December 9th, 2016, 11:50 AM  #1 
Newbie Joined: Dec 2016 From: Natal  Brazil Posts: 10 Thanks: 0  Linear combination
Let S = {p1, p2, p3} be the set formed by the polynomials p1 (x) = 1 + x  x ^ 2 + x ^ 3, p2 (x) = 1 + x ^ 2 + x ^ 3 and p3 (x ) = 1  x + x ^ 2. Find p (x) = 1 + 3x  17x ^ 2  5x ^ 3 as a linear combination of S. The coordinate vector of p (x) written on base S is?

December 9th, 2016, 04:35 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,646 Thanks: 836 
let $bx = (1,~x,~x^2,~x^3)$ $S = bx \begin{pmatrix} 1 &1 &1 \\ 1 &0 &1 \\ 1 &1 &1 \\ 1 &1 &0 \end{pmatrix}$ $p = bx\cdot (1, 3, 17, 5)$ we are looking for a coefficient matrix $c$ such that $Sc = p$ $ c = bx \cdot (1, 3, 17, 5)$ $bx \begin{pmatrix} 1 &1 &1 \\ 1 &0 &1 \\ 1 &1 &1 \\ 1 &1 &0 \end{pmatrix} c = bx \cdot (1, 3, 17, 5)$ $\begin{pmatrix} 1 &1 &1 \\ 1 &0 &1 \\ 1 &1 &1 \\ 1 &1 &0 \end{pmatrix} c = (1, 3, 17, 5)$ you can solve this matrix equation in the usual fashion. I leave it to you so show that $c = (9, 14, 6)$ and thus $p =1+3x17x^25x^3= 9(1 + x  x^2 + x^3)  14(1 + x^2 + x^3) + 6(1  x + x^2)$ the coordinate vector of p with respect to the basis S should be pretty obvious 
December 10th, 2016, 07:07 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,872 Thanks: 766 
Equivalently: You want numbers a, b, and c such that p= ap1+ bp2+ cp3 That is, you want 1+ 3x 17x^2 5x^3= a(1+ x x^2+ x^3)+ b(1+ x^2+ x^3)+ c(1 x+ x^2) = a+ ax ax^2+ ax^3+ b+ bx^2+ bx^3+ c cx+ cx^2 = (a+ b+ c)+ (a c)x+ (a+ b+ c)x^2+ (a+ b)x^3 In order that two polynomials be the same for all x, the "corresponding coefficient" must be the same so you must have a+ b+ c= 1 a c= 3 a+ b+ c= 17 a+ b= 5 That is four equations for only three unknowns it might be that this vector, p, is not in the three dimensional subspace of the four dimensional space of polynomials of order three or less spanned by these three vectors. (Notice that the second equation can be written c= a 3 and the fourth as b= a 5. Put those into the first and third equations.) p1 (x) = 1 + x  x ^ 2 + x ^ 3, p2 (x) = 1 + x ^ 2 + x ^ 3 and p3 (x ) = 1  x + x ^ 2. Find p (x) = 1 + 3x  17x ^ 2  5x ^ 3 

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base, combination, linear 
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