Consider the following subspaces of R³ and R³, respectively: =< ().() > 0 K > and R =< (¹) > Is there a linear transformation f: R³ → R² such that ker f = K and Im f = R? If yes, find a formula for f. If there is none, justify.

Let denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: 03 → be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) E 3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + aº and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in 3 and c E R. a. D(p(x) + q(x)) = D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = ? choose b. D(cp(x)) = c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c ER and all p(x) E 3? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.)

Prove that T(x1,x2,?3) = [x1,x2] is a linear transformation.

Find a linear functional ℓ on R3 that satisfies the following conditions:ℓ(1, 2, 2) = 1ℓ(1, 2, 3) = 2.