Gauss Elimination

A System of Linear Equations

Two Equations, Two Unknowns: Lines in a Plane

Three Possible Types of Solutions 1. No solution

Three Possible Types of Solutions 1. A unique solution

Three Possible Types of Solutions 1. Infinitely many solutions

Three Equations, Three Unknowns: Planes in Space

Intesections of Planes What type of solution sets are represented?

Solve the System

Elementary Operations Interchange the order in which the equations are listed. Multiply any equation by a nonzero number. Replace any equation with itself added to a multiple of another equation.

Augmented Matrix

Row Operations Switch two rows. Multiply any row by a nonzero number. Replace any row by a multiple of another row added to it.

Solve the System

Echelon Form A rectangular matrix is in echelon form if it has the following properties: 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry of a row is in a column to the right of the leading entry of the row above it.

Echelon Form

Echelon Form

Pivot Positions and Pivot Columns The positions of the first nonzero entry in each row are called the pivot positions. The columns containing a pivot position are called the pivot columns.

Types of Solutions 1. No solution – the augmented column is a pivot column. 2. A unique solution – every column except the augmented column is a pivot column. 3. An infinite number of solutions – some column of the coefficient matrix is not a pivot column. The variables corresponding to the columns that are not pivot columns are assigned parameters. These variables are called the free variables. The other variables may be solved in terms of the parameters and are called basic variables or leading variables.

Example

Example

Example

Solve the System

Solve the System

Echelon Form A rectangular matrix is in row reduced echelon form if it has the following properties: 1. It is in echelon form. 2. All entries in a column above and below a leading entry are zero. 3. Each leading entry is a 1, the only nonzero entry in its column.

Reduced Row Echelon Form

Reduced Row Echelon Form

Solve the System

Solve the System

Example Estimate the temperatures T1, T2, T3, T4, T5, and T6 at the six points on the steel plate below. The value Tk is approximated by the average value of the temperature at the four closest points. 20 20 20 T1 T2 T3 T4 T5 T6 20 20 20

Number of free variable = n – rank(A) The rank of a matrix is the number of nonzero rows in its row echelon form. Rank Theorem Let A be the coefficient matrix of a system of linear equations with n variables. If the system is consistent, then Number of free variable = n – rank(A)

Homogeneous System

Theorem