Question

Is there a linear map T : P2(R) ->R2 such that T(1) = (2, 3), T(1+x) = (-2, 7), and T(1+x+x^2) =
(0, 9)? Justify your answer.

Answer 1

Clearly $1, 1+x,1+x+x^2$ forms a basis of $P^2(\mathbb{R}).$ This follows from the fact that $x=1+x-1, x^2=1+x+x^2 -1-(1+x)$ and $1,x,x^2$ forms a basis. Now mapping any basis to any set of vectors gives a linear map. Hence the given map is linear. We need to extend the map by linearity on the entire space.