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Homework 3
Due Wednesday April 26, in class.
- Section 12.1
- Homework: 1, 2, 6, 8 and the following problem:
- 23. Let \(R\) be an integral domain with field of quotients \(F\) (so \(R\ \subset F\)). Regard \(F\) as an \(R\)-module via the field operations. Prove that \(F\) is torsion-free as an \(R\)-module, while \(F/R\) is a torsion module. (Source: Derek J.S. Robinson, "Abstract Algebra (Second Edition)", De Gruyter, Berlin/Boston (2015))
- Hand In: 23
- Homework: 1, 2, 6, 8 and the following problem:
- Section 12.2
- Homework: 2, 4, 9, 10 and the following problems:
- 26. Find all similarity types (in rational canonical form) of rational \(3 \times 3\) matrices \(A\) that satisfy the equation \(A(A - 2I)^2 = 0\). (Source: Derek J.S. Robinson, "Abstract Algebra (Second Edition)", De Gruyter, Berlin/Boston (2015))
- 27. Find the invariant factors and rational canonical form of the rational matrix \[ \left[ \begin{array}{ccc}2&3&1\\ 1&2&1\\ 0&0&-4 \end{array}\right].\] (Source: Derek J.S. Robinson, "Abstract Algebra (Second Edition)", De Gruyter, Berlin/Boston (2015))
- Hand In: 26, 27
- Homework: 2, 4, 9, 10 and the following problems:
- Section 12.3
- Homework: 1, 5, 8, 11
- Hand In: 11
- Section 10.4
- Homework: TBD
- Hand In: TBD