Math 8530, Fall 2020

"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

Essentials

Syllabus
Class schedule
Canvas (for submitting homework)
Free textbook, Algebra, Topology, Differential Calculus, and Optimization Theory for Computer Science and Machine Learning, by Gallier and Quaintance

Lecture Notes

Section 1: Vector spaces. (12 pages, last updated Jan 23, 2013. slides)
Section 2: Duality. (6 pages. Updated Jan 30, 2013.)
Section 3: Linear mappings. (14 pages. Updated Feb 14, 2011. slides)
Section 4: Matrices. (9 pages. Updated Feb 19, 2015)
Section 5: Determinant and trace. (17 pages. Updated February 19, 2015)
Section 6: Spectral theory. (22 pages. Updated June 20, 2017)
Section 7: Euclidean structure. (23 pages. Updated April 2, 2013)
Section 8: Self-adjoint mappings. (17 pages. Updated April 16, 2013)
Section 9: Positive definite mappings. (19 pages. Updated May 3, 2013)

Homework

HW 1: pdf | tex. Topics: Vector spaces, bases, subspaces. Due Wednesday, August 26.
HW 2: pdf | tex. Topics: Dual vector spaces. Due Wednesday, September 2.
HW 3: pdf | tex. Topics: Linear maps. Due Wednesday, September 9.
HW 4: pdf | tex. Topics: Transposes and matrices of linear maps. Due Wednesday, September 16.
HW 5: pdf | tex. Topics: Multilinear forms and determinants. Due Wednesday, September 23.
HW 6: pdf | tex. Topics: Trace and tensors. Due Friday, October 2.
HW 7: pdf | tex. Topics: Eigenvalues, eigenvectors, and generalized eigenvectors. Due Wednesday, October 7.
HW 8: pdf | tex. Topics: Spectral theory and Jordan canonical form. Due Wednesday, October 14.
HW 9: pdf | tex. Topics: Rational canonical form. Due Friday, October 23.
HW 10: pdf | tex. Topics: Inner products. Due Wednesday, November 4.
HW 11: pdf | tex. Topics: Real inner product spaces and adjoints. Due Friday, November 13
HW 12: pdf | tex. Topics: Complex inner product spaces and normal operators. Due Monday, November 23.
HW 13: pdf | tex. Topics: The Rayleigh quotient, positive-definite maps. Due Monday, December 6.

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)

Lecture 1.1: Vector spaces and linearity [YouTube (36:24) | Slides]
Lecture 1.2: Spanning, independence, and bases [YouTube (39:25) | Slides]
Lecture 1.3: Direct sums and products [YouTube (19:23) | Slides]
Lecture 1.4: Quotient spaces [YouTube (43:57) | Slides]
Lecture 1.5: Dual vector spaces [YouTube (23:36) | Slides]
Lecture 1.6: Annihilators [YouTube (29:07) | Slides]

Section 2: Linear maps (7 lectures, 4 hrs 13 min)

Lecture 2.1: Rank and nullity [YouTube (31:31) | Slides]
Lecture 2.2: Applications of the rank-nullity theorem [YouTube (38:40) | Slides]
Lecture 2.3: Algebra of linear maps [YouTube (35:23) | Slides]
Lecture 2.4: The four subspaces [YouTube (43:09) | Slides]
Lecture 2.5: The transpose of a linear map [YouTube (41:07) | Slides]
Lecture 2.6: Matrices [YouTube (41:51) | Slides]
Lecture 2.7: Change of basis [YouTube (21:08) | Slides]

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 and least squares [YouTube | Slides]
Lecture 5.5: Isometries [YouTube | Slides]
Lecture 5.6: The norm of a linear map [YouTube | Slides]
Lecture 5.7: Sequences and convergence [YouTube | Slides]
Lecture 5.8: Complex inner product spaces [YouTube | Slides]

Section 6: Self-adjoint mappings

Lecture 6.1: Quadratic forms [YouTube (36:11) | Slides]
Lecture 6.2: Spectral resolutions [YouTube (38:56) | Slides]
Lecture 6.3: Normal linear maps [YouTube (35:07) | Slides]
Lecture 6.4: The Rayleigh quotient [YouTube | Slides]
Lecture 6.5: Self-adjoint differential operators [YouTube | Slides]

Section 7: Positive linear maps

Lecture 7.1: Definiteness and indefiniteness [YouTube | Slides]
Lecture 7.2: Nonstandard inner products [YouTube | Slides]
Lecture 7.3: Gram matrices [YouTube | Slides]
Lecture 7.4: Polar decomposition [YouTube | Slides]
Lecture 7.5: Singular value decomposition [YouTube | Slides]
Lecture 7.6: The partial order of positive maps [YouTube | Slides]
Lecture 7.7: Monotone matrix functions [YouTube | Slides]