7, DUE Friday, March 10th, 2017 Hand in Part A and Part B as two separate assignments. Include the following information in the top left corner of every assignment: your full name, instructor's last name and section number, homework number, whether they are Part A problems or Part B problems. A few words about solution writing: Unless you are explicitly told otherwise for a particular problem, you are always expected to show your work and to give justification for your answers. In particular, when asked if a statement is true or false, you will need to explain why it is true or false to receive full credit. Write down your solutions in full, as if you were writing them for another student in the class to read and understand. Don't be sloppy, since your solutions will be judged on precision and completeness and not merely on \"basically getting it right." Cite every theorem or fact from the book that you are using. (\"By Theorem 1.10. . . ", etc.) If you compute something by observation, say so and make sure that your imaginary fellow student who is reading your proof can also clearly see what you are claiming. Justify each step in writing and leave nothing to the imagination. Part A (15 points) Solve the following problems from the book: Section 3.4: 56, 62 Section 4.3: 7, 8, 42, 23, 24, 46, 60 Part B (25 points) Problem 1. Let B = (b1 , . . . , bn ) be a basis of the vector space V , let T : V V be a linear transformation of V , and let B be the B-matrix of T . (a) Prove that v ker(T ) if and only if [v]B ker(B). (b) Prove that v im(T ) if and only if [v]B im(B). (c) Prove that T is an isomorphism if and only if B is invertible. Problem 2. Let V be a vector space of dimension 3, and suppose that T : V V is a linear transformation of V that has the same matrix representation with respect to every basis of V . Prove that T must be a scalar multiple of the identity transformation. Problem 3. Let B = (~v1 , . . . , ~vn ) and C = (w ~ 1, . . . , w ~ n ) be two bases of Rn , and let T : Rn Rn be the unique isomorphism satisfying T (vk ) = wk for all k = 1, . . . , n. Prove that the B-matrix of T is the same as the C-matrix of T . That is, [T ]B = [T ]C . 1 HOMEWORK 7, DUE Friday, March 10th, 2017 2 Problem 4. Let V be a vector space with two bases B and C. Let T : V V be a linear transformation with B as its B-matrix and C as its C-matrix. Prove or disprove each of the following statements. (a) dim(ker(B)) =dim(ker(C)). (b) ker(B) =ker(C). (c) The sum of all entries in B is equal to the sum of all entries in C. (d) The trace (sum of the diagonal entries) of B is equal to the trace of C. Hint: It may be useful to remember the fact previously proven on HW #3 that trace(AB)=trace(BA) for any two n n matrices A and B. (e) If there exists a vector ~v1 such that B~v1 = k~v1 for some scalar k R, then there exists a vector ~v2 such that C~v2 = k~v2 for the same value of k. Problem 5. Let B be an invertible n n matrix. Define B : Rnn Rnn by B (A) = B 1 AB. (a) Prove that B is an isomorphism by showing that B is linear and bijective. What is its inverse transformation? (b) Now, define TB : Rnn Rnn by TB (A) = B (A) A. Give a nice property that determines if A ker(TB ). (c) Suppose now that n = 3 and that every diagonal n n matrix D is in ker(TB ). What can you say about the matrix B