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. . xn= 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: .
2.1.2Geometric 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. 2.2Definition − System of linear equations. A system of m linear equations with n unknownshas the form : a11x1+ a12x2+ ... + a1nxn= b1. a21x1+ a22x2+ ... + a2nxn = b2. ... am1x1+ am2x2+ ... + amnxn= bm. where b1,b2, ..., bm, are given numbers and x1, x2, ..., xnare unknowns. The aij's, 1 ≤ i ≤ m and 1 ≤ j ≤ n, are numbers which we call coefficients. 2.2.1Definition −The matrix A⎡a11a12... ... a1n⎤⏐a21a22... a2n⏐⏐a31a32... : ⏐⏐: : : : ⏐⏐: : : : ⏐⎣am1am2... ... amn⎦often written as A = [aij]m x n