| Week | Description |
| week 34 | Mathematical Reasoning and Basic Set Theory |
| week 35 Will be revised soon | Mononomial basis and Bernstein basis for the space of polynomials, Interpolation polynomial (Vandermonde matrix, Lagrange basis, Newton basis), basics of quantum calculus |
| Additional material for week 36, see itslearning for further material | Review of basic notions of linear algebra: sum and intersection of vector spaces, spanning sets and bases, quotient space, dimension of a vector space, isomorphisms between vector spaces |
| week 37: Chapter 2 of Lax | Linear functionals, dual of a finite-dimensional vector space, linear mappings |
| week 38: Chapter 3 and 4 of Lax | Eigenvectors and eigenvalues, characteristic polynomial, similar matrices, Theorem of Cayley-Hamilton |
| week 39: Chapter 3 and 4 of Lax, Terry Tao's proof of the Jordan Normal Form | Generalized eigenvectors, Spectral Theorem for finite-dimensional vector spaces in terms of generalized eigenspaces, nilpotent operators, Jordan Normal Form |
| week 40: Chapter 2 of Mats Ehrnstom's notes on TMA4145, Kreyszig Chapter 1 | Metric spaces, basic examples including the space of n-tuples with the 1,2 and sup-distance, sequence spaces with 1,2 and sup-distance, Hamming distance, limits of sequences, completeness axiom for the real numbers, supremum and infimum of a set |
| week 41: TMA4145 notes: Chapter 2, Kreyszig Chapter 1 | closed and open sets, Cauchy sequences, completeness, Bolzano-Weierstrass, compact sets |
| week 42: TMA4145 notes: Chapter 2, Kreyszig Chapter 1 and Chapter 5, K. Conrad Notes on Banach Fixed Point Theorem Google Pagerank | Completeness of C[0,1] with respect to the sup-metric, Banach fixed point theorem, Completions |
| week 43 | Normed spaces, equivalent norms, Schauder bases, innerproduct spaces, Cauchy-Schwarz inequality, polarization identity, projection theorem |
| week 44 Hilbert spaces | Orthogonal bases, Bessel's inequality, Parseval's identity, Gram-Schmidt |
| week 45 | Riesz representation theorem, bounded linear operators, operator norm, perturbations of linear systems, condition number |
| week 46 | Adjoint operator, QR decomposition, Least squares solution, spectral theorem for selfadjoint matrices |
| week 47 SVD | positive definite matrices, singular value decomposition, polar decompostion, pseudo inverse |