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# Coursebooks

## Advanced linear algebra II

#### MATH-115(b)

#### Lecturer(s) :

Viazovska Maryna#### Language:

English

#### Summary

The purpose of this course is to introduce the basic notions of linear algebra and to prove rigorously the main results of the subject.#### Content

- Linear forms, dual space, bilinear forms, sesquilinear forms, symmetric and hermitian matrices, Sylvester's theorem.

- Inner products: orthonormal bases, orthogonal projections, orthogonal and unitary matrices.

- Adjoint operator, self-adjoint and normal operators, spectral theorem, singular values.

- Systems of linear differential equations with constant coefficients.

-Basics of multilinear algebra

#### Keywords

inner product, bilinearity, orthogonality, scalar product, spectral theorem

#### Learning Prerequisites

##### Required courses

Linear algebra I

#### Learning Outcomes

By the end of the course, the student must be able to:- Give an example to illustrate the basic concepts of the course
- State all definitions and theorems from the course
- Reconstruct proofs from the course
- Apply techniques from the course to various problems in mathematics and physics
- Compute basechange for linear maps, bilinear forms, sesquilinear forms; Gram matrix of a bilinear or sesquilinear form, Sylvester basis for a symmetric form, orthonormal basis for a given symmetric or symplectic form, orthogonal projection on a vector subspace, singular values of a linear map, Jordan normal form of a matrix, exponential of a matrix.
- Formulate main ideas of the course
- Synthesize major results of the course to give a `big picture' of the material and its potential applications
- Create new proof of correct statements in linear algebra
- Design counterexamples for wrong statements in linear algebra

#### Transversal skills

- Use a work methodology appropriate to the task.
- Assess one's own level of skill acquisition, and plan their on-going learning goals.
- Continue to work through difficulties or initial failure to find optimal solutions.
- Access and evaluate appropriate sources of information.

#### Teaching methods

Ex cathedra course, exercises in classroom

#### Expected student activities

Understanding the course notes, solving the exercices

#### Assessment methods

Written exam

#### Supervision

Office hours | Yes |

Assistants | Yes |

Forum | No |

#### Resources

##### Bibliography

- R. Cairoli*, Algèbre linéaire , *PressesPolytechniques Universitaires Romandes

*2e édition 1999.*

**,**- K. Hoffman, R. Kunze, *Linear Algebra,*Prentice-Hall, second edition, 1971.

-** **R. Dalang, A. Chabouni**, ***Algèbre linéaire***, **PressesPolytechniques Universitaires Romandes, 2e édition, 2004.

##### Ressources en bibliothèque

### In the programs

**Semester**Spring**Exam form**Written**Coefficient**

6**Subject examined**

Advanced linear algebra II**Lecture**

3 Hour(s) per week x 14 weeks**Exercises**

3 Hour(s) per week x 14 weeks

### Reference week

Mo | Tu | We | Th | Fr | |
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8-9 | |||||

9-10 | |||||

10-11 | |||||

11-12 | |||||

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13-14 | |||||

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16-17 | |||||

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20-21 | |||||

21-22 |

### legend

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