Math 636 - Assignment 7 - Written Component Due: Saturday, July 2 at 4:00PM 1. Let f (x1 , x2 ) = (2x2 , x1 3x2 , 2x2 ). Prove that f is linear and then find the standard matrix of f . 2. Invent a linear mapping L : R3 P2 (R) that satisfies 1 0 L 0 = 1 + x2 , L 1 = 2 x, 0 0 0 L 0 = 1 + x + x2 1 (You do not need to give a proof that your mapping is linear.) 3. Let U be the subspace \u0012\u0014 of upper \u0015\u0013triangular matrices in M22 (R) and let T : U P2 (R) be the linear a b mapping defined by T = (a + b) + (a + b + c)x + (a + b)x2 . Find a basis for the range and 0 c kernel of T and verify that they satisfy the Rank-Nullity Theorem. 4. Explain the difference between the codomain and the range of a linear mapping. Demonstrate the difference with an example. 5. Let {~v1 , . . . , ~vk } be a basis for a subspace S of an n-dimensional vector space V. Prove that there exists a linear mapping L : V V such that ker(L) = S. (You do not need to prove that the mapping is linear.) 1