L92 - Fall 2003 Math 308/501502 9 Matrix Methods for Linear...

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Fall 2003 Math 308/501–502 9 Matrix Methods for Linear Systems 9.2 Linear Algebraic Equations Fri, 07/Nov c ± 2003, Art Belmonte Summary Linear system of n algebraic equations This is a system in the n unknown functions x 1 , x 2 ,..., x n ,that has the form a 11 x 1 + a 12 x 2 +···+ a 1 n x n = b 1 a 21 x 1 + 2 12 x 2 a 2 n x n = b 2 . . . a n 1 x 1 + a n 2 x 2 a nn x n = b n , where the a ij and b i are constants . In matrix-vector form, Ax = b , where A is an n × n constant matrix and b is n × 1 column vector. The unknowns are x 1 , x 2 x n . Or one could say the unknown column vector is x = [ x 1 ; x 2 ; ..., ; x n ]. Note that the system is linear since the unknowns only occur to the first power. Gauss-Jordan elimination algorithm This consists of systematically eliminating variables from one equation to the next, so as to be able to readily read off the solution. More precisely, it involves transforming the augmented matrix M = [ A , b ] to reduced row echelon form (RREF) via elementary row operations. Interchanging two rows of a matrix; Multiplying a row of a matrix by a nonzero scalar (a real [or complex] number); Adding a nonzero scalar multiple of one row to another row. MATLAB Examples You want hand examples? Take a linear algebra class. Here we employ MATLAB, which was literally invented to tackle these sorts of problems! (NOTE: format rat means rational format.) 512/2 Find all solutions to the system x 1 + 2 x 2 + 2 x 3 = 6 2 x 1 + x 2 + x 3 = 6 x 1 + x 2 + 3 x 3 = 6 .
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This note was uploaded on 03/29/2008 for the course MATH 308 taught by Professor Comech during the Spring '08 term at Texas A&M.

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L92 - Fall 2003 Math 308/501502 9 Matrix Methods for Linear...

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