Let P2 a0 a1x a2x2 a0 a1 a2 elements of Ropf be the vect

Let P_2 = {a_0 + a_1x + a_2x^2 | a_0, a_1, a_2 elements of Ropf} be the vector space of polynomials with real coefficients and of degree at most 2. Consider elements f_1 = 1 - x + x^2, f_2 = 1 + x^2, f_3 = 1 - x of this vector space. Show that {f_1, f_2, f_3} is a basis of P_2. Write an arbitrary element a_0 + a_1x + a_2x^2 of P_2 as a linear combination of f_1, f_2 and f_3. That is, find the coordinates of a_0 + a_1x + a_2x^2 with respect to the basis {f_1, f_2, f_3}.

Solution

a)

If we can show that standard basis vectors belng to span of {f1,f2,f3} then we are done

f2-f1=x

f1-f3=x^2

f2+f3-f1=1

HEnce proved

b)

a0+a1x+a2x^2

=a0*(f2+f3-f1)+a1(f2-f1)+a2(f1-f3)

=(a0+a1)f2+(-a0-a1+a2)f1+(a0-a2)f3