MAT 2401Linear Algebra

1.2 Part II Gauss-Jordan Elimination

http://myhome.spu.edu/lauw

HW

Written Homework

Time

Part I may be a bit longer. Part II will be shorter.

Preview

System with No solutions. System with Infinite Number of

solutions.

How many solutions?

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

Inconsistent System Consistent System

Example 4

Use Gauss-Jordan Elimination to solve the system. 3 1

2 5 2 2

2 3 1

x y z

x y z

x y z

Conclusion: 1 0 0 *

0 1 0 *

0 0 1 *

1 2 3

Example 5

Use Gauss-Jordan Elimination to solve the system. 0

3 1

3 1

y z

x z

x y

Example 5

0

3 1

3 1

y z

x z

x y

1 0 0 *

0 1 0 *

0 0 1 *

1 2 3

Geometric Interpretation

We are looking for the intersection points of 3 planes (linear equations)

It turns out that they are the same intersection points of (another) 2 planes A straight line

Remarks

The solution set is the collection of the points on the intersection line of the planes: x-3z=-1 and y-z=0.

Expectations

Some descriptions are necessary to help your audience to follow your solutions

Here, I suggested “The system becomes”. You can use similar wordings if you want.

Expectations

Since the intersection is a line, it is customary to represent it as a parametric equations (Calculus III)

3 1

,

x t

y t t R

z t

Expectations

Instead of “t is any real number”, we use the set notation

i.e. t is an element of the real numbers.

t R

System of Linear Equations (LE)

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

1 1 2 2

Linear System with equations and unknowns

n n

n n

i i in n i

m m mn n m

a x a x a x b

a x a x a x b

a x a x a x b

a x

m

b

n

x a a x

Homogeneous System of LE

11 1 12 2 1

21 1 22 2 2

1 1 2 2

1 1 2 2

0

0

0

0

n n

n n

i i in n

m m mn n

a x a x a x

a x a x a x

a x a x a x

a x a x a x

Homogeneous System of LE

Q: Is the system consistent? A:

11 1 12 2 1

21 1 22 2 2

1 1 2 2

1 1 2 2

0

0

0

0

n n

n n

i i in n

m m mn n

a x a x a x

a x a x a x

a x a x a x

a x a x a x

Trivial and Non-Trivial Solutions

Trivial Solutions

Non-Trivial Solutions At least one of the xi is non-zero.

(You need this for today’s HW)

1 2 0nx x x

Visual Summary

2 2 5 4 6 8

Linear System

x y zx y zx y z

1 1 1

2 2 5

4 6 8

Agumented Matrix

1 0 0

0 1 0

0 0 1

Unique Solution

1

0 1

0 0 0 0

No Solution

1 0 2 1

0 1 3 2

0 0 0 0

# of Solutions

.

23 ,

Parametric Solutions

Let z t

x ty t t Rz t