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 xxxxx xx x 12345 12 34 5 100 2 3 4 250 + + + + = ++ = A. The matrix operations used to reduce this matrix to a row reduced form are 10 21 111111 0 0 231412 5 0 11 01 1 1 1 1 1 100 01 12 1 5 0 10 2 1 2 5 0 01 1 2 15 0 −− 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 zero-level, variables we can form a homogeneous set of equations ( Ax =0): 22 0 20 5 5 x x + = −+ −=
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Lecture04-2003 - Lecture IV: The Canonical Form and Null...

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