Name: Math 308 J, Winter 2017 Proof HW #2 Due 2/6/17 Proof Homework 2 WARNING: Aside from 3c.), REFERRING TO THE MATRICES OF LINEAR MAPS WILL NOT HELP YOU. It will just make your proof harder than it needs to be. BE SURE TO USE ALL OF THE HYPOTHESES. If you do not, your proof is wrong. Note that T being linear is a hypothesis you must use at some point. Although problems 1.) and 2.) sound similar to results we have seen to be true, they are more general because the vectors T (v~1 ), T (v~2 ), . . . T (v~n ) do not need to be related to the columns of the matrix for T . Be sure to do both directions of if and only if. 1. Suppose T : Rn Rm is a linear map, and suppose {v~1 , . . . v~n } spans Rn . Prove that T is onto IF AND ONLY IF {T (v~1 ), . . . T (v~n )} spans Rm . 2. Let T : Rn Rm be a linear transformation. Suppose {v~1 , . . . v~n } spans Rn and is linearly independent. Show that T is one-to-one IF AND ONLY IF {T (v~1 ), . . . , T (v~n )} is linearly independent. (You need to show either one of these hypotheses implies the other.) 3. Suppose T : Rn Rk , and S : Rk Rm are linear maps. Then, S T : Rn Rm is a linear map given by (S T )(x) = S(T (x)). Show the following: (a) If both T and S are one-to-one, S T is one-to-one. (b) If both T and S are onto, S T is onto. (c) If S T is one-to-one, then T is one-to-one. (d) If S T is onto, then S is onto. (e) Give an example of such linear maps T : Rn Rk , and S : Rk Rm so that T is NOT onto, S is NOT one-to-one, but S T is both one-to-one and onto. You get to pick these maps, but all of these conditions must hold. 1