Algèbre Linéaire (Partie 2)

Course Description

Embark on an exciting journey into the world of Linear Algebra with this comprehensive course offered by EPFLx. This course, "Algèbre Linéaire (Partie 2)," is the second part of a three-part series designed to provide a solid foundation in linear algebra. Whether you're an engineering, economics, physics, or statistics student looking to enhance your mathematical toolkit, or simply someone fascinated by the beauty of mathematics, this course is tailored for you.

Linear algebra is not just a powerful complementary tool for various scientific disciplines; it's the essence of mathematics itself, bridging algebra and geometry. This course will introduce you to the more abstract realms of mathematics while equipping you with practical problem-solving skills applicable to real-world scenarios.

What students will learn from the course

• Define and illustrate key theoretical concepts introduced in the course
• Recognize linear applications and define associated basic notions (e.g., kernel, image)
• Study linear applications of finite-dimensional vector spaces using matrix representations
• Calculate the determinant of a given matrix and effectively use its properties
• Master concepts related to diagonalization (e.g., eigenvalues/vectors/spaces, algebraic/geometric multiplicity, characteristic polynomial)
• Determine whether a given linear operator is diagonalizable
• Apply the diagonalization method to a given linear operator

Prerequisites

• A baccalaureate/federal maturity diploma
• Mastery of algebra taught at high school level
• Solid understanding of the material taught in the MOOC "Algèbre Linéaire (Partie 1)"

Course Coverage

• Linear equation systems
• Matrix algebra
• Vector spaces
• Bases and dimensions
• Linear applications
• Matrices and linear applications
• Determinants
• Eigenvectors, eigenvalues, diagonalization
• Scalar products and Euclidean spaces
• Orthogonal and symmetric matrices

Who this course is for

• First-year engineering students at the École Polytechnique Fédérale de Lausanne
• University students in economics, physics, or statistics
• Anyone interested in building a strong foundation in linear algebra
• Math enthusiasts looking to explore the beauty of abstract mathematics

Real-world applications

• Engineering: Solving complex systems and optimizing processes
• Economics: Analyzing market trends and economic models
• Physics: Understanding quantum mechanics and relativity
• Statistics: Performing data analysis and machine learning
• Computer Graphics: Creating 3D animations and game design
• Artificial Intelligence: Developing algorithms for image recognition and natural language processing

Syllabus

The course is organized into ten chapters, with this second part focusing on chapters 5-8:

1. Linear equation systems
2. Matrix algebra
3. Vector spaces
4. Bases and dimensions
5. Linear applications
6. Matrices and linear applications
7. Determinants
8. Eigenvectors, eigenvalues, diagonalization
9. Scalar products and Euclidean spaces
10. Orthogonal and symmetric matrices

The course material includes video lectures, multiple-choice quizzes, PDF exercise sets with solutions, and detailed video corrections for selected problems. Students are encouraged to work regularly and diligently to keep up with the course content.