| Date |
Topics (Section numbers are from Treil's book) |
|
| 9/3/2026 |
- 1.1: Vector spaces
- 1.2: Linear combinations, bases
- 1.1.2: Matrix notation
- 1.3: Linear transformations
- 1.4: Space of linear transformations
- 1.5: Composition of linear transformations
- 1.6: Invertible transformations and matrices
- 1.7: Subspaces and quotient spaces
|
|
| 9/10/2026 |
- 2.1: Systems of linear equations
- 2.2: Solving a linear system
- 2.3: Pivots
- 2.4: Inverting matrix by row reduction
- 2.5: Dimension of a vector space
- 2.7: Kernel, image, rank of linear map
- 2.8: Change of basis
|
|
| 9/17/2026 |
- Oriented Area of a Parallelogram
- Oriented Volume of a Parallelepiped
- 3.2: Properties of a determinant
- 3.3: Construction of a determinant
- 3.4: Definition of a determinant
|
Slides
|
| 9/24/2026 |
- 4.1: Eigenvalues and eigenvectors
- 4.2: Diagonalization
|
Slides
|
| 10/1/2026 |
- 5.1: Inner product on \(\mathbb{R}^n\) and \(\mathbb{C}^n\)
- 5.2: Orthogonality
- 5.3: Orthogonal projection and Gram-Schmidt
- 5.4: Least square solution
- 5.5: Adjoint of linear transformation
- 5.6: Unitary and orthogonal transformations
|
Slides
|
| 10/8/2026 |
- 6.1: Schur representation
- 6.2: Spectral theorem for self-adjoint operators
- 6.3: Polar and singular value decompositions
- 6.4: Applications of the singular value decomposition
|
Slides
|
| 10/15/2026 |
- 7.1: Bilinear and quadratic forms
- 7.2: Diagonalization of quadratic forms
- 7.3: Sylvester's law of inertia
- 7.4-7.5: Positive definite forms
|
Slides
|
| 10/22/2026 |
- 8.1: Dual spaces
- 8.2: Dual of inner product space
- 8.3: Dual transformations
- 8.5: Tensors
- 8.6: Change of coordinates for tensors
|
Slides
|
| 10/29/2026 |
- 9.1: Cayley-Hamilton Theorem
|
Slides
|
| 11/5/2026 |
Midterm
|
|
| 11/12/2026 |
- 9.2: Spectral mapping theorem
|
Slides
|
| 11/19/2026 |
- 9.3: Generalized eigenspaces
- 9.4: Nilpotent maps
- 9.5: Jordan decomposition
|
Slides
|
| 11/26/2026 |
No class
|
|
| 12/3/2026 |
- Normal Form of Linear Map
- Quasi-solutions of systems of linear equations
- Moore-Penrose pseudoinverse operators
|
Slides
|
| 12/10/2026 |
- Affine spaces
- Affine maps and transformations
- Euclidean space
- Rigid motions
|
Slides
|
| TBA |
Final
|
|
