CIVIL DEPARTMENT SEMESTER-2 2015-16 G H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGY LINEAR ALGEBRA & VECTOR CALCULUS RAKSHIT KATHIRIYA 150110106025 KRUNAL KEDARIYA 150110106026 SHIVANI KETANI 150110106027 KARAN LADANI 150110106028 ROHAN LAKHANI 150110106029 LALIT SHARMA 150110106030
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system of non-linear equation (linear algebra & vector calculus)
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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
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.