CHAPTER ONE
LINEAR SYSTEM OF EQUATIONS
A wide variety of problems lead ultimately to the need to solve a linear system of equation linear system of equations are associated with many problems in engineering and science as well as with applications of mathematics to the social sciences and the quantitative study of business and economic problems.
In 1985, according to Atkinson, system of Simultaneous linear equation occur in solving problems in a wide variety of areas with respect to mathematics, statistics, physical quantities (examples are temperature, voltage, population management and displacement). Social sciences, engineering and business. They arise directly in solving real life problems.
The world sometimes reveals itself to us as observable relationships among the relevant variables what it does make evident are relationship that describe how both the variable and their rate of change affect each other.
Apparently, such life changing problem gives rise to systems of simultaneous linear equation. In almost every human activities, man seems to be compelled to uncover fundamental relationship that exist among the objects he observes. According to Maron in 1982, he said in order to make the relationship that exist between variables explicit, we frequently attempt to make a mathematical model that will accurately reflect real life situation. Many mathematical model that will accurately reflect real life situation. Many mathematical models have the same basic structure although disparity in Symbolic rotation may be utilized, which can arise from economics, transportation, which need may arise to make efficient allocation among several points or to solve the growth of population in which units of x1, x2 ...., xn arises from net flow from one point to another or in relationship to population growth, that is, number of individuals in a particular age group at a particular time.
There are various methods in solving linear system of simultaneous equations. In numerical analysis the techniques and methods for solving system of linear equations belongs to two categories: Direct and Iterative methods. The direct methods obtain the exact solution (in real arithmetic) in finitely many operations where as iterative method generate a sequence of approximations that only converge in the limit to the solution. The direct method falls into two categories or clam that is the Gaussian elimination method and cholesky decomposition method. Some others are matrix inverse method and LU factorization method and the Cramer’s rule method.
The elimination approach reduces the given system of equations to a form from which the solution can be obtained by simple substitution since calculators and computers have some limit to the number of digits for their use this may lead to round-off errors and produces poorer results. Generally, the direct method are best for full or bounded matrices where as iterative methods are best for very large and sparse matrices. The iterative method provide an alternative to the direct methods for solving systems of linear equations. This method involves assumption of some initial values which are then refined repeatedly till they reach some accepter rang of accuracy. The Jacobi and Gawn-siedel methods are good examples of the iterative method.
Systems of linear equations may be grouped as follows
|
System of linear equations |
|
Inconsistent |
|
No solution |
|
Consistent |
|
Unique solution |
|
Infinite no solution |
The system of linear equations are divided into consistent and inconsistent and inconsistent.
The inconsistent equation is an equation that has numbers that are not all zero that is the equation has no solution.
For example
X + 2y – 3z = 4
Y + 2z = 5
The consistent equation is an equation that has numbers that are all zero, that is, the equation has a solution. There are two cases
CASE I
r = n, that is, there are so many non-zero equations as unknowns. Then we can successfully solve uniquely for the unknowns and there exist a unique solution for the system.
CASE II
r < n, m that is, there are more unknowns than there are non-zero equations. Thus, we arbitrarily assign values to the unknowns and then solve uniquely for the unknowns. Accordingly, there exist an infinite number or solutions. For example
Since there is no equation of the form 0 = c with c
0 the system is consistent. Furthermore since there are three unknowns and three non zero equations the system has a unique solution.
Also, the system
X + 2y – 3z + w = 4
y + 2z +3w = 5
is consistent and since there are more unknowns than non-zero equations, the system has an infinite number of solution. On the other hand, a linear system can be written in matrice form as follows
A x = b.
A linear equation X1, X2, ..., Xn is obtained by requiring the linear combination to assume a prescribed value b, that is
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