Gaussian Elimination

Definition

Gaussian Elimination is a method used to solve systems of linear equations by transforming the system into an equivalent system that is easier to solve. It involves a series of row operations to reduce the system of equations into an upper triangular form, allowing for the systematic solution of the variables.

5 Must Know Facts For Your Next Test

1. Gaussian Elimination is a systematic method for solving systems of linear equations by performing row operations to transform the system into an equivalent system that is easier to solve.
2. The goal of Gaussian Elimination is to reduce the augmented matrix of the system of equations into row echelon form, which allows for the systematic solution of the variables.
3. Gaussian Elimination involves three basic row operations: row scaling, row swapping, and row addition, which are used to eliminate the entries below the leading 1 in each column.
4. The process of Gaussian Elimination can be applied to systems of linear equations with two or more variables, and it is particularly useful for solving larger systems of equations.
5. Gaussian Elimination is a fundamental technique in linear algebra and is widely used in various fields, including numerical analysis, computer science, and engineering.

Review Questions

• Explain the purpose of Gaussian Elimination in the context of solving systems of linear equations with two variables.
  • The purpose of Gaussian Elimination in the context of solving systems of linear equations with two variables is to transform the system into an equivalent system that is easier to solve. By performing a series of row operations, the augmented matrix of the system is reduced to row echelon form, which allows for the systematic solution of the variables. This process involves eliminating the entries below the leading 1 in each column, ultimately leading to a system that can be solved using back-substitution.
• Describe how Gaussian Elimination can be used to solve systems of linear equations with three variables.
  • When solving systems of linear equations with three variables, Gaussian Elimination can be used to transform the augmented matrix of the system into an upper triangular form. This is achieved through a series of row operations, such as row scaling, row swapping, and row addition, which eliminate the entries below the leading 1 in each column. Once the augmented matrix is in upper triangular form, the system can be solved using back-substitution, where the values of the variables are determined one by one, starting from the last equation.
• Analyze the relationship between Gaussian Elimination and Cramer's Rule in the context of solving systems of linear equations.
  • Gaussian Elimination and Cramer's Rule are two distinct methods for solving systems of linear equations, but they are related in the sense that both techniques rely on the properties of matrices. While Gaussian Elimination focuses on transforming the augmented matrix of the system into row echelon form, Cramer's Rule utilizes the determinants of the coefficient matrix and the augmented matrix to find the values of the variables. Gaussian Elimination is generally more efficient for solving larger systems of equations, as it does not require the computation of multiple determinants, as in the case of Cramer's Rule. However, Cramer's Rule can provide additional insights into the properties of the system, such as the existence and uniqueness of solutions.

Related terms

Row Echelon Form:

A matrix is in row echelon form when the leading entry in each row is 1, and all entries below it in that column are 0.

Augmented Matrix:

The augmented matrix of a system of linear equations is a matrix formed by combining the coefficient matrix and the constant terms on the right-hand side of the equations.

Back-Substitution:

The process of solving a system of linear equations in upper triangular form by substituting the known values of the variables to find the remaining unknown variables.