## Numerical Linear Algebra And Optimization |

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active Algorithm applied associated assume basic bound calculation called columns complete components computed condition Consider constraints contains corresponding decreasing defined definition denote dependent direction discussion dual element elimination equality equations exact example Exercise exists Explain expressed factorization feasible direction feasible point feasible region Figure given gives implies indices inequality infeasible involving iteration least least-squares problem Lemma linear combination linear program linearly independent matrix means minimize move multiplier n-vector negative nonnegative nonsingular nonzero normal null space objective function obtain optimal original orthogonal orthogonal matrix perturbation phase-1 pivot positive primal problem properties range rank reduced relation remaining residual respect result rows satisfying Show shown side simplex method singular value solution solving standard-form step strictly Suppose transformations unique variables vector vertex zero