MAT1341-L08-SystmEqnP13

MAT1341-L08-SystmEqnP13 - SYSTEMS OF LINEAR EQUATIONS -...

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SYSTEMS OF LINEAR EQUATIONS - PART 1 JOS ´ E MALAG ´ ON-L ´ OPEZ A recurrent situation so far has been to determine the solution set of a system of linear equations. In this lecture we start the study of systems of linear equations so we can improve our techniques used so far. A Frst step will be to express the solution set in terms of vector spaces. Basic Defnitions A linear equation in n unknowns is an algebraic expression of the form (*) a 1 x 1 + a 2 x 2 + · · · + a n x n = b, where x 1 , . . . , x n are unknowns (variables), a 1 , . . . , a n are real numbers called the coe±cients o² the equation , and b is a real number called the constant term o² the equation . A solution to the equation (*) is a vector s 1 . . . s n R n such that the equation holds when the x i ’s are substituted by the num- bers s i ’s: a 1 ( s 1 ) + a 2 ( s 2 ) + · · · + a n ( s n ) = b. Example. A solution to the equation 2 x 1 + 3 x 2 x 3 = 5 is the vector 2 0 1 R n since 2( 2) + 3(0) (1) = 5 . 1
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A system of m linear equations in n unknowns is a collection (S) a 11 x 1 + a 12 x 2 + · · · + a 1 n x n = b 1 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 consisting of m linear equations in n unknowns. The real numbers a ij , 1 i m , 1 j n , are called the coeFcients of the system (S). The real numbers b 1 , . . . , b m are called the constant terms of the system (S). A solution to the system (S) is a vector s 1 . . . s n R n that is a solution for EACH of the linear equations in the system (S). If the system has at least one solution we say that the system is con- sistent . A system with no solution is called inconsistent . Example. The vector p 2 3 P is a solution for the system b x 1 +4 x 2 = 14 2 x 1 x 2 = 7 Example. The vector 9 13 1 is a solution for the system b 5 x 1 +3 x 2 +7 x 3 = 1 2 x 1 x 3 = 19 Example. Unlike the systems in the previous two examples, the fol- lowing system is inconsistent. b x 1 + x 2 = 1 x 2 + x 2 = 2 2
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Example. The vectors 3 0 1 and 1 6 1 are solutions for the system b x 1 x 2 +2 x 3 = 1 x 1 + x 3 = 2 A linear system is called homogeneous if all the constant terms are equal to zero: a 11 x 1 + a 12 x 2 + · · · + a 1 n x n = 0 a 21 x 1 + a 22 x 2 + · · · + a 2 n x n = 0 . . . a
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MAT1341-L08-SystmEqnP13 - SYSTEMS OF LINEAR EQUATIONS -...

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