Math 8530, Fall 2021
Math 8530, Fall 2021
"We share a philosophy about linear algebra: we think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury." --Irving Kaplansky, writing of himself and Paul Halmos
About the class
This course will be a comprehensive survey of finite dimensional vector spaces and linear mappings at the graduate level. The goals are two-fold -- to prepare students for the linear algebra part of the prelim, and to build up a solid linear algebra foundation necessary for other classes and research, as linear algebra is a beautiful subject that appears in nearly all areas of mathematics, especially in applied fields such as statistics, operations research, and computational mathematics. Some advanced linear algebra classes, especially those geared to engineers and often called "matrix analysis," focus on computations involving matrices and under-emphasize the theory of vector spaces. Yet many other treatments take the opposite approach by presenting the theory of modules, which is too far detached from most applications that an applied mathematician, statistician, or optimizer will need. I will try to take a "Goldilocks" approach, and emphasize the theory of vector spaces (not modules), while being grounded in applications. The book that I will loosely follow, despite being quite theoretical, was written by a foremost expert on computational PDEs, containing that material that thought was most important for his applied mathematics graduate students to learn over the course of a year-long class.
Essentials
Lectures notes, slides, and videos
Links to the individual lectures are listed below. Or, you can view the full YouTube playlist here.
Section 1: Vector spaces (6 lectures, 3 hrs 2 min) [full slides; 27 pages]
Section 2: Linear maps (7 lectures, 4 hrs 13 min) [full slides; 39 pages]
Section 3: Multilinear forms (7 lectures, 4 hrs 15 min)
[full slides; 32 pages]
- Lecture 3.1: Determinant prerequesites
[YouTube (27:11)
| Slides]
- Lecture 3.2: Symmetric and skew-symmetric multilinear forms
[YouTube (30:52)
| Slides]
- Lecture 3.3: Alternating multilinear forms
[YouTube (41:56)
| Slides]
- Lecture 3.4: Determinants of linear maps
[YouTube (33:30)
| Slides]
- Lecture 3.5: The determinant and trace of a matrix
[YouTube (33:38)
| Slides]
- Lecture 3.6: Minors and cofactors
[YouTube (31:34)
| Slides]
- Lecture 3.7: Tensors
[YouTube (56:25)
| Slides]
Section 4: Spectral theory
[full slides; 36 pages]
- Lecture 4.1: Eigenvalues and eigenvectors
[YouTube (39:44)
| Slides]
- Lecture 4.2: The Cayley-Hamilton theorem
[YouTube (49:20)
| Slides]
- Lecture 4.3: Generalized eigenvectors
[YouTube (29:29)
| Slides]
- Lecture 4.4: Invariant subspaces
[YouTube (41:31)
| Slides]
- Lecture 4.5: The spectral theorem
[YouTube (32:59)
| Slides]
- Lecture 4.6: Generalized eigenspaces
[YouTube (26:41)
| Slides]
- Lecture 4.7: Jordan canonical form
[YouTube (31:49)
| Slides]
- Lecture 4.8: Generalized eigenvectors of differential operators
[YouTube
| Slides]
- Lecture 4.9: Rational canonical form
[YouTube | Slides]
Section 5: Inner product spaces
[full slides; 50 pages]
- Lecture 5.1: Inner products and Euclidean structure
[YouTube (41:52)
| Slides]
- Lecture 5.2: Orthogonality
[YouTube (48:14)
| Slides]
- Lecture 5.3: Gram-Schmidt and orthogonal projection
[YouTube (52:29)
| Slides]
- Lecture 5.4: Adjoints
[YouTube (20:18)
| Slides]
- Lecture 5.5: Projection and least squares
[YouTube (36:31)
| Slides]
- Lecture 5.6: Isometries
[YouTube (32:19)
| Slides]
- Lecture 5.7: The norm of a linear map
[YouTube (47:06)
| Slides]
- Lecture 5.8: Sequences and convergence
[YouTube
| Slides]
- Lecture 5.9: Complex inner product spaces
[YouTube (29:54)
| Slides]
Section 6: Self-adjoint mappings 5 lectures, 3 hr 28 min
[full slides; 37 pages]
Section 7: Positive linear maps
[full slides; 19 pages]
- Lecture 7.1: Definiteness and indefiniteness
[YouTube (34:42)
| Slides]
- Lecture 7.2: Nonstandard inner products and Gram matrices
[YouTube (44:50)
| Slides]
- Lecture 7.3: Polar decomposition
[YouTube
| Slides]
- Lecture 7.4: Singular value decomposition [YouTube | Slides]
- Lecture 7.5: The partial order of positive maps
[YouTube
| Slides]
- Lecture 7.6: Monotone matrix functions
[YouTube
| Slides]
Homework
- HW 0: pdf |
tex. Topics: Academic policies, integrity, etc.. Due
Monday, August 23.
- HW 1: pdf |
tex. Topics: Vector spaces, bases, subspaces. Due
Friday, August 27.
- HW 2: pdf |
tex. Topics: Dual vector spaces. Due Friday, Septemer 3. (you can have until Sunday night)
- HW 3: pdf |
tex. Topics: Linear maps. Due Friday, September 10.
- HW 4: pdf |
tex. Topics: Transposes and matrices of linear maps. Due
Friday, September 17.
- HW 5: pdf |
tex. Topics: Multilinear forms and determinants. Due Friday, September 24.
- HW 6: pdf |
tex. Topics: Trace and tensors. Due Friday, October 1.
- HW 7: pdf |
tex. Topics: Eigenvalues, eigenvectors, and generalized eigenvectors. Due Friday, October 8.
- HW 8: pdf |
tex. Topics: Spectral theory and Jordan canonical form. Due Friday, October 15.
- HW 9: pdf |
tex. Topics: Differential operators, rational canonical form. Due Friday, October 22.
- HW 10: pdf |
tex. Topics: Inner product spaces, orthogonality, and Gram-Schmidt. Due Friday, October 29.
- HW 11: pdf |
tex. Topics: Adjoints, projection, and least squares. Due Friday, November 5.
- HW 12: pdf |
tex. Topics: Complex inner product spaces, unitary maps, and normal operators. Due Friday, November 12.
- HW 13: pdf |
tex. Topics: The Rayleigh quotient, positive-definite maps. Due Tuesday, November 23.
- HW 14: pdf |
tex. Topics: Polar and singular value decomposition. Due Friday, December 3.