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Proving in math 
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#1
Aug2704, 09:41 AM

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Let's say I wanted to prove that, given n points, it takes a maximum of a (n1)th degree polynomial to represent them all. How would I do it? My instinct is to just say because you need a max of (n1) max/mins ...



#2
Aug2704, 10:18 AM

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what does represent mean in this context? and are you sure you mean "maximum"
what degree one polynimial "represents" the two points 0 and 1. and point in what space? R, R^2, R^3...? 


#3
Aug2704, 10:26 AM

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I mean that, given 1 point that your equation must touch, you need a 0th degree equation. Given 2 you need a 1st, etc...
For example, if you are given a set of points with 2 elements: (a,b), (c,d) You need a 1st degree equation, or line. y = mx + e The correct value of m and e will hit both points. Similarly, if you have 3 points, you need a quadratic. 


#4
Aug2704, 10:33 AM

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Proving in math
In that case, given n points in the plane with distinct x coords, there exists a degree n1 polynomial passing through them, since a degree n1 poly has n coefficients and therefore you have a system of n linearly independent equations in n unknowns to solve.
you don't mean maximum at all since given n points then there is a polynomial of degree r=>n1 passing through those points (again with distinct x values) which is unique when r=n1. 


#5
Aug2704, 10:36 AM

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Alkatran,
If u go through Lagrange Interpolation method, u would see how lagrange came up with an extremely simple way to do it! 


#6
Aug2704, 10:41 AM

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#7
Aug2704, 10:46 AM

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Lagrange Interpolation Method works for any given n points.
Hence Proved! 


#8
Aug2704, 10:51 AM

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The proof that the equations formed by substituting in the n points are linearly independent is called the vandermonde determinant.



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