lect136_2_w09

lect136_2_w09 - Wednesday January 7 - Lecture 2 : The...

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Wednesday January 7 Lecture 2 : The Gauss-Jordan elimination algorithm and RREF. (Refers to sections 2.1 and 2.2) Expectations: 1. Define a system of m linear equations in n unknowns. 2. Define coefficient matrix, augmented coefficient matrix. 3. Define Row echelon form (REF),"Reduced row echelon form" (RREF), "leading- 1's" and "pivots" in a matrix. 4. Solve a system of linear equations by using the Gauss-Jordan elimination algorithm. 5. Recognize basic unknowns and free unknowns . 6. Define the rank of a coefficient matrix. 2.1 Introduction Linear equations are equations of the form a 11 x 1 + a 12 x 2 + . .. + a 1 n x n = b 1 . Solving simultaneously many linear equations means finding the set of all variables that satisfy all of the given linear equations. In this lecture we discuss a method for solving these which involves only the coefficients and the constants. 2.1.1 Example We give an example where we solve a system of 3 linear equations in 3 unknowns by using the mathematical principles above. (By linear equations we mean that the exponents of all variables are 0 or 1.) The solution of this system involves the application of the following principles: a = b c = d then a + c = b + d ac = bd . Please illustrate how these 2 principles are invoked over and over the solution of this system: 4 x 8 y + 4 z = 36 2 x y + z = 6 3 x 2 y + 2 z = 2. The system has solution ( x , y , z ) = ( 2, 7, 3). See details of solution .
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2.1.2 Geometric interpretation. The 3 equations described above, when plotted in 3-space, form planes. The solution indicates the only point that belongs to all 3 planes, i.e., the 3 planes intersect at that point. 2.2 Definition System of linear equations. A system of m linear equations with n unknowns has the form : 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 . where b 1 , b 2 , . .., b m , are given numbers and x 1 , x 2 , . .., x n are unknowns. The a ij 's, 1 i m and 1 j n , are numbers which we call coefficients . 2.2.1
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lect136_2_w09 - Wednesday January 7 - Lecture 2 : The...

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