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Unformatted text preview: 1.1 SYSTENS OF LINEAR EQUATIONS KIAM HEONG KWA pp. 1-5 A linear system of m equations in n unknowns or an m n linear system is a system of the form a 11 x 1 + a 12 x 2 + + a 1 n x n = b 1 , (I) a 21 x 1 + a 22 x 2 + + a 2 n x n = b 2 , . . . a m 1 x 1 + a m 2 x 2 + + a mn x n = b m , where a ij s and b i s are given real numbers. The x j s are the vari- ables . A solution of such a system is any ordered n-tuple of num- bers ( x 1 ,x 2 , ,x n ) that satisfies all the equations of the system. The solution set of the same system is the collection of all its solutions. The system is called consistent if its solution set is nonempty; otherwise, it is called inconsistent . Two linear systems involving the same variables are said to be equiv- alent if and only if they have the same solution set. Operationally, two systems are equivalent if one can be obtained from another by a finite sequence of the following elementary operations : (1) Interchange of the order in which two equations are written. (2) Multiplication of an equation by a nonzero scalar. (3) Addition of one equation to another. Multiplying an equation by- 1 and adding the resulting equation to another results in the following operation: (4) Substraction of one equation from another. This is also considered by some authors as an elementary operation. The relation that two linear systems are equivalent is an equivalence re- lation in the following sense. Let A , B , and C be three linear systems involving the same variables. Then the following conditions hold: (1) A is equivalent to itself. (2) If A is equivalent to B , then so is B equivalent to A . Date : June 18, 2011. 1 2 KIAM HEONG KWA (3) If A is equivalent to B and B is equivalent to C , then A is equivalent to C ....
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