## Modules, Linear Algebra, and Groups, Fall 2019 ### Homework:

Assignment 1 (due 8/30): Section 7.1, problems 15, 21, 26; Section 7.2, problems 3, 5; Section 7.3, problems 13, 21, 29; Section 7.4, problems 8, 25, 27

Assignment 2 (due 9/6): Section 8.1, problems 3, 7, 8 part a (but only for -3); Section 8.2, problems 2, 3, 5, 6; Section 8.3, problems 4, 8

Assignment 3 (due 9/13): Section 9.2, problems 4, 5; Section 9.3, problems 2, 3, 4; Section 9.4, problems 11, 13; Section 9.5, problem 7; also, show that given a field and n+1 ordered pairs (a_i,b_i) with entries in the field and all a_i distinct, we can find a unique polynomial L(x) of degree n so that L(a_i) = b_i for all i=1,…,n+1

Assignment 4 (due 9/20): Section 11.1, problems 9, 13, 14; Section 11.2, problems 9, 11 (for part b, it might help to show that a space is isomorphic to the direct sum of two subspaces iff the union of the subspaces span, and they intersect trivially); Section 11.3, problems 1 (for the last sentence, replace “F-algebra” with “(unnatural) ring”), 2, 4; plus the 3 problems from lecture (direct sum of linear maps, uniqueness of tensor product, properties of tensor product)

Assignment 5 (due 9/25): Section 10.1, problems 5, 8, 9; Section 10.2, problems 3, 6, 9, 11, 12; Section 10.3, problems 7, 9

Assignment 6 (due 10/2): Section 10.3, problems 2, 15, 16, 17, 27 (also, now is the time to review the Chinese Remainder Theorem, see Section 7.6); Section 10.4, problems 2, 3, 8, 10, 11, 13, 14, 15, 16

Assignment 7 (due 10/9): Section 12.1, problems 1, 2, 3, 5, 15, 16, 17, 18, 19

Assignment 8 (due 10/23): Section 12.1, problems 9, 11; Section 12.2, problems 8, 9, 13, 15, 18

Assignment 9 (due 10/31): Section 12.3, problems 2, 5, 6, 17, 21, 22, 24, 32, and solve problem 4 here: http://www.math.lsa.umich.edu/~kesmith/593hmwk6.pdf