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Lecture

Linear Systems Resolution

Description

This lecture covers the resolution of linear systems using the Gauss method to transform the augmented matrix into a row echelon form. By analyzing the resulting matrix, it is possible to determine if the system has a unique solution, infinite solutions, or no solution. The process involves identifying pivot elements and free variables, which are crucial in understanding the nature of the system's solutions.

In MOOCs (2)

Warm-up for EPFL

Linear Algebra (Part 1)

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (29)

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.

Linear system

In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be modeled by linear systems.

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.

Augmented matrix

In linear algebra, an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices. Given the matrices A and B, where the augmented matrix (A|B) is written as This is useful when solving systems of linear equations. For a given number of unknowns, the number of solutions to a system of linear equations depends only on the rank of the matrix representing the system and the rank of the corresponding augmented matrix.

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.

Related lectures (38)

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

Covers the resolution of linear systems through the Gaussian elimination method.

System of Equations: Parameters and Solutions MOOC: Linear Algebra (Part 1)

Explores solving systems of equations with parameters to find different types of solutions.

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.

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Column Rank and Systems of Equations MOOC: Linear Algebra (Part 1)