#### 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.

#### 9.5

Determinants: definition, existence and properties.

Matrix is singular iff its determinant vanishes.

#### 9.7

Determinants andÂ \(LU\) decomposition.

Example: tridiagonal matrices.

Kramer’s rule.

#### 9.10

Remarkable determinants: VanderMonde, Cauchy.

Applications: Lagrange interpolation formula.

#### 9.12

Minors, adjoint matrices.

Cauchy-Binet formula.

#### 9.14

Matrix tree theorem.

#### 9.17

General linear spaces.

Linear operators.

#### 9.19

Bases. Bases changes.

#### 9.21

Normal forms: mappings between different spaces.

#### 9.24

Endomorphisms: classification problem.

#### 9.26

Eigenvectors. Characteristic polynomial.

#### 9.28

Jordan normal forms.

#### 10.1

Cayley-Hamilton.

#### 10.3

Functions of matrices.

Matrix exponentials.

#### 10.5

Midterm 1

#### 10.8

Euclidean and Hermitean spaces.

#### 10.10

Self-adjoint operators and their spectra.

#### 10.12

Orthogonalization. Orthogonal polynomials.

#### 10.15

Rayleigh min-max theorem

#### 10.17

Spectra interlacing.

#### 10.19

Exterior forms.

#### 10.22

Exterior differential forms. Line integrals

#### 10.24

Integrals over higher-dimensional chains.

Exterior differentiation

Stokes and relatives.

#### 10.26

Classical identities in 3D; scalar, vector products.

Div, grad, rot.

#### 10. 29

Applications: winding numbers.

Complex analysis. Complex analytic functions.

#### 10. 31

Analytic functions. Cauchy-Riemann.

Integrals along a curve.

#### 11. 2

Laurent series. Domains of convergence.

#### 11. 5

Integrals via residues.

#### 11. 7

Midterm 2

#### 11.9

Integrals involving multivalued functions.

#### 11.12

Fresnel integral. Logarithmic residues. Rouche theorem.

#### 11.14.

Elements of Fourier transform.

Parseval theorem.

#### 11.16.

Inverse transform.

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

#### Week of 11.19

no classes

#### Week of 11.26

Elements of asymptotic analysis.

Laplace method. Steepest descent.

Tauber theorems.

Asymptotics for coefficients of generating functions; Darboux transfer.

#### Week of 12.10

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.

Harmonic functions.

Dirichlet, Neumann.

#### Additional material

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

Roll, pitch, yaw.

Commutators, Jacobi identity.Vector calculus.

Metric spaces: completeness, examples.

Contractions and their fixpoints.

Hyperbolic spaces.

Applications: search and navigation.

Applications: Ideal fluids.

Fractional-linear mappings.

Course recap.