Please give me a detailed solution and answer so that I can compare it with my solving process.

1. Let P denote the vector space of all polynomials and f'(x) denote the derivative of f E P. State "True" or "False" for each of the following statements. You do not need to justify your answers. (1) T : R2 -> R3 defined by T(x, y) = (72x, 0, y/2) is a linear transformation. (2) T : R2 -> R3 defined by T(x, y) = (xty, x-y, xy) is a linear transformation. (3) T : R2 -> R3 defined by T(x, y) = (1, 0, 0) is a linear trans- formation. (4) T : R2 -> R3 defined by T(x, y) = (y, x, y) is a linear trans- formation. (5) T : R2 -> R3 defined by T(x, y) = (x, x2, x3) is a linear transformation. (6) T : R2 -> R3 defined by T(x, y) = (2x + 3y, 3x + 4y, 4x + 5y) is a linear transformation. (7) T : P -+ P defined by T(f(x)) = f(0) + f'(0)x + f"(0)x2 is a linear transformation. (8) T : P -> P defined by T(f(x)) = f(1) + f'(2)x + f"(3)x2 is a linear transformation. (9) T : P -> P defined by T(f(x)) = f(x - 3) is a linear trans- formation. (10) T : P -> P defined by T(f(x)) = f(x) - 3 is a linear trans- formation. 2. Find a basis for the kernel of the linear transformation T : RS -> R3 defined by T(x) = Ax, where vectors in R' and R' are written as columns, A = 0 1 0 1 0 and Ax denotes matrix multiplication.3. Let Pr denote the vector space of all polynomials with degree less than or equal to k. Define a linear transformation T : PA -> P3 by T(f(x)) = f(0) +f'(1)(x -1) + f"(2)(x-2)2+ f"(3)(x -3)3. Find the matrix representation for T relative to the standard basis {1, x, x2, x3, x ] of PA and the reversed standard basis {23, x2, x, 1} of P3