Gauss Elimination
This presentation is the property of its rightful owner.
Sponsored Links
1 / 31

Gauss Elimination PowerPoint PPT Presentation


  • 36 Views
  • Uploaded on
  • Presentation posted in: General

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.

Download Presentation

Gauss Elimination

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Gauss elimination

Gauss Elimination


A system of linear equations

A System of Linear Equations


Two equations two unknowns lines in a plane

Two Equations, Two Unknowns: Lines in a Plane


Three possible types of solutions

Three Possible Types of Solutions

1. No solution


Three possible types of solutions1

Three Possible Types of Solutions

1. A unique solution


Three possible types of solutions2

Three Possible Types of Solutions

1. Infinitely many solutions


Three equations three unknowns planes in space

Three Equations, Three Unknowns:Planes in Space


Intesections of planes

Intesections of Planes

What type of solution sets are represented?


Solve the system

Solve the System


Elementary operations

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

Augmented Matrix


Row operations

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 system1

Solve the System


Echelon form

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 form1

Echelon Form


Echelon form2

Echelon Form


Pivot positions and pivot columns

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

Types of Solutions

1. No solution – the augmented column is a pivot column.

2. A unique solution – every column except theaugmented 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


Example1

Example


Example2

Example


Solve the system2

Solve the System


Echelon form3

Echelon Form

A rectangular matrix is in row reducedechelon 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


Reduced row echelon form1

Reduced Row Echelon Form


Solve the system3

Solve the System


Solve the system4

Solve the System


Example3

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

0

0

T6

T4

T5

0

0

20

20

20


Gauss elimination

Rank

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

Homogeneous System


Theorem

Theorem


  • Login