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systems_linear_equations

# systems_linear_equations - Math 568 Systems of Linear...

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Math 568 Systems of Linear Equations and Matrices In these notes, we define a linear system and their associated matrices. We also indicate the algebra which can be preformed on these objects. 1. Definitions and Notation A linear equation in n variables is an equation of the form: a 1 x 1 + a 2 x 2 + · · · + a n x n = b and a system of m linear equations in n variables is a collection of linear equations in the same n variables: 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 A solution to a system of linear equations in n variables is an vector [ s 1 , s 2 , . . . , s n ] such that the components satisfy all of the equations in the system when we set x i = s i . We say that a system of linear equations is consistent if it has at least one solution; otherwise we say that it is inconsistent . A system of linear equations may have more than one solution (we will see later that it must have infinitely many solutions in this case) and the collection of all solutions of a linear system is called its solution set . Consider the following two linear systems: x 1 + x 2 = 2 x 1 - x 2 = 0 and x 1 = 1 x 2 = 1 Notice that they have exactly the same solution set, namely { [1 , 1] } . We say that these two systems are equivalent. More generally, two systems of linear equations (in the same variables) are said to be

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systems_linear_equations - Math 568 Systems of Linear...

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