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Lecture

# Solving Systems of Linear Equations

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School
Department
Mathematics
Course
MATH 2331
Professor
Rita Jimenez Rolland
Semester
Spring

Description
Linear Algebra Notes Linear Equations January 31, 2014 Example. Find the solutions of the system: 2x + 4y + 6z = 0 (▯2) 4x + 5y + 6z = 3 7x + 8y + 9z = 6 x + 2y + 3z = 0 4x + 5y + 6z = 3 (-4I) 7x + 8y + 9z = 6 (-7I) x + 2y + 3z = 0 ▯3y ▯ 6z = 3 (▯ ▯ 3) ▯6y ▯ 12z = 6 (▯ ▯ 6) x + 2y + 3z = 0 (-2II) y + 2z = ▯1 y + 2z = ▯1 (-II) x ▯ z = 2 y + 2z = ▯1 ! x = 2 + z ;z 2 R ) 1 solutions y = ▯1 ▯ 2z 0 = 0 Example. Find the solutions of the system: x + 2y + 3z = 0 4x + 5y + 6z = 3 (-4I) 7x + 8y + 9z = 0 (-7I) x + 2y + 3z = 0 ▯3y ▯ 6z = 3 (▯ ▯ 3) ▯6y ▯ 12z = 0 (▯ ▯ 6) x + 2y + 3z = 0 (-2II) y + 2z = ▯1 y + 2z = 0 (-II) x ▯ z = 2 y + 2z = ▯1 0 = 1 ! this false statement implies no solution. Steps to Solve a System of Linear Equations: 1. Leading variable of equation I should have coe▯cient of 1; if not, divide. 2. Eliminate leading variable from other equations by subtracting scalar multiples of I. 3. Repeat with remaining equations. Solution set ) single point solution; true statement (0 = 0) ) 1 solutions; false statement (0 = 1) ) no solutions. 1 For e▯ciency, split into arrays. 0 1 0 1 1 2 3 . 0 1 2 3 B C @ 4 5 6A ! B . C - 3 ▯ 4 augmented matrix @ 4 5 6 . 3A 7 8 9 . 7 8 9 . 0 0 . 1 1 0 0 . # B . C ! goal:B 0 1 0 . # C @ . A 0 0 1 . # An n ▯ m matrix A is the array of real numbers with n ▯ m entries 0 1 a11 a12 a13 ▯▯▯ a1m B a21 a22 a23 ▯▯▯ a2mC A = B . . . . . C - aij= entry in the row and jh column @ . . . .. . A an1 an2 an3 ▯▯▯ anm if n=m, A is a square matrix. A = B if and only if they have the same size and, for alij= bij j, a ▯ ▯ 0 0 A = - zero matrix; every element is zero. 0 0 for square matrices: 0 1 0 1 a11 a12 ▯▯▯ a1n a11 0 0 ▯▯▯ 0 B a12 a22 ▯▯▯ a2nC B 0 a22 0 ▯▯▯ 0 C B . . . . C - diagonal of matrix; entries ofiioBm.a ; . . . . C - diagonal matrix @ . . .. . A
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