lect136_2_w09 - Wednesday January 7 Lecture 2 The Gauss-Jordan elimination algorithm and RREF(Refers to sections 2.1 and 2.2 Expectations 1 Define a

# 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 mlinear equations in nunknowns. 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 unknownsand free unknowns6.Define the rank of a coefficient matrix. . x n = 2.1.1Example We give an example where we solve a system of 3linearequations in 3 unknowns by using the mathematical principles above. (Bylinearequations 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 = bc = dthen a + c = b + d ac = bd. Please illustrate how these 2 principles are invoked over and over the solution of this system: .
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 Definition The matrix A a 11 a 12 ... ... a 1 n a 21 a 22 ... a 2 n a 31 a 32 ... : : : : : : : : : a m 1 a m 2 ... ... a mn often written as A = [ a ij ] m x n