T : P2(R) → P3(R) defined by T(f(x)) = xf(x) + f′(x)

Let T be the linear transformation whose standard matrix is given. Decide if T is a one-to-one mapping. Justify your answer. Choose the correct answer below. -7 6 9 5 2 11 -5 2 -6 - 3 - 1 10 3-15 9-25 O A. The transformation T is one-to-one because the equation T(x) = 0 has only the trivial solution. B. The transformation T is one-to-one because the equation T(x) = 0 has a nontrivial solution. O C. The transformation T is not one-to-one because the equation T(x) = 0 has a nontrivial solution. O D. The transformation T is not one-to-one because the equation T(x) = 0 has only the trivial solution.

Please box answers and write it in the same format of the question.

Let T be the linear transformation represented by the matrix (The matrix is shown in the image). Find the kernel and image of this transformation. Does the dimension theorem hold? It is this invertible linear transformation, if so find the inverse linear transformation. Note: In the image the exercise is clearer, do not skip any step to arrive at the result.