My Math Forum lagrange polynomials and polynomials...

 Linear Algebra Linear Algebra Math Forum

 October 10th, 2009, 01:57 PM #1 Senior Member   Joined: Apr 2009 Posts: 201 Thanks: 0 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
 October 10th, 2009, 01:58 PM #2 Senior Member   Joined: Apr 2009 Posts: 201 Thanks: 0 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
October 13th, 2009, 12:03 PM   #3
Senior Member

Joined: Oct 2007
From: Chicago

Posts: 1,701
Thanks: 3

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 $\sum_{i=0}^n c_ix^i$

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).

October 15th, 2009, 04:14 PM   #4
Senior Member

Joined: Apr 2009

Posts: 201
Thanks: 0

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 $\sum_{i=0}^n c_ix^i$

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

 Tags lagrange, polynomials

,

### lagrange polynomial in c

Click on a term to search for related topics.
 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post chocolate Algebra 25 July 13th, 2012 07:10 AM omare702 Algebra 2 July 2nd, 2012 06:21 PM Tutu Algebra 1 June 26th, 2012 03:13 PM Dr.Nick Linear Algebra 7 May 23rd, 2011 05:49 PM SAT_frinzzy New Users 0 September 1st, 2008 07:08 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top