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NullSpaceExample-2003

# NullSpaceExample-2003 - AEB 6533 Some Examples of...

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AEB 6533 Some Examples of Nullspaces I. Starting with a simple 2*3 example, assume that we have the matrix equation 1 2 3 1 1 1 5 0 1 1 6 x Ax b x x = = A. Note that by row operations the A matrix can be transformed to 1 1 1 1 1 1 0 2 0 1 0 1 1 0 1 1  =   B. This expression implies the following homogeneous relationships: 1 3 2 3 2 0 0 x x x x + = + = 1. Setting 3 1 x = yields 1 2 2 1 x x = − = − 2. Or in vector form: 2 1 1 z = C. Next we confirm that z is the nullspace. We do this by confirming that both vectors of the A matrix are orthogonal to the nullspace: ( ) ( ) 2 1 1 1 1 2 1 1 0 1 2 0 1 1 1 0 1 1 0 1 = − + + = = − + = D. Implications of the nullspace: 1. Start by defining a feasible solution: 5 1 1 1 1 1 5 1 0 2 11 6 0 1 0 1 1 6 0 1 1 6 11 1 1 1 11 6 0 5 6 0 1 1 6 0 6 0 Ax  = =   + = = + 2. Next, we would like to generate another feasible point based on

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NullSpaceExample-2003 - AEB 6533 Some Examples of...

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