Linear Systems: Resolution and Solutions

This lecture covers the resolution of linear systems, focusing on finding the solutions through the concepts of principal and free unknowns. It explains the distinction between homogeneous and non-homogeneous linear equations, emphasizing the importance of echelons. The lecture also delves into the process of obtaining solutions by fixing arbitrary values for free unknowns, leading to the definition of equations with and without a right-hand side. Additionally, it explores the concept of column spaces and the reduced row echelon form, providing insights into the fundamental principles of linear algebra.

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MATH-110(a): Advanced linear algebra I

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

Power purchase agreement

A power purchase agreement (PPA), or electricity power agreement, is a long-term contract between an electricity generator and a customer, usually a utility, government or company. PPAs may last anywhere between 5 and 20 years, during which time the power purchaser buys energy at a pre-negotiated price. Such agreements play a key role in the financing of independently owned (i.e. not owned by a utility) electricity generators, especially producers of renewable energy like solar farms or wind farms.

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.

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.

Row and column spaces

In linear algebra, the column space (also called the range or ) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the or range of the corresponding matrix transformation. Let be a field. The column space of an m × n matrix with components from is a linear subspace of the m-space . The dimension of the column space is called the rank of the matrix and is at most min(m, n). A definition for matrices over a ring is also possible.

Homogeneous differential equation

A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written where f and g are homogeneous functions of the same degree of x and y. In this case, the change of variable y = ux leads to an equation of the form which is easy to solve by integration of the two members. Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives.

Linear Algebra: Elementary Row Operations MATH-110(a): Advanced linear algebra I

Covers elementary row operations, equivalent matrices, invertibility, and bases in linear algebra.