Write the vector of P4 1 2x 2x2 x3 as a linear combinatio

Write the vector of P_4, 1 + 2x - 2x^2 + x^3 as a linear combination of the set S = {x^4, x^4 - x^3 + x^2,x^4 - x^3 + x^2 - x, x^3 - 1}.

Solution

LEt,

1+2x-2x^2+x^3=ax^4+b(x^4-x^3+x^2)+c(x^4-x^3+x^2-x)+d(x^3-1)

Comparing coefficients, first we get:d=-1 as x^3-1 is the only terms with a constant term

-2c=2

Hence, c=-1

a+b+c=0

-b-c+d=1

Hence, b=-c+d-1=1-1-1=-1

a+b+c=0

a-1-1=0

a=2

Hence,

1+2x-2x^2+x^3=2x^4-(x^4-x^3+x^2)-(x^4-x^3+x^2-x)-(x^3-1)