.

 

[micromass]

Also, you need to give more information.
1) What is S?
2) This defines an inner product on which set?

 

[smith345]

what are the properties that can define the inner product? I know this only that
inner product is a generalization of the dot product. tell me more about this

 

[HallsofIvy]

If you really don't know what an inner product is, why are you asking this question?

An inner product, defined on a vector space, V, is a function that associates every pair of vectors, (u, v), a real number <u, v> satisfying
1) Linearity: <au+ bv, w>= a<u, w>+ b<v, w>
(If the vector space is over the complex numbers, a and b on the right should be their complex conjugates.)
2) <u, u>= 0 if and only if u= 0.
3) <u, u> is greater than 0 if u is not 0.

Here, you are defining <P, Q> to be int, from -1 to 1, P(x)Q(x)dx where P and Q are polynomials.

If int from -1 to 1 of P(x)^2 dx= 0, what can you say about P? What if P(x)= x^3?

 

[henry_m]

If you really don't know what an inner product is, why are you asking this question?

An inner product, defined on a vector space, V, is a function that associates every pair of vectors, (u, v), a real number <u, v> satisfying
1) Linearity: <au+ bv, w>= a<u, w>+ b<v, w>
(If the vector space is over the complex numbers, a and b on the right should be their complex conjugates.)
2) <u, u>= 0 if and only if u= 0.
3) <u, u> is greater than 0 if u is not 0.

Here, you are defining <P, Q> to be int, from -1 to 1, P(x)Q(x)dx where P and Q are polynomials.

If int from -1 to 1 of P(x)^2 dx= 0, what can you say about P? What if P(x)= x^3?

This question was from a new poster.
The wikipedia article on http://en.wikipedia.org/wiki/Inner_product_space" is fairly good. To get familiar with them, try taking some of the examples they give there, and proving that they are indeed inner product spaces. You should also try to prove that every finite-dimensional inner product space over the field \mathbb{F} is isomorphic to \mathbb{F}^n.