Question

Show your work! No work, no credit! 1. Given T[c x, y, z >)-< x-z, y >. Complete the following a. Check if T is a linear transformation. Show your work! b. Find the domain and range of T. c. If T is a linear transformarion, find the matrix A that induced T. (6 points) 1)

Answer 1

T: R^2 + R^2 T(x, y, z) = (x-z, y) To Check whether T is Linear Transformation Let U, upsilon epsilon R^3 implies U = (x_1, y_1, z-1) v = (x_2, y_2, z_2) alpha, beta epsilon R alpha U + beta v = (alpha x_1 + beta x_2, alpha y_1 + beta y_2, alpha z_1 + beta z_2) T(alpha U + beta U) = (alpha x_1 + beta x_2) -(alpha z_1 + beta z_2) , alpha y_1 + beta y_2) = (alpha(x_1-z_1) + beta(x_2 - z_2) , alpha y_1 - beta y_2) = alpha(x_1 - z_1, y_1) + beta(x_2 - z_2, y_2) implies = alpha T(x_1, y_1, z_1) + beta T(x_2, y_2, z_2) T(alpha U + beta V) = alpha T(U) + beta T(V) Hence T is L.T \r\n To Find domain, Clearly where of R^3 is domain For Range apply T(e_1) = T(1, 0, 0) = (1, 0) T(e_2) = T(0, 1, 0) = (0, 1) T(e_3) = T(0, 0, 1) = (-1, 0) Now Form a matrix [1 0 -1 0 1 0] implies [1 0 0 0 1 0] implies basis of Ranse space is alpha (1, 0) , (0, 1) and dimension of Range space is 2 Hence