The problems below are from the text Linear Algebra by Hoffman and Kunze.
HW #1 : Section 1.2, #2, 4, 7 and Section 1.3, #2, 4, 6, 8
HW #2 : Section 1.3, #2, 4, 5, 7 and Section 1.4, #1, 2
HW #3 : Section 1.4, #3, and Section 1.5, #4, 6, 8
HW #4 : Section 1.6, #2, 5, 9, 11, 12
HW #5 : Section 2.1, #3, 4, 6 and Section 2.2, #2, 3, 5
HW #6 : Section 2.3, #3, 5, 7, 9
HW #7 : Section 2.3, #8, 10, 11
HW #8 : Section 2.3, #9, 13, 14
HW #9 : Section 2.6, #2, 3, 7
HW #10 : Section 3.1, #3, 7
HW #11 : Section 3.1, #10, 12, 13
HW #12 : Section 3.2, #3, 7, 10, 11
HW #13 : Section 3.3, #3, 7
HW #14 : Section 3.4, #6, 7, 8, 12
HW #15 : Let F be a field and f(x), g(x) polynomials with coefficients in F. The greatest
common
divisor of f(x) and g(x) is a polynomial d(x) with the following two properties : (i) d(x) divides
both
f(x) and g(x) and (ii) any polynomial dividing both f(x) and g(x) also divides d(x).
(A) Use the division
algorithm to show that any two polynomials in F[x] have a GCD that is unique up to unit multiple.
(B) Now define the least common multiple of f(x) and g(x) in a similar manner and show that least
common multiples
exist and are unique up to unit multiple. (C) Find a formula relating the GCD
and the LCM of f(x) and g(x).
HW #16 : Use the divison algorithm to find the GCDs for the pairs of polynomials in Section 4.4, #2
and then express each GCD in terms of the given polynomials as in Bezout's principle.
HW #17 : Section 6.3, #6, 8, 10 and Section 6.4, #1
HW #18 : Section 6.6, #1, 2 and Section 6.7, #9
HW #19 : Section 6.8, #1, 10
HW #20 : Section 7.1, #2, 4, 5
HW #21 : Section 7.2, #1, 3, 19
HW #22 : Section 7.2, #2, 7 (just find the minimal polynomials)
HW #23 : Section 7.2, #7 (find RCFs), 9
HW #24 : Section 7.2, #7 (find the JCFs assuming the field is C), 12, 16
HW #25 : Section 7.3, #5, 6, 10, 11
HW #26 : Section 7.3, #15, 16
HW #27 : Section 5.3, #2, 6, 11
HW #28 : Section 5.4, #1, 4, 9
HW #29 : Section 5.4, #5, 8, 11
HW #30 : Section 6.2, #1, 4, 11
HW #31 : Section 6.2, #13, 14
HW #32 : Section 3.5, #2, 4, 11, 14
HW #33 : Section 3.5, #3, 16
HW #34 : Section 8.1, #1, 5, 6
HW #35 : Section 8.1, #8, 10, 11
HW #36 : Section 8.2, #1, 3, 9, 10
HW #37 : Section 8.2, #6, 8, 14
HW #38 : Section 8.3, #2, 4, 8, 9
HW #39 : Section 8.3, #3, 10, 12
HW #40 : Section 8.4, #3, 4, 6, 7