Linear Algebra Find image rank kernel and nullity of the tra

Linear Algebra

Find image, rank, kernel, and nullity of the transformation in Exercise 22. T(f(t)) = integral_-2^3 f(t) dt from P_2 to R Do the polynomials f(t) = 1 + 2t + 9t^2 + t^3, g(t) = 1 + 7t + 7t^3, h(t) = 1 + 8t + t^2 + 5t^3, and k(t) = 1 + 8t + 4t^2 + 8t^3 form, a basis of P_3? In Exercises 5 through 40, find the tin- ear transformation T with respect to the basis. If no basis is specified, use the standard basis: for P_2.U = ([1 0 0 0], [0 1 0 0], [0 0 1n 0], [0 0 0 1]) for R^2 times 2, and U = (1, i) for C. For the space of upper triangular 2 times 2 matrices, use the basis U = ([1 0 0 0], [0 1 0 0], [0 0 0 1]) unless another basis is given. In each case, determine whether T is an isomorphism. If T isn\'t an isomorphism, find bases of the kernel and image of T, and thus determine the rank of T. T (M) = [1 1 2 2] M from R^2 times 2 to R^2 times 2, with respect to the basis B = ([1 0 -1 0], [0 1 0 -1], [1 0 2 0], [0 1 0 2])

Solution

(54)

Image of T: All continuous function f such that f has continuous first derivative.

Nullspace of T or Kernel of T: trivial.

Rank of T: infinite

Nullity of T: zero