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By Kallenrode

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The assumption of optimization runs via so much elements of regulate thought. the easiest optimum controls are preplanned (programmed) ones. the matter of creating optimum preplanned controls has been widely labored out in literature (see, e. g. , the Pontrjagin greatest precept giving valuable stipulations of preplanned keep watch over optimality).

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Example text

This has to do with finite number fields (try F2 ), and the only available example is rather dull. 5 All bases have equally many vectors We have seen that any linearly independent set of n vectors in an n-dimensional space is a basis. The following statement shows that a basis cannot have fewer than n vectors. The proof is somewhat long and can be skipped unless you would like to gain more facility with coordinate-free manipulations. Theorem: In a finite-dimensional vector space, all bases have equally many vectors.

10) j=1 j=1 where λj , µj are some constants, not all equal to zero. Suppose all λj are zero; then fl+1 would be a linear combination of other fj ; but this cannot happen for a basis {fj }. Therefore not all λj , 1 ≤ j ≤ k are zero; for example, λs = 0. This gives us the index s. Now we can replace es in the set S by fl+1 ; it remains to prove that the resulting set T defined by Eq. 9) is linearly independent. 2 Linear maps in vector spaces where ρj , σj are not all zero. In particular, σl+1 = 0 because otherwise the initial set S would be linearly dependent, l k s−1 σj fj = 0.

However, it does not mean that matrices are always needed. e. g. ” In this book I concentrate on general properties of linear transformations, which are best formulated and studied in the geometric (coordinate-free) language rather than in the matrix language. Below we will see many coordinate-free calculations with linear maps. In Sec. 8 we will also see how to specify arbitrary linear transformations in a coordinate-free manner, although it will then be quite similar to the matrix notation. Exercise 1: If V is a one-dimensional vector space over a field K, prove that any linear operator Aˆ on V must act simply as a multiplication by a number.