Math 8530, Spring 2021
Math 8530, Spring 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
- Syllabus
- Calendar
- Canvas (for submitting homework)
- Free textbook, Algebra, Topology, Differential
Calculus, and Optimization Theory for Computer Science and Machine
Learning, by Gallier and Quaintance
Lectures
Links to the individual lectures are listed below. Or, you can view the full YouTube playlist here. To avoid blurriness, these are best viewed by changing the settings to 720p (High Definition) rather than the default of 240p. This can be easily done by clicking the "wheel" on the lower right corner; right next to the "cc" button.
Section 1: Vector spaces (6 lectures, 3 hrs 2 min)
Section 2: Linear maps (7 lectures, 4 hrs 13 min)
Section 3: Multilinear forms (7 lectures, 4 hrs 15 min)
- 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
- 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
- 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
Section 7: Positive linear maps
- 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 1: pdf |
tex. Topics: Vector spaces, bases, subspaces. Due
Friday, January 15.
- HW 2: pdf |
tex. Topics: Dual vector spaces. Due
Friday, January 22.
- HW 3: pdf |
tex. Topics: Linear maps. Due
Friday, January 29.
- HW 4: pdf |
tex. Topics: Transposes and matrices of linear maps. Due
Friday, February 5.
- HW 5: pdf |
tex. Topics: Multilinear forms and determinants. Due Friday, February 12.
- HW 6: pdf |
tex. Topics: Trace and tensors. Due Friday, February 19.
- HW 7: pdf |
tex. Topics: Eigenvalues, eigenvectors, and generalized eigenvectors. Due Friday, February 26.
- HW 8: pdf |
tex. Topics: Spectral theory and Jordan canonical form. Due Friday, March 5.
- HW 9: pdf |
tex. Topics: Differential operators, rational canonical form. Due Friday, March 12.
- HW 10: pdf |
tex. Topics: Inner product spaces, orthogonality, and Gram-Schmidt. Due Friday, March 26.
- HW 11: pdf |
tex. Topics: Least squares, adjoints, and norms of linear maps. Due Friday, April 2
- HW 12: pdf |
tex. Topics: Complex inner product spaces, unitary maps, and normal operators. Due Friday, April 9.
- HW 13: pdf |
tex. Topics: The Rayleigh quotient, positive-definite maps. Due Friday, April 16.