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

Lecture

Linear Systems Resolution: Echelon & Reduced Echelon Forms

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 covers the concepts of echelon and reduced echelon forms of matrices, focusing on the leading entries and their positions. It explains the conditions for a matrix to be in scaled or stepped form, and introduces the pivot method for solving linear systems. The instructor demonstrates how to write systems of equations from matrices and determine their compatibility, emphasizing the importance of unique solutions.

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)

Jérôme Scherer

Nicolas Boumal

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (29)

Row echelon form

In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination. A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and column echelon form means that Gaussian elimination has operated on the columns. In other words, a matrix is in column echelon form if its transpose is in row echelon form. Therefore, only row echelon forms are considered in the remainder of this article. The similar properties of column echelon form are easily deduced by transposing all the matrices.

Gaussian elimination

In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of an invertible matrix. The method is named after Carl Friedrich Gauss (1777–1855).

System of linear equations

In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by the ordered triple since it makes all three equations valid. The word "system" indicates that the equations should be considered collectively, rather than individually.

Canonical form

In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an object and allows it to be identified in a unique way. The distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a unique representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness.

Elementary matrix

In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GLn(F) when F is a field. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations. Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form.

Related lectures (43)

Row-echelon and reduced row-echelon matrices MOOC: Warm-up for EPFL MOOC: Linear Algebra (Part 1)

Explains row-echelon and reduced row-echelon matrices and their role in simplifying system resolution.

Linear Systems Resolution MOOC: Warm-up for EPFL MOOC: Linear Algebra (Part 1)

Covers the resolution of linear systems using the Gauss method to determine unique, infinite, or no solutions.

Linear Algebra: Exercises

Covers exercises on matrices, vectors, and linear systems.

Linear algebra: systems of equations

Covers linear algebra, focusing on systems of equations and their solutions.

Factorisation LU