Math 110 Homework Assignment 8 due date: Nov. 11, 2016 1. Determine if the linear transformations described by the following matrices are invertible. If not, explain why, and if so, find the matrix of the inverse transformation. (a) \u0014 4 0 0 3 \u0015 (b) 3 1 5 (e) 6 3 1 0 0 0 \u0014 2 0 6 0 3 1 1 0 3 1 (f) 5 6 7 10 \u0015 0 0 1 4 (c) \u0014 7 3 9 4 \u0015 (d) \u0014 3 6 2 4 \u0015 0 0 0 1 2. Suppose that A is the matrix 5 2 4 A = 2 3 1 . 5 6 3 (a) Find the inverse of A. (b) Explain why, for any values of a, b, and c, the equations 5x + 2y + 4z = a 2x + 3y + z = b 5x + 6y + 3z = c always have a unique solution. (c) Find this unique solution (in terms of a, b, and c). 3. Suppose that T1 : Rn Rm and T2 : Rm Rp are linear transformations. (a) If T1 and T2 are injective, prove that T2 T1 is injective. (b) If T1 and T2 are surjective, prove that T2 T1 is surjective. (c) If T1 and T2 are invertible, prove that T2 T1 is invertible. 1 4. For each of the following subsets W of R3 , either show that they are subspaces, or show why they aren't subspaces by explaining which of the three conditions don't hold. (a) W = {(x, y, z) R3 | x2 + y 2 = z 2 }. (b) W = {(x, y, z) R3 | (x, y, z) is orthogonal to (3, 1, 2)}. (c) W = {(x, y, z) R3 | x + y + z > 0}. 2