Chapter 1.1 SystemsLinearEquations

The reduced row echelon form of a matrix is unique

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Unformatted text preview: elon form of a matrix is unique. The columns containing leading 1’s are called pivot columns. The variables corresponding to pivot columns are called leading variables. The the variables that are not leading variables are called free variables. Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Examples Example Use Gauss-Jordan elimination to solve the linear systems. 1 3x1 + 2x2 − x3 = 4 x1 − 2x2 + 2x3 = 1 11x1 + 2x2 + 4x3 = 14 2 x1 + x2 + x3 + x4 + x5 = 5 x1 + x2 + 3x3 + x4 + 3x5 = 1 x1 + x2 + 2x3 + x4 + 3x5 = 5 (This system is underdetermined) Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Homogeneous Linear Systems A homogeneous system of linear equations is a system of linear equations with zero constant term, so its augmented matrix has the form: a11 a12 · · · a1n 0 a21 a22 · · · a2n 0 (6) . . .. . . . . . . . . . .. am 1 am 2 · · · amn 0 Every homogeneous system of linear equations is consistent because the trivial solution x1 = x2 = · · · = xn = 0 is a solution of the system. If there are other solutions, they are called nontrivial solutions. Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Homogeneous Linear Systems There are two possibilities for the solutions of a homogeneous system of linear equations: There is only the trivial solution. There is the trivial solution and infinitely many nontrivial solutions. Proposition A homogeneous system of linear equations having more unknowns than equations has infinitely many nontrivial solutions....
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This note was uploaded on 02/10/2014 for the course MATH 2270 taught by Professor Kenyonj.platt during the Spring '14 term at Snow College.

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