33. Let T: P₂→ P₂ be defined by T(ax² + bx + c) = ax^2 + (a-2b)x + b a. Determine whether p(x) = 2x^2 - 4x +6 is in the range of T. b. Find a basis for R(T).

a. To determine whether p(x) = 2x^2 - 4x + 6 is in the range of T, we need to find some polynomial q(x) = dx^2 + ex + f such that T(q(x)) = p(x).

T(q(x)) = T(dx^2 + ex + f) = dx^2 + (d-2e)x + e

Setting this equal to p(x), we get:

dx^2 + (d-2e)x + e = 2x^2 - 4x + 6

Comparing coefficients, we get the system of equations:

d = 2 d - 2e = -4 e = 6

Solving this system, we find that there are no solutions. Therefore, p(x) is not in the range of T.

b. To find a basis for R(T), we need to find the set of all possible outputs of T.

T(ax^2 + bx + c) = ax^2 + (a-2b)x + b

This can be rewritten as:

T(ax^2 + bx + c) = a(x^2 + x) + b(-2x + 1)

So, the set of all possible outputs of T is the span of the polynomials {x^2 + x, -2x + 1}. Therefore, a basis for R(T) is {x^2 + x, -2x + 1}.