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**Unformatted text preview: **MATH 304, Fall 2011 Linear Algebra MATH 304 Linear Algebra Lecture 2: Gaussian elimination (continued). Row echelon form. Gauss-Jordan reduction. System of linear equations 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 Here x 1 , x 2 , . . . , x n are variables and a ij , b j are constants. A solution of the system is a common solution of all equations in the system. A system of linear equations can have one solution, infinitely many solutions, or no solution at all. x y braceleftbigg x y = 2 2 x + 3 y = 6 x = 0, y = 2 x y braceleftbigg 2 x + 3 y = 2 2 x + 3 y = 6 inconsistent system (no solutions) x y braceleftbigg 4 x + 6 y = 12 2 x + 3 y = 6 2 x + 3 y = 6 Solving systems of linear equations Elimination method always works for systems of linear equations. Algorithm: (1) pick a variable, solve one of the equations for it, and eliminate it from the other equations; (2) put aside the equation used in the elimination, and return to step (1). x y = 2 = x = y + 2 2 x y z = 5 = 2( y + 2) y z = 5 After the elimination is completed, the system is solved by back substitution . y = 1 = x = y + 2 = 3 Gaussian elimination Gaussian elimination is a modification of the elimination method that allows only so-called elementary operations . Elementary operations for systems of linear equations: (1) to interchange two equations. (2) to multiply an equation by a nonzero scalar; (3) to add an equation multiplied by a scalar to another equation; Theorem (i) Applying elementary operations to a system of linear equations does not change the solution set of the system. (ii) Any elementary operation can be undone (inverted) by another elementary operation. Operation 1: multiply the i th equation by r negationslash = 0. a 11 x 1 + a 12 x 2 + + a 1 n x n = b 1 a i 1 x 1 + a i 2 x 2 + + a in x n = b i a m 1 x 1 + a m 2 x 2 + + a mn x n = b m = a 11 x 1 + a 12 x 2 + + a 1 n x n = b 1 ( ra i 1 ) x 1 + ( ra i 2 ) x 2 + + ( ra in ) x n = rb i a m 1 x 1 + a m 2 x 2 + + a mn x n = b m To undo the operation, multiply the i th equation by r 1 . Operation 2: add r times the i th equation to the j th equation. a i 1 x 1 + a i 2 x 2 + + a in x n = b i a j 1 x 1 + a j 2 x 2 + + a jn x n = b j = a i 1 x 1 + + a in x n = b i ( a j 1 + ra i 1 ) x 1 + + ( a jn + ra in ) x n = b j + rb i To undo the operation, add...

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