stnote131-117-ch1.pdf - Chapter 1 Systems of Linear...

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Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as “ linear algebra. ” In the first section we shall introduce some basic terminology and discuss a method for solving such system. 1.1 Systems of Linear Equations Linear Equations Any straight line in the xy -plane can be represented algebraically by an equation of the form a 1 x + a 2 y = b where a 1 , a 2 , and b are real constants and a 1 and a 2 are not both zero. An equation of this form is called a linear equation in the variables x and y . More generally, we define a linear equation in the n variables x 1 , x 2 , . . . , x n to be one that can be expressed in the form a 1 x 1 + a 2 x 2 + · · · + a n x n = b where a 1 , a 2 , . . . , a n , and b are real constants. The variables in a linear equation are sometimes called unknowns . For example, the equations 3 x + y = 7 , y = 1 5 x + 2 z + 4 , and x 1 + 3 x 2 - 2 x 3 + 5 x 4 = 7 are linear. The equations x + 3 y = 2 , 3 x - 2 y - 5 z + yz = 4 , and y = cos x are not linear. Observe that a linear equation does not involve any products or roots of variables. All variables occur only to the first power and do not appear as arguments for trigonometric, logarithmic, or exponential functions. A solution of a linear equation a 1 x 1 + a 2 x 2 + · · · + a n x n = b is a sequence of n numbers s 1 , s 2 , . . . , s n such that the equation satisfied when we substitute x 1 = s 1 , x 2 = s 2 , . . . , x n = s n . The set of all solutions of the equation is called its solution set or sometimes the general solution of the equation. 1
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MA131 (Section 750002): Prepared by Asst.Prof.Dr.Archara Pacheenburawana 2 Find the solution set of (a) 5 x - 2 y = 3 and (b) x 1 - 3 x 2 + 4 x 3 = 2. Example 1.1. Solution Linear Systems A finite set of linear equations in the variables x 1 , x 2 , . . . , x n is called system of linear equations or a linear system . A sequence of numbers s 1 , s 2 , . . . , s n is called a solution of the system if x 1 = s 1 , x 2 = s 2 , . . . , x n = s n is a solution of every equation in the system. For example, the system 4 x 1 - x 2 + 3 x 3 = - 1 3 x 1 + x 2 + 9 x 3 = - 4 the the solution x 1 = 1, x 2 = 2, and x 3 = - 1 since these values satisfy both equations. However, x 1 = 1, x 2 = 8, x 3 = 1 is not a solution since these values satisfy only the first equation in the system. Thus, not all system of linear equations have solutions. A system of equations that has no solutions is said to be inconsistent ; if there is at least one solution of the system, it is called consistent . To illustrate the possibilities that can occur in solving systems of linear equations, consider a general system of two linear equations in the unknowns x and y : a 1 x + b 1 y = c 1 ( a 1 , b
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