Is integration of polynomial a bilinear for

In summary, to prove that \int f(x)g(x)dx is bilinear, you need to show that it satisfies the definition of "bilinear" by checking if the two given equations hold. Also, to show that it is non-degenerate, you need to determine the definition of "non-degenerate" and prove it for this specific bilinear form. Additionally, someone may be able to explain the significance of kroenecker delta.
  • #1
jut24
1
0
Hello,

I am begin billinear form and need help with a proof

say you have an integral from 0 to 1 f(x)g(x) is it bilinear if show how do you prove that it is.
2) Can someone explain the significance of kroenecker delta.



Jut24
 
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  • #2
jut24 said:
Hello,

I am begin billinear form and need help with a proof

say you have an integral from 0 to 1 f(x)g(x) is it bilinear if show how do you prove that it is.
Directly.
 
  • #3
In other words, you prove that [itex]\int f(x)g(x)dx[/itex] is bilinear by showing that it satisfies the definition of "bilinear":

Is [itex]\int_0^1 (af(x)+ bg(x))h(x)dx= a\int_0^1 f(x)h(x)dx+ b\int_0^1 g(x)h(x)dx[/itex]?
Is [itex]\int_0^1 f(x)(ag(x)+ bh(x))dx= a\int_0^1 f(x)g(x)dx+ b\int_0^1 f(x)h(x)dx[/itex]?
 
  • #4
on this topic:

this bilinear form is non-degenerate right?

I'm not exactly sure how to show that though...
 
  • #5
Well, what is the definition of "non-degenerate"?
 

1. Is integration of polynomial a bilinear form?

No, integration of polynomial is not a bilinear form. Bilinear forms are defined as functions that are linear in each of their two arguments. Integration of polynomial is a single variable operation and does not involve two arguments.

2. What is the definition of a bilinear form in integration of polynomial?

A bilinear form in integration of polynomial would refer to a function that is linear in each of its two arguments, which in this case would be the polynomial itself and the variable of integration.

3. Can a polynomial function be integrated bilinearly?

No, a polynomial function cannot be integrated bilinearly. Integration is a single variable operation and does not involve two arguments, thus it cannot be considered as a bilinear form.

4. How is integration of polynomial different from a bilinear form?

The main difference between integration of polynomial and a bilinear form is that integration is a single variable operation, while a bilinear form is defined as a function that is linear in each of its two arguments. Additionally, integration is a mathematical operation while a bilinear form is a type of function.

5. Are there any applications of bilinear forms in integration of polynomial?

No, there are no direct applications of bilinear forms in integration of polynomial. Bilinear forms are commonly used in linear algebra and optimization problems, while integration of polynomial is used in calculus and other fields of mathematics.

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