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 Nov 3, 2025.
|
Week
|
Date
|
Topics
|
Reading
|
Work
|
|
1
|
Tue 16 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 18 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 23 Sep
|
Exact sequences. Factoring through. Resolutions. Complexes. (Co)homology.
|
DF 17.1 (pp. 777-778)
|
Home Work 1
|
|
Thu 25 Sep
|
Dual space. Transpose. Some category theory.
|
FIS 2.6, 6.8, DF 11.3
|
|
3
|
Tue 30 Sep
|
Bilinear forms. Orthogonal and symplectic groups.
|
FIS 6.1-6.6, 6.8
|
Home Work 2
|
|
Thu 02 Oct
|
No class: Yom Kippur
|
|
|
4
|
Tue 07 Oct
|
Similitude and orthogonal groups. Orthogonality. Diagonalization.
|
FIS 6.1-6.6, 6.8
DF 11.5, 10.4, R, Ch. 14 pp. 355–378
|
Home Work 3
|
|
Thu 09 Oct
|
Inner product spaces.
Spectral theorem.
Tensor products.
|
FIS 6.1-6.6, 6.8
DF 11.5, 10.4, R, Ch. 14 pp. 355–378
|
|
5
|
Tue 14 Oct
|
Multilinear
forms. Tensor products. Determinants.
|
R, Ch. 14 pp. 379-406
|
Home Work 4
|
|
Thu 16 Oct
|
Symmetric and exterior powers.
|
DF 11.5, 10.4, R, Ch. 14 pp. 355–378
|
|
6
|
Tue 21 Oct
|
Algebras. Graded algebra and homogeneous ideals. Tensor, symmetric, exterior algebras.
|
R, Ch. 14 pp. 385-406, Ch. 18 pp. 451-469
|
Home Work 5
|
|
Thu 23 Oct
|
Modules. Free modules. Quotient modules. Isomorphism theorems.
|
FIS 7.1-7.4, DF 10.1-10.4, 12.1-12.3,
DF 10.5, 12.1
|
|
7
|
Tue 28 Oct
|
Projective modules.
|
DF 10.5
|
|
|
Thu 30 Oct
|
Modules over a PID. Smith normal form.
|
DF 12.1-12.3
|
|
8
|
Tue 04 Nov
|
More modules over a PID. Canonical form.
|
DF 12.1-12.3
|
|
|
Thu 06 Nov
|
Tensor products. Flat modules. Projective resolutions.
|
DF 10.4-5, 17.1
|
|
9
|
Tue 11 Nov
|
Chain homotopy category.
Derived functors. Tor and Ext.
|
DF 10.4-10.5, 17.1, L III.1-4
|
|
|
Thu 13 Nov
|
Representation theory.
|
DF 18.1-18.3, 19.1
|
|
10
|
Tue 18 Nov
|
Representation theory.
|
DF 18.1-18.3, 19.1
|
|
|