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MOOC# Algebra (part 1)

Description

Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.

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Related concepts (410)

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Related publications (73)

Related courses (453)

Lectures in this MOOC (145)

Matrix (mathematics)

In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.

Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.

Invertible matrix

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular, nondegenerate or (rarely used) regular), if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A, and is called the (multiplicative) inverse of A, denoted by A−1. Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.

MATH-111(e): Linear Algebra

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.

MATH-111(a): Linear Algebra

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.

MATH-115(b): Advanced linear algebra II

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et de démontrer rigoureusement les résultats principaux du sujet.

Systems of Linear Equations and MatricesMOOC: Algebra (part 1)

Covers the definition and solution of linear equations with real coefficients in 2D and 3D space.

Number of Solutions in Linear SystemsMOOC: Algebra (part 1)

Explores the number of solutions in linear systems of equations and the conditions for no solution, a unique solution, or an infinite number of solutions.

Matrix Operations: Triangular MatricesMOOC: Algebra (part 1)

Explores operations with triangular matrices and their role in solving linear systems.

Linear Equations and Matrices: Solution Methods, Elementary OperationsMOOC: Algebra (part 1)

Explores solution methods and elementary operations for linear equations and matrices, demonstrating techniques to simplify systems and obtain solutions.

Elementary Operations: Matrix VersionMOOC: Algebra (part 1)

Covers elementary operations in matrix form, matrix size, component notation, equality, augmented matrices, and row operations.

The field of computational topology has developed many powerful tools to describe the shape of data, offering an alternative point of view from classical statistics. This results in a variety of complex structures that are not always directly amenable for machine learning tasks. We develop theory and algorithms to produce computable representations of simplicial or cell complexes, potentially equipped with additional information such as signals and multifiltrations. The common goal of the topics discussed in this thesis is to find reduced representations of these often high dimensional and complex structures to better visualize, transform or formulate theoretical results about them. We extend the well known graph learning algorithm node2vec to simplicial complexes, a higher dimensional analogue of graphs. To this end we propose a way to define random walks on simplicial complexes, which we then use to design an extension of node2vec called k-simplex2vec, producing a representation of the simplices in a Euclidean space. Furthermore, the study of this method leads to interesting questions about robustness of graph and simplicial learning methods. In the case of graphs, we study node2vec embeddings arising from different parameter sets, analysing their quality and stability using various measures. In the topic of signal processing, we explore how discrete Morse theory can be used for compression and reconstruction of cell complexes equipped with signals. In particular we study the effect of the compression of a complex on the Hodge decomposition of its signals. We study how the signal changes through compression and reconstruction by introducing a topological reconstruction error, showing in particular that part of the Hodge decomposition is preserved. Moreover, we prove that any deformation retract over R can be expressed as a Morse deformation retract in a well-chosen basis, thus extending the reconstruction results to any deformation retract. In addition, we introduce an algorithm to minimize the loss induced by the reconstruction of a compressed signal. Finally, we use discrete Morse theory to compute an invariant of multi-parameter persistent homology, the rank invariant. We can restrict a multi-parameter persistence module to a one- dimensional persistence module along any line of positive slope and compute the one-dimensional analogue of the rank invariant, namely the barcode. Through a discrete Morse matching we can determine critical values in the multifiltration, which in turn allows us to identify equivalence classes of lines in the parameter space. In our main result, we explain how to compute the barcode along any given line of an equivalence class given the barcode along a representative line. This provides a way to fiber the rank invariant according to the critical values of a discrete Morse matching and to perform computations in the corresponding one-dimensional module, which is much better understood.

Let G be a simple algebraic group over an algebraically closed field F of characteristic p >= h, the Coxeter number of G. We observe an easy 'recursion formula' for computing the Jantzen sum formula of a Weyl module with p-regular highest weight. We also discuss a 'duality formula' that relates the Jantzen sum formula to Andersen's sum formula for tilting filtrations and we give two different representation theoretic explanations of the recursion formula. As a corollary, we also obtain an upper bound on the length of the Jantzen filtration of a Weyl module with p-regular highest weight in terms of the length of the Jantzen filtration of a Weyl module with highest weight in an adjacent alcove.

Let k be an algebraically closed field of arbitrary characteristic, let G be a simple simply connected linear algebraic group and let V be a rational irreducible tensor-indecomposable finite-dimensional kG-module. For an element g of G we denote by $V_{g}(x)$ the eigenspace corresponding to the eigenvalue x of g on V. We define N to be the minimum difference between the dimension of V and the dimension of $V_{g}(x)$, where g is a non-central element of G. In this thesis we identify pairs (G,V) with the property that $N\leq \sqrt{\dim(V)}$. This problem is an extension of the classification result obtained by Guralnick and Saxl for the condition $N\leq \max\bigg\{2,\frac{\sqrt{\dim(V)}}{2}\bigg\}$. Moreover, for all the pairs (G,V) we had to consider in our classification, we will determine the value of N.