CIV IL DEPARTMENT SEMESTER-22015-16

G H PATEL COLLEGE OF ENGINEERING

AND TECHNOLOGY

L I N E A R A L G E B R A & V E C T O R C A L C U L U S

RAKSHIT KATHIRIYA150110106025KRUNAL KEDARIYA150110106026SHIVANI KETANI150110106027KARAN LADANI150110106028ROHAN LAKHANI150110106029LALIT SHARMA150110106030

SYSTEM OF NON-LINEAR EQUATIONS

CONTENT ROW-ECHELON FORM

REDUCED ROW-ECHELON FORM

AUGMENTED MATRIX

SYSTEM OF NON-LINEAR EQUATIONS

SOLUTION FOR SYSTEM OF NON-LINEAR EQUATIONS

EXAMPLES

ROW-ECHELON FORM A matrix is said to be in row-echelon form if it satisfies the following properties:

1. If the matrix has any zero rows, then they are at the bottom of the matrix.

2. In any nonzero row, the first nonzero entry is 1. it is named as leading 1.

3. Each leading 1 is to the right of the leading 1 in the previous row.

EXAMPLES :

1 5 6 -40 1 3 -70 0 1 -5

0 1 -8 9 30 0 1 5 40 0 0 1 -90 0 0 0 10 0 0 0 0

REDUCED ROW-ECHELON FORM A matrix is said to be in reduced row-echelon form if it satisfies the following properties:

1. It is in row-echelon form.2. Each leading 1 is the only nonzero entry in its column.

EXAMPLES :

1 0 0 00 1 0 00 0 1 00 0 0 1

1 7 0 0 20 0 1 0 30 0 0 1 40 0 0 0 50 0 0 0 0

AUGMENTED MATRIX In linear algebra, an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices. Given the matrices A and B, where

A=

the augmented matrix (A|B) is written as 1 3 2 4

(A|B)= 2 0 1 3 5 2 2 1

1 3 2 2 0 1 5 2 2

4 3 1

B=

SYSTEM OF NON-LINEAR EQUATIONSA non-linear system of equations is a system in which at least one of the variables has an exponent other than 1 and/or there is a product of variables in one of the equations. EXAMPLES :

sinα + 2cosβ + 3tanγ= 0

2sinα + 5cosβ +3tanγ = 0

-sinα - 5cosβ + 5tanγ= 0SYSTEM OF HOMOGENEOUS

NON-LINEAR EQUATIONS

2sinα - cosβ + 3tanγ= 3

4sinα + 2cosβ - 2tanγ = 2

6sinα - 3cosβ + tanγ = 9SYSTEM OF NON-HOMOGENEOUS

NON-LINEAR EQUATIONS

SYSTEM OF NON-LINEAR EQUATIONS Note that in a nonlinear system, one of our equations can be linear, just not all of them.

System of following equations: X2 + Y2 = 100

(Nonlinear) Y – X = 2 (Linear)E

xa

mp

le:

X2 + Y2 = 100Y –

X =

2

METHODS TO SOLVE NON-LINEAR SYSTEM

There are two methods to solve the system of non-linear equations:1. Gauss Elimination Method2. Gauss-Jordan Elimination Method

GAUSS ELIMINATION METHOD

Write a system of linear equations as an augmented matrix. Perform the elementary row operations to put the matrix into row-echelon form. convert the matrix back into a system of linear equations. Use back substitution to obtain all the answer.

GAUSS-JORDAN ELIMINATION METHOD

Write a system of linear equations as an augmented matrix. Perform the elementary row operations to put the matrix into reduced row-echelon form. convert the matrix back into a system of linear equations. No back substitution is necessary.

SOLUTION OF A SYSTEM In general, a solution of a system in two variables is an ordered

pair that makes BOTH equations true. 

In other words, it is where the two graphs intersect, what they have in common.  So if an ordered pair is a solution to one equation, but not the other, then it is NOT a solution to the system.

For nonlinear systems, in some cases, there may be more than one ordered pair that satisfies all equations in the system.

SOLUTION OF A SYSTEM A consistent system is a system that has at least one solution. An inconsistent system is a system that has no solution.

CONSISTENT SYSTEM

INCONSISTENT SYSTEM

SOLUTION OF A SYSTEM The equations of a system are dependent if all the solutions of

one equation are also solutions of the other equation.  In other words, they end up being the same graph.

SOLUTION OF A SYSTEM The equations of a system are independent if they do not share

all solutions.  They can have one point in common, just not all of them.

SOLUTION OF A SYSTEM There are three possible outcomes that we may

encounter when working with these systems:1)A finite number of solutions2)No solution3)Infinite solutions

A FINITE NUMBER OF SOLUTIONS If the system in two variables has one solution, it is an ordered pair that is a solution to BOTH equations.  

In nonlinear systems, in some cases there may be more than one ordered pair that satisfies all equations in the system.

If we get finite number of solution for the system then the system would be

Consistent

Independent

A FINITE NUMBER OF SOLUTIONS The graph below illustrates a system of two equations and two unknowns that has four solutions:

Finite number of solution

Consistence system Independent

equations

NO SOLUTION In some cases, the equations in the system will not have any points in common.  In this situation, you would have no solution. 

If we don’t get any solution for the system then the system would be

Inconsistent

Independent

NO SOLUTION The graph below illustrates a system of two equations and two unknowns that has no solution:

No intersection – No solution

Inconsistence system Independent equations

INFINITE SOLUTIONS If the two graphs end up lying on top of each other, then there is an infinite number of solutions. 

In this situation, they would end up being the same graph, so any solution that would work in one equation is going to work in the other.  If we get finite number of solution for the system then the system would be

Consistent

Dependent

INFINITE SOLUTIONS The graph below illustrates a system of two equations and two unknowns that has an infinite number of solutions:

Same circles – Infinite number of solutions

Consistence system Dependent equations

TIPS TO DETERMINE THE TYPE OF SOLUTION

Here are some tips that will allow us to determine what type of solutions we have based on either the reduced-row echelon form.

1. If we have a leading one in every column, then we will have a unique solution. 2. If we have a row of zeros equal to a number for a nonhomogeneous system,then the system has no solution. 3. If we don’t have a leading one in every column in a homogeneous system, then we have infinite solutions.

E X A M P L E - 1 :

Solve the following nonlinear system for the unknown angles α, β and γ, where 0≤α≤2π, 0≤β≤2π, 0≤γ≤π.

2sinα – cos β + 3tanγ = 3

4sinα + 2cosβ - 2tanγ = 2

6sinα - 3cosβ + tanγ = 9

SOLUTION: Augmented matrix = Applying we get

Applying and we get

Applying and we get

Applying we get

SOLUTION:Applying and +2 we get

This matrix is in reduced row-echelon form.

So, sinα = 1 cos β = -1 tanγ = 0

(0≤α≤2π) (0≤β≤2π) (0≤γ≤π)

THANK YOU