Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Linear Algebra: Abstract Concepts

In course

MATH-111(e): Linear Algebra

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

Description

This lecture introduces abstract concepts in linear algebra, focusing on operations with vectors and matrices. The instructor discusses the transition from concrete to abstract topics, such as polynomials as vectors. The lecture covers properties of vector spaces, including addition, scalar multiplication, and the existence of a zero vector. Through formal demonstrations, the instructor illustrates how to verify the properties of vector spaces, emphasizing the importance of understanding the definitions and operations involved.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructors (2)

Nicolas Boumal

Jérôme Scherer

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Analysis: Numerical analysis

Related concepts (32)

Polynomial

In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1. Polynomials appear in many areas of mathematics and science.

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.

Laguerre polynomials

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's differential equation: which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. Sometimes the name Laguerre polynomials is used for solutions of where n is still a non-negative integer. Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin).

Legendre polynomials

In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a vast number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions.

Orthogonal polynomials

In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases.

Related lectures (47)

Vector Spaces: Properties and Examples

Explores vector spaces, focusing on properties, examples, and subspaces within a practical exercise on polynomials.

Vector Spaces: Definitions and Applications

Introduces vector spaces, subspaces, linear maps, and evaluation maps, with examples and exercises for better comprehension.

Important Examples: Operations and Multiplication MOOC: Linear Algebra (Part 1)

Covers important examples related to operations and multiplication in real vector spaces.

Signal Representations

Covers the representation of signals in vector spaces and inner product spaces, including the Projection Theorem.

Vector Spaces: Operations and Linear Transformations