questions below:

1.)

Matrix A is factored in the form PDP '. Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace. 2 0 -24 -6 0 -1 6 0 0 0 0 1 A = 12 6 72 0 1 3 0 6 0 3 1 18 0 0 1 0 0 0 0 2 -10 -6 Select the correct choice below and fill in the answer boxes to complete your choice. (Use a comma to separate vectors as needed.) A. There is one distinct eigenvalue, 2 = . A basis for the corresponding eigenspace is O B. In ascending order, the two distinct eigenvalues are My = and 12 = . Bases for the corresponding eigenspaces are { } and { }, respectively. O C. In ascending order, the three distinct eigenvalues are My = , 2= , and 23 = . Bases for the corresponding eigenspaces are { }, { }, and { }, respectively.Define T:P2-RS as shown to the right. p( - 3) a. Find the image under T of p(t) = - 3 - 2t. T(p) = P(0) b. Show that T is a linear transformation. p(3) c. Find the matrix for T relative to the basis B= {b1, b2, b3} = {1, t, to) for P2 and the standard basis E = fe1, e2, e3) for RS. a. The image under T of p(t) = - 3 - 2t is b. Let p(t) and q(t) be polynomials in P2. Show that T(p(t) + q(t)) = T(p(t)) + T(q(t)). First apply the definition of T. T(p(t) + q(t)) = Next apply the definition of (p + q)(t). What is the result? p(3) + q(3) p( - 3) + q(-3) O A. p(0) + q(0) O B. p(0) + q(0) p( -3) + q( -3) p(3) + q(3) p( - 3) + q(3) p(t) + q(t) O C. p(0) + q(0) O D. p(t) + q(t) p(3) + q( -3) p(t) + q(t)Rewrite this as the sum of two vectors. What is the result? p( - 3) q(3) p (t ) q(t) O A. p(0) + q(0) O B. p(t) + q(t) p(3) q( -3) p(t) q(t) p( - 3) q( -3) p(3) q(3) O C. p(0) + q(0) O D. p(0) + q(0) p(3) q(3) p(-3) q(-3) Now apply the definition of T again. What is the result? O A. p(t) + T(q(t)) O B. T(p(t)) + q(t) O c. T(p(t)) + T(q(t))Let p(t) be a polynomial in P2 and let c be a scalar. Show that T(c . p(t)) = c . T(p(t)). First apply the definition of T. T(c . p(t)) = Next apply the definition of (c . p)(t). What is the result? c . p( -3) c . p(3) O A. C . p(0) c . p(t) O B. c . P(0) c . p(3) O c. c . p(t) c . p( -3) c . p(t) Remove a common factor from this vector. What is the result? p(3) p(t) O A. C. p(0) p( -3) OB. C. p(t) p( -3) O C. C. p(0) p(t) p(3)Now apply the definition of T again, thus completing the proof that T is a linear transformation. What is the result? O A. C . T(p(t)) O B. T(p(t)) + c O C. T(p(t)) c. The matrix for T relative to B and E is