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 – a i  a real number that is a coefficient of x i – b  another number called a constant term

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?

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

Number of Solution Solution can have – No solution – One solution – Infinite solutions

Example 1 Show that – For any value of s and t – x i is the solution to the system

Example 1 Solution

Parametric Form

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

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

As a point No intersection As a line

3D Case

A plane

Higher Space? Somewhat difficult to imagine – But Linear Algebra will, at least, provides some characteristic for us Cogito, ergo sum I also speak Calculus

MANIPULATING THE SYSTEM

Augmented Matrix Augmented matrix Coefficient matrix Constant matrix

Equivalent System System 1 System 2 System 3 Solution preserve operation

Elementary Operation Solved!

Elementary Operation

Theorem 1 Suppose that an elementary operation is performed on a linear equation system – Then, there solution are still the same

Proof

Elementary Row Operation

Goal of Elementary Operation To arrive at an easy system

GAUSSIAN ELIMINATION

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

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

Example

Echelon? Diagonal Formation

Reduced Row Echelon The leading 1 is the only nonzero element in that column row echelon Reduced row echelon

Theorem 2 Every matrix can be manipulated into a (reduced) row echelon form by a series of elementary row operations

Using (Reduced) Row Echelon Form

No solution

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

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

Gauss?

Redundancy Subtract 2 time row 1 from row 2 And Subtract 7 time row 1 from row 3 Subtract 2 time row 2 from row 1 And Subtract 3 time row 2 from row 3

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

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

Example

Another Example Try it

Solution Must be 0

Rank

Theorem 3

Homogeneous Equation When b = 0 What is the solution?

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

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

GEOMETRICAL VIEW OF LINEAR EQUATION

Geometrical Point of View A system of Linear Equation

Column Vector view

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

Network Flow Problem Rules – For each node, traffic in equals traffic out

Formulate the System

Five equations, six vars

Solve it