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### LinAlgSys_ArbUnknowns

Course: EGN 3420, Summer 2008
School: UCF
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Word Count: 1590

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UNKNOWNS The ARBITRARY echelon form of the augmented matrix confirms the existence of arbitrary unknowns, i.e. a consistent system of equations in which one or more variables can be chosen arbitrarily There are several ways to establish if indeed a certain variable can be included in the subset of arbitrary unknowns. A few simple examples illustrate the point. Example 1 For the system of equations below, establish...

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UNKNOWNS The ARBITRARY echelon form of the augmented matrix confirms the existence of arbitrary unknowns, i.e. a consistent system of equations in which one or more variables can be chosen arbitrarily There are several ways to establish if indeed a certain variable can be included in the subset of arbitrary unknowns. A few simple examples illustrate the point. Example 1 For the system of equations below, establish the existence of 1 arbitrary unknown and determine if x, y, and z can each be arbitrary. x 2x x x + y y + + z = z = 4z = 2z = 0 3 3 3 + 4y 2y bA|bg = F1 GG 2 GG 1 H1 1 1 0 1 1 3 4 4 3 3 2 2 I JJ JJ K ~ F1 GG 0 GG 0 H0 1 1 0 3 3 3 3 3 3 3 3 3 I JJ JJ K ~ F1 GG 0 GG 0 H0 1 1 0 0 1 0 1 1 0 0 0 0 I JJ JJ K The 4 by 4 system has been reduced to a 2 by 3 system. That is, there are only 2 linearly independent equations in the 3 variables (unknowns) and hence one of the variables is arbitrary. These equations are x + ...
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