MATH:5010 - Spring 2017
Homework
Assignment 6
Reading (from Dummit and Foote):
Read Sections 12.2, 12.3, 13.1, 13.2.
Please look through the sections for EACH lecture before the lecture.
See the Lecture Schedule for the sections that will be covered.
Then after EACH lecture, review your lecture notes and
thoroughly read the sections covered in class.
Part A Problems (to be handed in on
Tuesday, March 07, at the beginning of the discussion section):
- Section 12.2: # 14, 18.
- Section 12.3: # 16, 21, 31.
IMPORTANT for # 16: Work over the rational numbers.
IMPORTANT for # 21, 31: Work over any field.
Part B Problems (to be handed in on
Friday, March 10, at the beginning of the lecture):
If you hand in ALL Part B Problems together with the Part A Problems already on
Tuesday, March 07, 2 extra points will be added to your homework score.
- Section 12.2: # 22, 23, 24, 25.
IMPORTANT: Read the text on p. 490 before # 22, just after # 21.
IMPORTANT for # 25: For the conclusion explain how you use the exercises from
Section 12.1 to prove Theorem 21 on p. 479 and how the Invariant Factor Decomposition
Algorithm follows.
Please note that we allow a third row/column operation in Section 12.2 given by
multiplying a row or column by a unit in F. This is OK since we could have also allowed
a third row/column operation in Section 12.1 given by multiplying a row/column
by a unit in R. Just describe what this third operation does to the basis elements of
Rn, respectively to the generators for Ker ϕ in the set-up of # 17, 18
in Section 12.1. You may use # 17, 18 of Section 12.1 without proof.
Additional Practice Problems (you have
to know how to do these, but you do NOT have to hand these in):
- Section 12.2: # 1, 3, 4, 6, 8, 9, 10, 11, 12.
- Section 12.3: # 1, 2, 4 - 15, 17, 18, 19, 20, 22, 23.
Frauke Bleher
Mar 01 2017