Math 101 Graduate Algebra:
Linear and Multilinear Algebra
Syllabus
Main course texts:
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[DF] David S. Dummit and Richard M. Foote, Abstract algebra,
3rd ed., Wiley, 2003; see their errata
List of other useful texts:
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[FIS] Stephen Friedberg, Arnold Insel, and Lawrence Spence, Linear algebra, 4th. ed., Pearson, 2002
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[R] Steven Roman, Advanced linear algebra, 3rd. ed., GTM vol. 135, 2007
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[L] Serge Lang, Algebra, 3rd. ed., GTM vol. 211, Springer-Verlag, 2005
Weekly Syllabus and Homework
Updated Oct 03, 2024.
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Week
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Date
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Topics
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Reading
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Work
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1
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Tue 17 Sep
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Finite dimensional vector space review. Bases.
Change of basis. Subspaces. Sums and direct sums.
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FIS 1.1-1.6, DF 11.1, R Ch. 1, L III.5, XIII.1-XIII.2
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Thu 19 Sep
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Linear maps and matrices. Quotient spaces.
Rank-nullity.
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FIS 2.1-2.5, DF 11.2, R Ch. 2, L XIII.3
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2
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Tue 24 Sep
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Exact sequences. Factoring through. Resolutions. Complexes. (Co)homology.
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DF 17.1 (pp. 777-778)
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Home Work 1
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Thu 26 Sep
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Dual space. Transpose. Annihilators.
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FIS 2.6, 6.8, DF 11.3
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3
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Tue 01 Oct
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Bilinear forms. Orthogonal and symplectic groups.
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FIS 6.1-6.6, 6.8
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Home Work 2
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Thu 03 Oct
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Orthogonality. Diagonalization. Inner product spaces.
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FIS 6.1-6.6, 6.8
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4
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Tue 08 Oct
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Spectral theorem. Tensor products.
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DF 11.5, 10.4, R, Ch. 14 pp. 355–378
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Home Work 3
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Thu 10 Oct
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More tensor products.
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5
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Tue 15 Oct
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Symmetric and exterior powers. Multilinear
forms. Determinants.
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R, Ch. 14 pp. 379-406
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Home Work 4
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Thu 17 Oct
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Algebras. Graded algebra and homogeneous ideals. Tensor, symmetric, exterior algebras.
Categories.
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R, Ch. 14 pp. 385-406, Ch. 18 pp. 451-469, DF Appendix II, L I.11
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6
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Tue 22 Oct
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Functors. Natural transformations.
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DF Appendix II, L I.11
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Home Work 5
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Thu 24 Oct
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Universal properties. Adjoints.
Modules.
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DF Appendix II, L I.11,
DF 10.1-10.4
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7
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Tue 29 Oct
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Free modules. Quotient modules. Isomorphism theorems.
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FIS 7.1-7.4, DF 12.1-12.3
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Home Work 6
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Thu 31 Oct
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Projective modules. Modules over PID.
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DF 10.5, 17.1
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8
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Tue 05 Nov
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Modules over PID. Smith normal form. Canonical form.
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DF 10.5, 17.1
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Thu 07 Nov
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Tensor products. Flat modules. Projective resolutions.
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DF 10.4-5, 17.1
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9
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Tue 12 Nov
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Chain homotopy category.
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DF 10.4-10.5, 17.1, L III.1-4
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Thu 14 Nov
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Derived functors. Tor and Ext.
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DF 17.1, L III.1-4
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10
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Tue 19 Nov
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Representation theory.
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DF 18.1-18.3, 19.1
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