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Lecture04-2003 - Lecture IV The Canonical Form and Null...

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Lecture IV: The Canonical Form and Null Spaces I. To demonstrate the meaning of null space matrices, I first want to discuss the use of canonical forms. A canonical form is a solution to an underidentified system of equations. For example in a crop selection model we may have a land and a labor constraint x x x x x x x x x x 1 2 3 4 5 1 2 3 4 5 100 2 3 4 250 + + + + = + + + + = A. The matrix operations used to reduce this matrix to a row reduced form are 1 0 2 1 1 1 1 1 1 100 2 3 1 4 1 250 1 1 0 1 1 1 1 1 1 100 0 1 1 2 1 50 1 0 2 1 2 50 0 1 1 2 1 50 1. This representation is a canonical form representing a solution to the system of equations. Specifically, x 1 =50, x 2 =50, x 3 =0, x 4 =0, x 5 =0 solves the system of equations. 2. In addition, the solution says something about maintaining feasibility in the nonbasic variables. Taking the columns from the nonbasic, or
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