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System of Linear Equations Nattee Niparnan
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System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Dec 26, 2015

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Page 1: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

System of Linear Equations

Nattee Niparnan

Page 2: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

LINEAR EQUATIONS

Page 3: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Linear Equation

• An Equation– Represent a straight line– Is a “linear equation” in the variable x and y.

• General form– ai a real number that is a coefficient of xi

– b another number called a constant term

Page 4: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

System of a Linear Equation

• A collection of several linear equations– In the same variables

• What about– A linear equation• in the variables x1, x2 and x3

– Another equation• in the variables x1, x2,x3 and x4

– Do they form a system of linear equation?

Page 5: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Solution

• A linear equation

• Has a solution

• When

• It is called a solution to the system if it is a solution to all equations in the system

Page 6: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Number of Solution

• Solution can have– No solution

– One solution– Infinite solutions

Page 7: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Example 1

• Show that– For any value of s and t

– xi is the solution to the system

Page 8: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Example 1 Solution

Page 9: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Parametric Form

• Solution of the system in Equation 1 is described in a parametric form– It is given as a function in parameters s and t– It is called a general solution of the system

• Every linear equation system having solutions– Can be written in parametric form

Page 10: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Try another one

• Solve it using parametric form

• In term of x and z

• In term of y and z

There are several general

solutions

Page 11: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Geometrical Point of View

• In the case of 2 variables– Each equation is represent a line in 2D– Every point in the line satisfies the equation

• If we have 2 equations– 3 possibilities• Intersect in a point• Intersect as a line• Parallel but not intersect

Page 12: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

As a point No intersection

As a line

Page 13: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

3D Case

• What does Ax + By + Cz = D represent?

Page 14: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

3D Case

• A plane

Page 15: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Higher Space?

• Somewhat difficult to imagine– But Linear Algebra will, at least, provides some

characteristic for us

Cogito, ergo sum

I also speak Calculus

Page 16: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

MANIPULATING THE SYSTEM

Page 17: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Augmented Matrix

Augmented matrix

Coefficient matrix

Constantmatrix

Page 18: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Equivalent System

• System a set of linear equations– Two systems having the same

solution is said to be “equivalent”

• Some system is easier to identify the solution

• To solve a system, we manipulate it into an “easy” system that is still equivalent to the original system

System 1

System 2

System 3

Solution preserve operation

Solution preserve operation

Page 19: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Elementary Operation

Solved!

Page 20: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Elementary Operation

• Interchange two equations• Multiply one equation with a nonzero number• Add a multiple of one equation to a different

equation

Page 21: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Theorem 1

• Suppose that an elementary operation is performed on a linear equation system– Then, there solution are still the same

Page 22: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Proof

Page 23: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Elementary Row Operation

• We don’t really do the elementary operation• We write the system as an augmented matrix

and then perform “elementary row operation” on that matrix

Page 24: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Goal of Elementary Operation

• To arrive at an easy system

Page 25: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

GAUSSIAN ELIMINATION

Page 26: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Gaussian Elimination

• An algorithm that manipulate an augmented matrix into a “nice” augmented matrix

Page 27: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Row Echelon Form

• A matrix is in “Row Echelon Form” (called row echelon matrix) if– All zero rows are at the bottom– The first nonzero entry from the left in each

nonzero row is 1 • (that 1 is called a leading 1 of that row)

– Each leading 1 is to the right of all leading 1’s in the row above it

Page 28: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Example

Page 29: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Echelon?

• Diagonal Formation

Page 30: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Reduced Row Echelon

• The leading 1 is the only nonzero element in that column

row echelon

Reduced row echelon

Page 31: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Theorem 2

• Every matrix can be manipulated into a (reduced) row echelon form by a series of elementary row operations

Page 32: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Using (Reduced) Row Echelon Form

Page 33: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Using (Reduced) Row Echelon Form

No solution

Page 34: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Solution to (c)

Variable corresponding to the leading 1’s is called “leading variable”

The non-leading variables end up as a parameter in the solution

Page 35: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Gaussian Elimination

• If the matrix is all zeroes stop• Find the first column from the left containing a

non zero entry (called it A) and move the row having that entry to the top row

• Multiply that row by 1/A to create a leading 1• Subtract multiples of that row from rows below

it, making entry in that column to become zero• Repeat the same step from the matrix consists of

remaining row

Page 36: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Gauss?

Page 37: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Redundancy

Subtract 2 time row 1 from row 2AndSubtract 7 time row 1 from row 3

Subtract 2 time row 2 from row 1AndSubtract 3 time row 2 from row 3

Page 38: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Redundancy

redundancy

Observe that the last row is the triple of the second row

Page 39: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Back Substitution

• Gaussian Elimination brings the matrix into a row echelon form– To create a reduced row echelon form• We need to change step 4 such that it also create zero

on the “above” row as well• Usually, that is less efficient

• It is better to start from the row echelon form and then use the leading 1 of the bottom-most row to create zero

Page 40: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Example

Page 41: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Example

Page 42: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Another Example

Try it

Page 43: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Solution

Must be 0

Page 44: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Rank

• It is (later) shown that, for any matrix A, it has the same “Reduced row echelon form”– Regardless of the elementary row operation performed

• But it s not true for “row echelon form”– Different sequence of operations leads to different row

echelon matrix• However, the number of leading 1’s is always the same

– Will be proved later• Hence, the number of leading 1’s depends on A• The number of leading 1’s is called rank of A

Page 45: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Theorem 3

• Suppose a system of m equation on n variables has a solution, if the rank of the augmented matrix is r – the set of the solution involve exactly n-r

parameters

Page 46: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Homogeneous Equation

When b = 0What is the solution?

Page 47: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Homogeneous Linear System

• Xi = 0 is always a solution to the homogeneous system– It is called “trivial” solution

• Any solution having nonzero term is called “nontrivial” solution

Page 48: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Existence of Nontrivial Solution to the homogeneous system

• If it has non-leading entry in the row echelon form– The solution can be described as a parameter

• Then it has nonzero solution!!!– Nontrivial

• When will we have non-leading entry?– When we have more variable than equation

Page 49: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

GEOMETRICAL VIEW OF LINEAR EQUATION

Page 50: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Geometrical Point of View

• A system of Linear Equation

A line in 2D

A line in 2D

Page 51: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Column Vector view

2D vector

2D vector

2D vector

Page 52: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.
Page 53: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Network Flow Problem

• A graph of traffic– Node = intersection– Edge = road– Do we know the flow at each road?

Page 54: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Network Flow Problem

• Rules– For each node, traffic in equals traffic out

Page 55: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Formulate the System

Page 56: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

• Five equations, six vars

Page 57: System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.

Solve it