mat 2401 linear algebra 1.1, 1.2 part i gauss- jordan elimination

MAT 2401Linear Algebra

1.1, 1.2 Part I Gauss-Jordan Elimination

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HW

Written Homework

Time

Part I may be a bit longer today. Part II will be shorter next time.

Preview

Introduce the Matrix notations. Study the Elementary Row

Operations. Study the Gauss-Jordan Elimination.

Example 1

2 5Solve

4 6

x y

x y

Elimination:

Example 1

2 5Solve

4 6

x y

x y

Elimination: Geometric Meaning

How many solutions?

Q: Given a system of 2 equations in 2 unknowns, how many solutions are possible?

A:

How many solutions?

Q: Given a system of 3 equations in 3 unknowns, how many solutions are possible?

A:

How many solutions?

Q: Given a system of 3 equations in 3 unknowns, how many solutions are possible?

______ System ______ System

Unique Solution

We will focus only on systems of unique solution in part I.

Such systems appear a lot in applications.

Example 2

Elimination:

4

Solve 2 2 5 11

4 6 8 24

x y z

x y z

x y z

Observation 1

Q: Why eliminations are not good?

A:1.2.3.

Observation 2

Compare the 2 systems:

Q: Are the 2 systems the same?

A:

4 4

2 2 5 11 1 , 3 2 4 8

4

6 8 24 ,

2

1 3 3

x y z x y z

x y z

Before Afte

y

r

z

x y z z

Observation 2

Compare the 2 systems:

Q: What do the 2 systems have in common?

A:

4 4

2 2 5 11 1 , 3 2 4 8

4

6 8 24 ,

2

1 3 3

x y z x y z

x y z

Before Afte

y

r

z

x y z z

Observation 2

Compare the 2 systems:

4 4

2 2 5 11 1 , 3 2 4 8

4

6 8 24 ,

2

1 3 3

x y z x y z

x y z

Before Afte

y

r

z

x y z z

Observation 2

Compare the 2 systems:

Q: Which system is easier to solve?

A:

4 4

2 2 5 11 1 , 3 2 4 8

4

6 8 24 ,

2

1 3 3

x y z x y z

x y z

Before Afte

y

r

z

x y z z

Extreme Makeover?

We want a solution method that it is systematic, extendable, and

easy to automate it can transform a complicated

system into a simple system

Extreme Makeover?

We want a solution method that it is systematic, extendable, and

easy to automate it can transform a complicated

system into a simple system

4 4

2 2 5

11 2 4 8

4 6 8 2 3 3

4

x y z x y z

x y z

Before Afte

z

x

r

y

y z z

Extreme Makeover?

We want a solution method that it is systematic, extendable, and

easy to automate it can transform a complicated

system into a simple system

4 4

2 2 5

11 2 4 8

4 6 8 2 3 3

4

x y z x y z

x y z

Before Afte

z

x

r

y

y z z

Extreme Makeover?

We want a solution method that it is systematic, extendable, and

easy to automate it can transform a complicated

system into a simple system

4 1

2 2 5 11 2

4 6 8 2

4

1

x y z x

x y z y

Before After

x y z z

Gauss-Jordan Elimination

4 1

2 2 5 11 2

4 6 8 2

4

1

x y z x

x y z y

Before After

x y z z

Gauss-Jordan Elimination

Before we can describe our systematic solution method, we need the matrix notations.

Essential Information

A system can be represented compactly by a “table” of numbers.

4 6 4 1 6

2 5 1 2 5

x y

x y

Matrix

A matrix is a rectangular array of numbers.

If a matrix has m rows and n columns, then the size of the matrix is said to be mxn.

1 2

1

2

n

m

Example 2

4

2 2 5 11

4 6 8 24

x y z

x y z

x y z

Write down the (Augmented) matrix representation of the given system.

Coefficient Matrix

4

2 2 5 11

4 6 8 24

x y z

x y z

x y z

The left side of the Augmented matrix is called the Coefficient Matrix.

Elementary Row Operations

We can perform the following operations on the matrix

1. Switching 2 rows.2. Multiplying a row by a constant.3. Adding a multiple of one row to another.

Elementary Row Operations

We can perform the following operations on the matrix

1. Switching 2 rows.

44

2 2 5 11 11

4

1 1 1

52 2

6 8 2 84 2464

x y z

x y z

x y z

Elementary Row Operations

We can perform the following operations on the matrix

2. Multiplying a row by a constant.

44

2 2 5 11 11

4

1 1 1

52 2

6 8 2 84 2464

x y z

x y z

x y z

Elementary Row Operations

We can perform the following operations on the matrix

3. Adding a multiple of one row to another.

44

2 2 5 11 11

4

1 1 1

52 2

6 8 2 84 2464

x y z

x y z

x y z

Elementary Row Operations

Theory: We can use the operations to simplify the system without changing the solution.1. Switching 2 rows.2. Multiplying a row by a constant.3. Adding a multiple of one row to another.

Elementary Row Operations

Notations (examples)

1. Switching 2 rows.2. Multiplying a row by a constant.3. Adding a multiple of one row to another.

1 2R R

2 1 23R R R

3 3

1

3R R

Gauss-Jordan Elimination

Main Idea: We want to use elementary row operations to get the matrix into the form (reduced row-echelon form RREF)

1 0 0 *

0 1 0 *

0 0 1 *

Gauss-Jordan Elimination

Main Idea: We want to use elementary row operations to get the matrix into the form (reduced row-echelon form RREF)

The order of creating “0” and “1” is extremely important!

1 0 0 *

0 1 0 *

0 0 1 *

1 2 3

Example 2

1 1 1 4

52 2 11

6 84 24

x

y

z

1 0 0 *

0 1 0 *

0 0 1 *

1 2 3

Remarks

Notice sometimes 2 “parallel” row operations can be done in the same step.

The procedure (algorithm) is designed so that the exact order of creating the “0”s and “1”s is important.

Remarks

Try to avoid fractions!!

How do I Confirm My Answers?

Example 3

Use Gauss-Jordan Elimination to solve the system.

3 3 1

2 3

2 3 4

x y z

x y z

x y z

Example 3

x

y

z

3 31 1

31 2 1

32 1 4

1 0 0 *

0 1 0 *

0 0 1 *

1 2 3