By H. Ronald Miller

ISBN-10: 0471322423

ISBN-13: 9780471322429

ISBN-10: 0471351695

ISBN-13: 9780471351696

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**Additional resources for Optimization: Foundations and Applications**

**Sample text**

Putting that into the top equation on the left, u = \ and so s = ]. Similarly, from the right pair of equations we find (top equation) t = -^v. Putting that into the bottom equation gives ν = - \ and so t = These are the four elements of the inverse, so L~3 which you can easily (and should) check by finding that AA" = I and Α" Ά = I. 9), 1 1 2 3 X = A-'B = . 6). These are the appropriate number of students to admit into each of the two job retraining programs if both the allocated budget is to be completely used up and the mayor's new tax target is to be met.

T 2 2 { '-'For three-element vectors (three-dimensional vector space) the differences between coordinates on all three axes must be used. Hence, a term ( c - d ) would be added under the square root. ) For ^-dimensional vectors, we have 2_ ( i ~ d,Y 3 2 2 c 14 T h e demonstration depends on the law of cosines from trigonometry, and we need not bother with it. 5 LINEAR DEPENDENCE A N D INDEPENDENCE 35 L I N E A R D E P E N D E N C E AND INDEPENDENCE The concepts of linear dependence and independence between vectors and of the rank of a matrix provide a general framework within which to study the kinds of solutions that are possible for linear equation systems of any size.

When A is of order 4, we have y u t k «11 |Α,|=α , «II «12 π «21 «22 «12 «21 «22 ^23 «31 «32 «33 |A | = |A| 4 These η determinants, as a group, are the leading principal minors of A. "Leading" serves to remind us that they are found from a particular sequence of matrices of increasing size, beginning in the northwest corner and expanding toward the southeast, simultaneously adding the next row and column at each step. "Principal" is used to indicate that they are related to elements located on the main diagonal of A, so that if a given row is added the associated column is added also.

### Optimization: Foundations and Applications by H. Ronald Miller

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