Kunze Linear Algebra · Problem 3.4.7

Linear Transformations

Mathematics — Methods · matrix-in-different-bases-r3

Problem

Let TT be the linear operator on R3R^3 defined by T(x1,x2,x3)=(3x1+x3,  −2x1+x2,  −x1+2x2+4x3).T(x_1, x_2, x_3) = (3x_1 + x_3,\; -2x_1 + x_2,\; -x_1 + 2x_2 + 4x_3). (a) What is the matrix of TT in the standard ordered basis for R3R^3? (b) What is the matrix of TT in the ordered basis {α1,α2,α3}\{\alpha_1, \alpha_2, \alpha_3\} where α1=(1,0,1)\alpha_1 = (1, 0, 1), α2=(−1,2,1)\alpha_2 = (-1, 2, 1), and α3=(2,1,1)\alpha_3 = (2, 1, 1)? (c) Prove that TT is invertible and give a rule for T−1T^{-1} like the one which defines TT.

Related theory

Browse the problem library