Let V be the vector space of real polynomials of degree <=2. Find?

Question

the coordinates of the polynomial f(t)=2-t+t^2 relative to the basis B={1, 1+t, 1+t^2}.

Answer 1

f(t) = 2 - t + t²

f(t) = a(1) + b(1 + t) + c(1 + t²) = 2 - t + t²

2 - t + t² = 2 - 1 - t + 1 + t² = 2 - (1 + t) + (1 + t²)

a = 2
b = -1
c = 1

The coordinates relative to basis B are (2, -1, 1).

Answer 2

2-t+t^2=1-t+(1+t^2)=2-1-t+(1+t^2)
=2-(1+t)+(1+t^2)
so the coordinates are (2,-1,1)