Final answer:
To find polynomials that span the kernel of the transformation T(p) = p(0), we need polynomials that evaluate to zero at x=0; p_1(x) = x and p_2(x) = x^2 span the kernel. The range of T is all real numbers R, as it includes the evaluation of any polynomial at x=0.
Step-by-step explanation:
The student's question involves defining a linear transformation T from the space P_2 of all polynomials of degree at most 2 to the Euclidean space R^2, where the transformation T is defined by T(p) = p(0). To find polynomials that span the kernel of T, we look for polynomials p such that T(p) = 0. Since T(p) evaluates polynomials at x = 0, any polynomial in the kernel must have a zero constant term. So, the polynomials p_1(x) = x and p_2(x) = x^2 would both evaluate to zero at x = 0 and thus are in the kernel of T. The range of T is simply the set of all possible outputs, which, since T evaluates polynomials at x = 0, is R (the real numbers). Therefore, the correct option is d) (p_1(x) = x, p_2(x) = x^2, Range of T = (R)).