Math 101 Graduate Algebra:
Linear and Multilinear Algebra

Syllabus

Main course texts:

  • [DF] David S. Dummit and Richard M. Foote, Abstract algebra, 3rd ed., Wiley, 2003; see their errata

List of other useful texts:

  • [FIS] Stephen Friedberg, Arnold Insel, and Lawrence Spence, Linear algebra, 4th. ed., Pearson, 2002
  • [R] Steven Roman, Advanced linear algebra, 3rd. ed., GTM vol. 135, 2007
  • [L] Serge Lang, Algebra, 3rd. ed., GTM vol. 211, Springer-Verlag, 2005

Weekly Syllabus and Homework

Updated Oct 03, 2024.

Week Date Topics Reading Work
1 Tue 17 Sep Finite dimensional vector space review. Bases. Change of basis. Subspaces. Sums and direct sums. FIS 1.1-1.6, DF 11.1, R Ch. 1, L III.5, XIII.1-XIII.2
Thu 19 Sep Linear maps and matrices. Quotient spaces. Rank-nullity. FIS 2.1-2.5, DF 11.2, R Ch. 2, L XIII.3
2 Tue 24 Sep Exact sequences. Factoring through. Resolutions. Complexes. (Co)homology. DF 17.1 (pp. 777-778) Home Work 1
Thu 26 Sep Dual space. Transpose. Annihilators. FIS 2.6, 6.8, DF 11.3
3 Tue 01 Oct Bilinear forms. Orthogonal and symplectic groups. FIS 6.1-6.6, 6.8 Home Work 2
Thu 03 Oct Orthogonality. Diagonalization. Inner product spaces. FIS 6.1-6.6, 6.8
4 Tue 08 Oct Spectral theorem. Tensor products. DF 11.5, 10.4, R, Ch. 14 pp. 355–378 Home Work 3
Thu 10 Oct More tensor products.
5 Tue 15 Oct Symmetric and exterior powers. Multilinear forms. Determinants. R, Ch. 14 pp. 379-406 Home Work 4-->
Thu 17 Oct Algebras. Graded algebra and homogeneous ideals. Tensor, symmetric, exterior algebras. Categories. R, Ch. 14 pp. 385-406, Ch. 18 pp. 451-469, DF Appendix II, L I.11
6 Tue 22 Oct Functors. Natural transformations. DF Appendix II, L I.11 Home Work 5
Thu 24 Oct Universal properties. Adjoints. Modules. DF Appendix II, L I.11, DF 10.1-10.4
7 Tue 29 Oct Free modules. Quotient modules. Isomorphism theorems. FIS 7.1-7.4, DF 12.1-12.3 Home Work 6
Thu 31 Oct Projective modules. Modules over PID. DF 10.5, 17.1
8 Tue 05 Nov Modules over PID. Smith normal form. Canonical form. DF 10.5, 17.1
Thu 07 Nov Injective modules. Tensor products. Resolutions. Derived functors. Tor and Ext. Homological algebra. DF 17.1
9 Tue 12 Nov Representation theory. Noetherian rings. Integral extensions. Localization.
Thu 14 Nov Primary decomposition. Zariski topology. Artinian rings.
10 Tue 19 Nov Special topics.


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