Prove Polynomials Can be Written Using the Dimension Theorem

In summary: Can you please clarify what you need help with regarding the dimension theorem and polynomial notation?In summary, the dimension theorem states that every polynomial p(x) in Pn can be written in the form p(x)=q(x+1)-q(x) for some polynomial q(x) in Pn+1. To understand how to do this, start by trying the cases n=1 and n=2. Then, consider if h(x) = q(x+1) is a polynomial in x and determine its degree and x-coefficients. It is important to follow the rules of this forum and attempt to solve the problem yourself before seeking help.
  • #1
linalg
4
0
Use the dimension theorem to show that every polynomial p(x) in Pn can be written in the form p(x)=q(x+1)-q(x) for some polynomial q(x) in Pn+1.

I need to see all the steps so that I understand how to do it.
PLease and Thank you
 
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  • #2


linalg said:
Use the dimension theorem to show that every polynomial p(x) in Pn can be written in the form p(x)=q(x+1)-q(x) for some polynomial q(x) in Pn+1.

I need to see all the steps so that I understand how to do it.
PLease and Thank you
Hello linalg. Welcome to PF !

What have you tried ?

Where are you stuck?


Please look at the rules for this Forum. We won't do your work for you but will try tu help you arrive the solution.
 
  • #3


I am actually very confused on where to start. I tried poving if it is linear independent and then kinda got lost.
Please help
 
  • #4


Try the cases n=1 and n=2 first. That will show you what to do in general.

RGV
 
  • #5


Thanks for your guys' help. I tried the two cases n=1 and n=2 but i am still lost. Its a homework question due tomorrow that I don't need to understand how to do it until the test in a couple weeks.
 
  • #6


If q(x) is a polynomial of degree 2, is h(x) = q(x+1) a polynomial in x? What is its degree? What are its x-coefficients?

RGV
 
  • #7


linalg said:
I need to see all the steps so that I understand how to do it.
This isn't how it words here at Physics Forums.
 
  • #8


Ok Mark44 just answer the question then
 
  • #9


linalg said:
Ok Mark44 just answer the question then
What question? You have not asked a single question.
 

Related to Prove Polynomials Can be Written Using the Dimension Theorem

What is the Dimension Theorem?

The Dimension Theorem, also known as the Dimension Formula, is a mathematical theorem that states the dimension of any vector space is equal to the number of vectors in a basis for that space.

How does the Dimension Theorem relate to polynomials?

The Dimension Theorem can be used to prove that polynomials can be written using a basis. This means that any polynomial can be expressed as a linear combination of a finite number of basis polynomials.

What is a basis for polynomials?

A basis for polynomials is a set of polynomials that can be used to create any other polynomial through linear combinations. A common basis for polynomials is the set of all monomials of a certain degree.

How is the Dimension Theorem used to prove that polynomials can be written using a basis?

The Dimension Theorem states that the dimension of a vector space is equal to the number of vectors in the basis. Since the space of polynomials of degree n has dimension n+1, this means that any polynomial of degree n can be expressed as a linear combination of n+1 basis polynomials. Therefore, polynomials can be written using a basis.

Can the Dimension Theorem be applied to other types of mathematical objects?

Yes, the Dimension Theorem can be applied to any vector space, not just polynomials. It can also be used to prove the existence of a basis for any vector space.

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