#### 8.27

Introduction and course overview

Systems of linear equations.

Gauss elimination

#### 8.29

Basic properties of matrices.

Regular matrices; \(LU\) decomposition of regular matrices.

#### 8.31

Notion of (matrix) group. Examples: groups of lower, upper triangular matrices. Matrix inverses.

Row swaps and permutation matrices. Group of permutation matrices.

Non-singular matrices and row-echelon normal form.

Determinants: definition.

#### 9.5

Determinant: existence and properties.

Matrix is singular iff its determinant vanishes.

Determinants andÂ \(LU\) decomposition.

Example: tridiagonal matrices.

Kramer’s rule.

#### 9.7

Remarkable determinants: VanderMonde, Cauchy.

Applications: Lagrange interpolation formula.

Minors, adjoint matrices.

#### 9.10

Cauchy-Binet formula and matrix tree theorem.

#### 9.12

General linear spaces.

Linear operators.

Bases. Bases changes.

#### 9.14

Normal forms: mappings between different spaces: normal forms

Endomorphisms: classification problem.

Eigenvectors. Characteristic polynomial.

#### 9.17

Applications: Markov chains; random walks; transition matrices; random walks on binary cubes and on cycles.

#### 9.19

Reminder: complex numbers. Matrix interpretation.

#### 9.21

Jordan normal forms.

Functions of matrices.

#### 9.23

Matrix exponentials.

Shift-invariant spaces of functions, linear ODEs.

#### 9.26

Conjugate spaces.

Mappings to conjugate spaces. Quadratic forms.

Normal forms for quadratic forms. Signature, inertia theorem.

#### 9.28

Applications of quadratic forms:

Statistics; variance, correlations.

Orthogonalization. Orthogonal polynomials.

#### 10.1

Self-adjoint operators and their spectra.

Rayleigh min-max theorem; spectra interlacing.

#### 10.3

Midterm 1

#### 10.5

Applications: low rank approximations (Netflix problem).

#### Week of 10.8

Rotation matrices in 3D spaces, SO(3).

Roll, pitch, yaw. Commutators; Jacobi identity.

Vector calculus.

Classical identities in 3D; scalar, vector products.

Line integrals.

Integrals over higher-dimensional chains.

Exterior forms.

#### Week of 10.15

Differential forms.

Stokes and relatives.

Applications: data analysis.

Div, grad, rot. Maxwell equations.

#### Week of 10.22

Metric spaces: completeness, examples.

Contractions and their fixpoints.

Hyperbolic spaces.

Applications: search and navigation.

Complex analysis.

Analytic functions. Cauchy-Riemann.

Laurent series. Domains of convergence.

Integrals via residues.

#### Week of 10. 29

Logarithmic residues, Rouche theorem.

Applications: Ideal fluids.

Fractional-linear mappings.

Midterm 2

#### Week of 11.5

Elements of Fourier transform.

Parseval theorem.

Inverse transform.

Duality (smoothness-localization). “Uncertainty principle”.

#### Week of 11.12.

Elements of asymptotic analysis.

Laplace method. Steepest descent.

Tauber theorems.

Asymptotics for coefficients of generating functions; Darboux transfer.

#### Week of 11.19

no classes

#### Week of 11.26

Partial differential equations.

First order equations; examples (eikonal).

Cauchy problem. Quasilinear PDEs.

#### Week of 12. 3

Wave equation: dâ€™Alembert solution.

Wave equation via Fourier series.

Elliptic problems.

#### Week of 12.10

Harmonic functions.

Dirichlet, Neumann.

Course recap.