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October 10th, 2009, 01:57 PM   #1
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lagrange polynomials and polynomials...

Hi, I have a couple of questions:

Why is the dimension of Pn(F) n+1? is it because polynomials of degree n is a linear combination of x^n + ....... + x + C?

and if you have a lagrange interpolation polynomial, it's unique because the set of generation polynomials is an linearly independent set of n+1 elements, so is that implying that some interpolating polynomial is the only way a polynomial with a certain set of points will turn out?

thanks
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October 10th, 2009, 01:58 PM   #2
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Re: lagrange polynomials and polynomials...

and I'm wondering why the basis of polynomials couldn't be 2 elements, x and 1? since you could generate any degree polynomial with combinations of these.. but then I guess the reason is that you can't use x as a coefficient.. but I'm not sure why?

thanks
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October 13th, 2009, 12:03 PM   #3
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Re: lagrange polynomials and polynomials...

Quote:
Originally Posted by ElMarsh
and I'm wondering why the basis of polynomials couldn't be 2 elements, x and 1? since you could generate any degree polynomial with combinations of these.. but then I guess the reason is that you can't use x as a coefficient.. but I'm not sure why?

thanks
How do you get x^2 from a linear combination of x and 1? (Answer: you can't )
Yes-- x cannot be a coefficient: a coefficient must be a well-defined element of the field the vector space is over (at this point, the field is probably R).

Quote:
Originally Posted by ElMarsh
Why is the dimension of Pn(F) n+1? is it because polynomials of degree n is a linear combination of x^n + ....... + x + C?
Yes; except "C" should be 1, since a linear combination (read: polynomial) is


Quote:
and if you have a lagrange interpolation polynomial, it's unique because the set of generation polynomials is an linearly independent set of n+1 elements, so is that implying that some interpolating polynomial is the only way a polynomial with a certain set of points will turn out?
I think... my linear algebra is rusty, so I'm not entirely sure how lagrange interpolation polynomials work, but a polynomial of degree n is a linear combination of n+1 linearly independent vectors (namely, 1,x,x^2,...x^n). If you have n+1 points, for any m>n there will be more than one (in fact, infinitely many) polynomials of degree m that will work interpolate your points-- hence using a degree n polynomial for n+1 points (this guarantees a unique solution).
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October 15th, 2009, 04:14 PM   #4
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Re: lagrange polynomials and polynomials...

Quote:
Originally Posted by cknapp
Quote:
Originally Posted by ElMarsh
and I'm wondering why the basis of polynomials couldn't be 2 elements, x and 1? since you could generate any degree polynomial with combinations of these.. but then I guess the reason is that you can't use x as a coefficient.. but I'm not sure why?

thanks
How do you get x^2 from a linear combination of x and 1? (Answer: you can't )
Yes-- x cannot be a coefficient: a coefficient must be a well-defined element of the field the vector space is over (at this point, the field is probably R).

Quote:
Originally Posted by ElMarsh
Why is the dimension of Pn(F) n+1? is it because polynomials of degree n is a linear combination of x^n + ....... + x + C?
Yes; except "C" should be 1, since a linear combination (read: polynomial) is


Quote:
and if you have a lagrange interpolation polynomial, it's unique because the set of generation polynomials is an linearly independent set of n+1 elements, so is that implying that some interpolating polynomial is the only way a polynomial with a certain set of points will turn out?
I think... my linear algebra is rusty, so I'm not entirely sure how lagrange interpolation polynomials work, but a polynomial of degree n is a linear combination of n+1 linearly independent vectors (namely, 1,x,x^2,...x^n). If you have n+1 points, for any m>n there will be more than one (in fact, infinitely many) polynomials of degree m that will work interpolate your points-- hence using a degree n polynomial for n+1 points (this guarantees a unique solution).

thanks for the reply, I realized how lagrange polynomials worked after reading the text book over again.. I was just wondering about using "x" as a coefficient. I guess those aren't concretely defined on R

thanks
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