Row-Echelon Form

 

In linear algebra, a matrix is in echelon form if it has the shape resulting of a Gaussian elimination. 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.

 

Specifically, a matrix is in row echelon form if

 

Example:

 A 3×5 matrix in row echelon form:

Elementary Row Operations

Three operations the can be used on a system of linear equations to produce equivalent systems.

1. Interchange two equations.
2. Multiply an equation by a nonzero constant.
3. Add a multiple of an equation to another equation.

Two types of Matrix

• Augmented Matrix - the matrix derived from the coefficients and constant terms of a system of linear equations.

Example:

• Coefficient Matrix - the matrix containing only the coefficients of the system.

Example:

Matrix

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries.
An example of a matrix with 2 rows and 3 columns is

 

 

 

 

 

 

 

What is Linear Algebra?

Linear algebra is the study of linear sets of equations and their transformation properties. Linear algebra allows the analysis of rotations in space, least squares fitting, solution of coupled differential equations, determination of a circle passing through three given points, as well as many other problems in mathematics, physics, and engineering. The solving of the simple linear equation ax + b = 0 may be
considered as the original problem of this subject.


History
The study of linear algebra first emerged from the study of determinants, which were used to solve systems of linear equations. Determinants were used by Leibniz in 1693, and subsequently, Gabriel Cramer devised Cramer's Rule for solving linear systems in 1750. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination, which was initially listed as an advancement in geodesy.

Welcome to my Blog!

Hello! This is Ara, 15 years of age and I'm studying at Regional Science High School III. This blog will have most our lessons or topics discussed in our school.