Notes 1 IntroLinSys 2010

Notes 1 IntroLinSys 2010 - File: Notes 1 IntroLinSys.Doc...

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File: Notes 1 IntroLinSys.Doc Solving Linear Systems of Equations We will have many situations where we need to solve a system of linear equations. These problems will vary from the smallest two equations and two unknowns to reasonably large systems. There are elements to be learned from studying small systems. Knowledge about the tools for solving these systems such as determinants and their use in Cramer’s rule, matrix inverses, as well as the more computationally oriented matrix reduction methods will come in handy. We start with the basic concepts of linear systems and more to the efficient methods as needed. The first important concept is that a linear system of equations has three possibilities for the number of solutions: no solution, a unique solution, and an infinite number of solutions. These situations are easily illustrated for a system of two equations and two unknowns. The following figure displays the three cases. The no solution case, case (a), occurs when the two equations represent two parallel lines. These equations or lines in two-space do not cross and hence, have no values in common and, therefore, no simultaneous solution. The second case, case (b), illustrates the unique solution situation; this case has a unique solution where the equations or lines intersect. The third case, case (c), occurs when the two lines are parallel but coincide. Then any point on the line is a solution, and there are an infinite number of these. (a) no solution (b) unique solution (c) overlapping multiple solutions Figure 1: Illustration of the three cases for the number of solution to a linear system. Consider the problem of obtaining a solution to a system of n linear equations in n unknowns. This problem in general can be written as: a x a x a x b ax ax ax b b nn n n n n 11 1 12 2 1 1 21 1 22 2 2 2 11 2 2 + + + = ++ += " " # " A more compact notation is the matrix form: Ax b = ,
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This note was uploaded on 10/26/2011 for the course ISEN 620 taught by Professor Curry during the Fall '10 term at Texas A&M.

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Notes 1 IntroLinSys 2010 - File: Notes 1 IntroLinSys.Doc...

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