"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

- 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

**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]

- 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]

- 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]

- 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]

- 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]

- 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 (53:09) | Slides]
- Lecture 6.5: Self-adjoint differential operators [YouTube (44:50) | Slides]

- 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]

- 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.