Lecture plan
The recordings of the lectures can be found in panopto. Note that it will take some time until they are published (as editing and processing takes some time).
Here are the lecture notes (they are essentially complete, but they will be updated if I find (or get informed of) errors:
- LN … Lecture notes, last updated on November 16, 2023.
In addition, I have made use of the following material:
- This note on Linear algebra by Franz Luef.
- The Lecture Notes from 2019 by Franz Luef.
| Date | Topics | Reading | Slides and Code |
|---|---|---|---|
| Topics in linear algebra | |||
| Week 34 | Introduction Vector spaces and subspaces Linear independence and bases Sums and direct sums of subspaces Linear mappings and matrices Invariant subspaces Matrices of linear mappings | LN, Sec. 1.1-1.4 Ax, Chap. 1, 2, 3A-D | Lecture 1 Lecture 2 |
| Week 35 | Matrix decompositions Eigenvalues and eigenvectors Existence of eigenvalues Schur triangulisation Upper triangular matrices Diagonalisation | LN, Sec. 1.4-1.8 Ax, Chap. 3A-D, 5 | Lecture 3 Lecture 4 |
| Week 36 | Nilpotent matrices and operators Generalised eigenspaces Decomposition of operators in blockdiagonal form Cayley-Hamilton Theorem Minimal polynomial Jordan normal form | LN, Sec. 1.9-1.11 Ax, Chap. 8 | Lecture 5 Lecture 6 |
| Inner product spaces | |||
| Week 37 | Inner product spaces Cauchy-Schwarz-Bunyakovsky inequality Orthogonality and Gram-Schmidt orthogonalisation Orthogonal projections Singular values and singular value decomposition Unitary transformations Adjoints | LN, Sec. 2.1-2.5 Ax, Chap. 6 | Lecture 7 Lecture 8 |
| Week 38 | Singular value decomposition Moore-Penrose inverse SVD of a matrix Relation between singular values and eigenvalues Self-adjoint and positive definite operators | LN, Sec. 2.5-2.9 | Lecture 9 Lecture 10 |
| Banach and Hilbert spaces | |||
| Week 39 | Metrics and metric spaces Convergence and continuity Open and closed sets Interior, closure and boundary Cauchy sequences Completeness | LN, Sec. 3.1-3.2 He, Sec. 2.1-2.6, 2.9 | Lecture 11 Discrete_Deconvolution.ipynb Lecture 12 |
| Week 40 | Completeness of \(C([0,1])\) Lipschitz functions and contractions Banach's fixed point Theorem Application to integral equations Picard-Lindelöf Theorem (existence for ODEs) Normed spaces | LN, Sec. 3.2-3.5 He, Sec. 3.1 | Lecture 13 Lecture 14 Fredholm_Integral_Equations.ipynb |
| Week 41 | Normed Spaces \(p\)-norms on \(\mathbb{K}^n\) Hölder's inequality Minkowski's inequality Sequence spaces Banach and Hilbert spaces | LN, Sec. 3.5-3.8 He, Sec. 3.1-3.2 | Lecture 15 p_norms.ipynb Lecture 16 |
| Week 42 | Completeness of \(\ell^p\)-spaces Completions | LN, Sec. 3.8-3.9 | Lecture 17 |
| Bounded linear operators | |||
| Bounded linear operators Relation between boundedness and continuity Norms of bounded linear mappings | LN, Sec. 4.1-4.2 He 6.1-6.5 | Lecture 18 | |
| Week 43 | Norms of bounded linear mappings Matrix norms Extensions of linear mappings Range and kernel Invertibility of bounded linear mappings | LN, Sec. 4.2-4.4 | Lecture 19 Lecture 20 |
| Week 44 | Completeness of \(L(U,V)\) Linear functionals The dual of a normed space | LN, Sec. 4.5 | Lecture 21 |
| Best approximations and projections | |||
| Projections on closed convex sets Variational characterisation of projections Projection onto closed linear subspaces | LN, Sec. 5.1 | Lecture 22 | |
| Week 45 | Orthogonal complements of linear subspaces Duals of Hilbert spaces Riesz representation theorem Adjoint operators | LN, Sec. 5.2-5.4 | Lecture 23 Lecture 24 |
| Week 46 | Countable and uncountable sets Schauder bases Orthonormal sequences in Hilbert spaces Hilbert bases Structure of separable Hilbert spaces Galerkin's method | LN, Sec. 5.5-5.6 | Lecture 25 Lecture 26 |
| Summary and repetition | |||
| Week 47 | Tuesday: Summary Thursday: Questions and answers | LN Sec. 1.1-5.6 | Summary |