Algèbre Linéaire (Partie 2) from edX

Vous voulez apprendre l'algèbre linéaire, un précieux outil complémentaire à vos connaissances acquises durant vos études en économie, ingénierie, physique, ou statistique? Ou simplement pour la beauté de la matière? Alors ce cours est fait pour vous! Outre remplir le rôle d'outil dans les différentes branches mentionnées ci-dessus (permettant la résolution de problèmes concrets), l'algèbre linéaire, qui capture l'essence des mathématiques -à savoir, l'algèbre et la géométrie- vous introduira au monde plus abstrait des mathématiques.

Proposé comme complément de cours aux ingénieurs de première année à l'Ecole Polytechnique Fédérale de Lausanne, ce MOOC (composé de trois parties) n'en est pas moins un cours à part entière et peut être considéré comme une base solide d'algèbre linéaire pour tout étudiant intéressé par l'apprentissage de cette matière.

Bien que les vidéos constituent le coeur du cours, des exercices de type QCM (Questions à choix multiples) ainsi que des séries au format PDF seront disponibles chaque semaine, ainsi que des corrigés appropriés. Plus précisément, les séries d'exercices seront accompagnées d'un corrigé au format PDF et certains problèmes bénéficieront d'une correction détaillée en vidéo, dans laquelle l'un des enseignants présentera la solution, étape par étape. Finalement, chaque vidéo de cours sera suivie d'un quiz, dont le but est de tester le degré d’assimilation des connaissances acquises.

Le cours est organisé en dix chapitres dans lesquels une approche très détaillée des concepts théoriques est proposée, ainsi que de multiples exemples illustratifs :

Systèmes d'équations linéaires.
Algèbre matricielle.
Espaces vectoriels.
Bases et dimensions.
Applications linéaires.
Matrices et applications linéaires.
Déterminants.
Vecteurs propres, valeurs propres, diagonalisation.
Produits scalaires et espaces euclidiens.
Matrices orthogonales et matrices symétriques.

Cette deuxième partie du cours sera dévouée à l'étude des chapitres 5 à 8 cités plus haut. Une bonne connaissance de la matière enseignée dans le MOOC Algèbre Linéaire (Partie 1) est requise. Aussi, il est conseillé de travailler régulièrement et de manière assidue, de façon à ne pas prendre de retard lors de l'apprentissage de la matière.

What's inside

Learning objectives

De définir les concepts théoriques introduits en cours et d'en donner des exemples illustratifs ;
De reconnaître une application linéaire et de définir les notions de base associées à un tel objet (e.g. noyau, image) ;
D’étudier les applications linéaires d'espaces vectoriels de dimension finie à l'aide des représentations matricielles ;
De calculer le déterminant d'une matrice donnée et d'utiliser les propriétés de cet objet à bon escient ;

De maîtriser les notions relatives à la diagonalisation (e.g. valeurs/vecteurs/espaces propres, multiplicité algèbrique/géométrique, polynôme caractéristique) ;
De déterminer si un opérateur linéaire donné est diagonalisable ou non ;
D’appliquer la méthode de diagonalisation à un opérateur linéaire donné.
A la fin du cours, l'étudiant sera capable:

De définir les concepts théoriques introduits en cours et d'en donner des exemples illustratifs ;
De reconnaître une application linéaire et de définir les notions de base associées à un tel objet (e.g. noyau, image) ;
D’étudier les applications linéaires d'espaces vectoriels de dimension finie à l'aide des représentations matricielles ;
De calculer le déterminant d'une matrice donnée et d'utiliser les propriétés de cet objet à bon escient ;
De maîtriser les notions relatives à la diagonalisation (e.g. valeurs/vecteurs/espaces propres, multiplicité algèbrique/géométrique, polynôme caractéristique) ;
De déterminer si un opérateur linéaire donné est diagonalisable ou non ;
D’appliquer la méthode de diagonalisation à un opérateur linéaire donné.
A la fin du cours, l'étudiant sera capable:

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Approfondit les vecteurs propres, les valeurs propres et la diagonalisation

Explore les applications linéaires et leurs propriétés

Enseigné par des professeurs reconnus dans le domaine de l'algèbre linéaire

Recommandé en tant qu'introduction à l'algèbre linéaire pour les étudiants universitaires

Requiert des connaissances préalables de la partie 1 du cours

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Algèbre Linéaire (Partie 2) with these activities:

Organize course materials

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Maintain a well-organized collection of notes, assignments, and resources for effective review and retrieval.

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Create a dedicated folder or notebook for course materials.
File notes, assignments, and other handouts in a logical and accessible manner.
Use color-coding or other methods to categorize materials for easy reference.

Review linear algebra concepts

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Fortify foundational knowledge of linear algebra to ease comprehension of advanced concepts in this course.

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Revisit matrix operations and properties.
Review vector spaces, subspaces, and linear independence.
Go over determinants and their properties.
Recall eigenvalues and eigenvectors, and diagonalization of matrices.
Practice solving systems of linear equations using various methods.

Review Basic Matrix Operations

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Get refreshed on the basics of matrix operations before starting the course to ensure a smooth transition and a better understanding of the more advanced concepts covered.

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Recall the definition of a matrix and its dimensions.
Review basic matrix operations such as addition, subtraction, multiplication, and scalar multiplication.
Practice finding the inverse of a matrix.
Solve systems of linear equations using matrices and determinants.

Eight other activities

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Solve Linear Systems Using Row Reduction

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Develop proficiency in solving linear systems using row reduction to prepare for the more complex concepts in the course.

Browse courses on Systems of Linear Equations

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Set up a linear system in matrix form.
Apply row operations to transform the matrix into row echelon form.
Solve the system of equations represented by the row echelon form.

Participate in study sessions

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Connect with peers, discuss concepts, and reinforce learning through collaborative problem-solving.

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Find a study group or connect with classmates for collaborative sessions.
Review course material and prepare questions or topics for discussion.
Engage in active discussions, explaining concepts and solving problems together.

Solve linear equations practice

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Reinforce understanding of solving linear equations and systems by practicing various techniques.

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Solve systems of linear equations using Gaussian elimination.
Apply Cramer's rule to find solutions.
Use matrix methods to solve linear systems.

Learn about Vector Spaces and Linear Independence

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Gain a strong understanding of vector spaces and linear independence, which are fundamental concepts for the remainder of the course.

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Define a vector space and its properties.
Identify whether a set of vectors forms a vector space or not.
Check for linear independence and dependence among vectors.
Apply these concepts to solve problems involving vector spaces.

Explore online tutorials on linear transformations

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Seek additional resources and explanations to enhance comprehension of linear transformations.

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Identify reliable online platforms or tutorials for linear transformations.
Watch videos or read articles to gain alternative perspectives.
Follow along with examples and practice problems.

Develop concept maps

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Visualize and connect key concepts, fostering a deeper understanding of the course material.

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Identify and list down essential concepts and their relationships.
Create a visual representation using diagrams or mind maps.
Link concepts with arrows or lines, indicating relationships.
Review and refine concept maps to strengthen understanding.

Discuss Applications of Linear Algebra in Real-World Scenarios

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Connect the theoretical concepts of the course to real-world applications to enhance comprehension and practical relevance.

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Brainstorm real-world examples where linear algebra is used.
Discuss how linear algebra is applied in specific fields such as computer graphics, data analysis, or engineering.
Share insights and experiences on the practical value of linear algebra.
Reflect on how these applications impact the understanding of the course material.

Create a Visual Representation of a Linear Transformation

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Deepen understanding of linear transformations by visualizing their geometric effects and matrix representations.

Browse courses on Linear Transformations

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Choose a linear transformation and its matrix representation.
Sketch the transformation as a geometric operation.
Create a visual representation using software or online tools.
Analyze the geometric effects of the transformation and relate them to the matrix representation.
Present the visual representation and insights to others.

Career center

Learners who complete Algèbre Linéaire (Partie 2) will develop knowledge and skills that may be useful to these careers:

Mathematician

Algèbre Linéaire (Partie 2) provides a comprehensive foundation in linear algebra, a core branch of mathematics. The course covers advanced topics such as vector spaces, linear transformations, and diagonalization, which are essential for understanding and contributing to the field of mathematics. By mastering these concepts, individuals can develop the mathematical skills necessary to conduct research, solve complex problems, and advance the field of mathematics.

See salaries and explore the career path for Mathematician

Machine Learning Engineer

Algèbre Linéaire (Partie 2) provides a solid foundation in linear algebra, a branch of mathematics that is widely used in machine learning. The course covers topics such as matrix algebra, determinants, and vector spaces, which are essential for understanding and applying machine learning algorithms, neural networks, and deep learning techniques. By mastering these concepts, individuals can develop the mathematical skills necessary to design and implement machine learning models, analyze data, and make predictions, enhancing their competitiveness in the field of machine learning engineering.

See salaries and explore the career path for Machine Learning Engineer

Quantitative Analyst

Algèbre Linéaire (Partie 2) provides a solid foundation in linear algebra, a branch of mathematics widely used in quantitative finance. The course covers topics such as matrix algebra, determinants, and linear transformations, which are essential for understanding and applying statistical models, risk management techniques, and portfolio optimization. By mastering these concepts, individuals can develop the mathematical skills necessary to analyze financial data, build predictive models, and make data-driven investment decisions, enhancing their competitiveness in the quantitative finance industry.

See salaries and explore the career path for Quantitative Analyst

Operations Research Analyst

Algèbre Linéaire (Partie 2) provides a strong foundation in linear algebra, a branch of mathematics that is widely used in operations research. The course covers topics such as systems of linear equations, matrix algebra, and linear programming, which are essential for understanding and applying optimization techniques used in logistics, supply chain management, and resource allocation. By mastering these concepts, individuals can develop the mathematical skills necessary to analyze complex systems, formulate mathematical models, and make data-driven decisions, enhancing their effectiveness in the operations research field.

See salaries and explore the career path for Operations Research Analyst

Statistician

Algèbre Linéaire (Partie 2) provides a strong foundation in linear algebra, a branch of mathematics that is widely used in statistics. The course covers topics such as matrix algebra, determinants, and vector spaces, which are essential for understanding and applying statistical models, data analysis techniques, and hypothesis testing. By mastering these concepts, individuals can develop the mathematical skills necessary to analyze data, build predictive models, and make informed decisions, enhancing their competitiveness in the statistics field.

See salaries and explore the career path for Statistician

Data Scientist

Algèbre Linéaire (Partie 2) provides a strong foundation in linear algebra, a branch of mathematics used extensively in data science. The course covers concepts such as matrices, determinants, vector spaces, and linear transformations, which are essential for understanding and applying machine learning algorithms, statistical modeling, and data analysis. By mastering these concepts, individuals can develop the mathematical skills necessary to extract insights from complex datasets, build predictive models, and make data-driven decisions, enhancing their competitiveness in the data science field.

See salaries and explore the career path for Data Scientist

Data Analyst

Algèbre Linéaire (Partie 2) provides a strong foundation in linear algebra, a branch of mathematics that is widely used in data analytics. The course covers topics such as matrix algebra, determinants, and vector spaces, which are essential for understanding and applying statistical models, machine learning algorithms, and data visualization techniques. By mastering these concepts, individuals can develop the mathematical skills necessary to analyze large datasets, extract meaningful insights, and make data-driven decisions, enhancing their competitiveness in the data analytics field.

See salaries and explore the career path for Data Analyst

Financial Analyst

Algèbre Linéaire (Partie 2) provides a solid foundation in linear algebra, a branch of mathematics that plays a vital role in financial analysis. The course covers concepts such as matrix algebra, determinants, and linear transformations, which are essential for understanding and applying statistical models, financial forecasting techniques, and portfolio optimization. By mastering these concepts, individuals can develop the mathematical skills necessary to analyze financial data, build predictive models, and make informed investment decisions, enhancing their competitiveness in the financial analysis field.

See salaries and explore the career path for Financial Analyst

Actuary

Algèbre Linéaire (Partie 2) provides a strong foundation in linear algebra, a branch of mathematics that plays a vital role in actuarial science. The course covers concepts such as systems of linear equations, matrix algebra, and vector spaces, which are essential for understanding and applying statistical models used in risk assessment, insurance pricing, and financial planning. By mastering these concepts, individuals can develop the mathematical skills necessary to analyze data, evaluate risks, and make informed decisions in the actuarial field.

See salaries and explore the career path for Actuary

Computer Scientist

Algèbre Linéaire (Partie 2) provides a strong foundation in linear algebra, a branch of mathematics that is widely used in computer science. The course covers topics such as matrix algebra, linear transformations, and vector spaces, which are essential for understanding and applying algorithms, data structures, and computer graphics. By mastering these concepts, individuals can develop the mathematical skills necessary to design and implement efficient algorithms, optimize performance, and create visually appealing applications.

See salaries and explore the career path for Computer Scientist

Physicist

Algèbre Linéaire (Partie 2) provides a strong foundation in linear algebra, a branch of mathematics that is widely used in physics. The course covers topics such as matrix algebra, determinants, and vector spaces, which are essential for understanding and applying quantum mechanics, electromagnetism, and classical mechanics. By mastering these concepts, individuals can develop the mathematical skills necessary to analyze physical systems, solve complex problems, and contribute to the advancement of physics.

See salaries and explore the career path for Physicist

Engineer

Algèbre Linéaire (Partie 2) provides a solid foundation in linear algebra, a branch of mathematics that is widely used in engineering. The course covers topics such as matrix algebra, linear transformations, and vector spaces, which are essential for understanding and applying concepts in fields such as structural analysis, fluid mechanics, and control systems. By mastering these concepts, engineers can develop the mathematical skills necessary to design and analyze complex systems, solve engineering problems, and advance the field of engineering.

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Software Engineer

Algèbre Linéaire (Partie 2) provides a solid foundation in linear algebra, a branch of mathematics used in various aspects of software engineering. The course covers topics such as matrix algebra, linear transformations, and vector spaces, which are fundamental concepts for understanding computer graphics, image processing, and numerical simulations. By gaining expertise in these concepts, software engineers can develop robust and efficient algorithms, optimize performance, and create visually appealing applications.

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Math Teacher

The course Algèbre Linéaire (Partie 2) provides a foundational understanding of linear algebra, which is a branch of mathematics that is widely used in various fields, including education. Linear algebra is essential for developing the skills needed to teach mathematics at the secondary level. The course covers topics such as systems of linear equations, matrix algebra, vector spaces, and linear transformations, which are fundamental concepts in linear algebra. By gaining a solid understanding of these concepts, individuals can enhance their knowledge and pedagogical skills in teaching mathematics, particularly in areas such as geometry, algebra, and statistics.

See salaries and explore the career path for Math Teacher

Economist

Algèbre Linéaire (Partie 2) may be useful for economists seeking to advance their understanding of the mathematical foundations of economics. The course covers topics such as matrix algebra, linear transformations, and vector spaces, which are used in economic modeling, econometrics, and optimization. By gaining a solid grasp of these concepts, economists can enhance their ability to analyze economic data, build mathematical models, and make informed economic decisions.

See salaries and explore the career path for Economist