Show that T is a linear transformation by finding a matrix t

Show that T is a linear transformation by finding a matrix that implements the mapping. T(x_1,x_2,x_3) = (x_1 - 3x_2 + 7x_3, x_2 -8X_3) A =

Solution

Let 1 = r cos 8, 1 = r sin 8, . \'. r2 = 1 + 1 = 2 or, r = v\'2 and tan 8 = 1 or, 8 = l Then 1 + i = rcos 8 + irsin8 = v\'2(cos ~ + i.sin ~), Hence one cube root of 1 + i is {v\'2(cos ~ + i sin ~ )}1/3 = 21/6(cos ~ + i sin ~) [by DeMoivre\'s theorem]