Linear Algebra First show that the set S p1 p2 p3 is a basi

Linear Algebra

First show that the set S = {p1, p2, p3} is a basis for p2, then express p as a linear combination of the vectors in S, and then find the coordinate vector of p relative to S.

p1 = 1+2x+x^2; p2 =2+9x; p3 = 3+3x+4x^2; p = 2+17x-3x^2

Solution

p1 = 1+2x+x^2;

p2 =2+9x;

p3 = 3+3x+4x^2;

p = 2+17x-3x^2

first we need to show S = {p1, p2, p3} is a basis for p2

Now take 4p1 -p3 + p2 = 4+8x+4x^2 - 3 - 3x - 4x^2 + 2 + 9x

= 3 + 14x is the form of p2

so we can say S = {p1, p2, p3} is a basis for p2

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express p as a linear combination of the vectors in S

take a, b, and c

a*p1 + b*p2 + c*p3 = p

a*(1+2x+x^2) + b*(2+9x) + c*(3+3x+4x^2) = 2+17x-3x^2

Now compare coefficient

a + 2b + 3c = 2 ..................(1)

2a + 9b + 3c = 17...................(2)

a+ 4c = -3.......................(3)

Now after solving these equation we got

a = 1

b = 2

c = -1

coordinate vector of p relative to S is [1, 2, -1]..............................Answer

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