Linear or Vector Space

In optimal control, we desire to find the minimum or maximum value of a functional defined over a specified domain. The analytical procedure is to continuously change the associated function from some reference form and examine the corresponding change in the functional. The new form of the function is, in fact, the result of a linear combination of the reference form and some other form of the function in the same domain. This examination can continue only if the new form of the function lies within the specified domain each time the function is changed. Otherwise, the corresponding new functional may not exist or be valid. The validity of the functional is ensured by having the specified domain be a linear or vector space. This space holds within itself all linear combinations of its elements (functions), which are called vectors. A precise definition of linear space is provided in Section 9.19 (p.278). 

In this book, we will deal with functions or vectors that belong to linear or vector spaces. Examples of these spaces include a space of vectors of specified n components, a space of continuous functions dependent on an independent variable varying in a specified interval, etc. 

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This chapter introduces the fundamental concepts of optimal control. Beginning with a functional and its domain of associated functions, we learn about the need for them to be in linear or vector spaces and be quantified based on size measures or norms. With this background, we establish the differential of a functional and relax its definition to variation in order to include a broad spectrum of functionals. A number of examples are presented to illustrate how to obtain the variation of an objective functional in an optimal control problem. 

We require domains that are linear or vector spaces. Such a space contains all linear combinations of its elements. The details and the rationale of linear spaces are as follows. 

A linear or vector space is a set of elements known as vectors for which the following two operations ... 

A set of complete orthonormal functions ipfx) of a single variable x may be regarded as the basis vectors of a linear vector space of either finite or infinite dimensions, depending on whether the complete set contains a finite or infinite number of members. The situation is analogous to three-dimensional cartesian space formed by three orthogonal unit vectors. In quantum mechanics we usually (see Section 7.2 for an exception) encounter complete sets with an infinite number of members and, therefore, are usually concerned with linear vector spaces of infinite dimensionality. Such a linear vector space is called a Hilbert space. The functions ffx) used as the basis vectors may constitute a discrete set or a continuous set. While a vector space composed of a discrete set of basis vectors is easier to visualize (even if the space is of infinite dimensionality) than one composed of a continuous set, there is no mathematical reason to exclude continuous basis vectors from the concept of Hilbert space. In Dirac notation, the basis vectors in Hilbert space are called ket vectors or just kets and are represented by the symbol tpi) or sometimes simply by /). These ket vectors determine a ket space. 

Support Vector Machines (SVMs) generate either linear or nonlinear classifiers depending on the so-called kernel [149]. The kernel is a matrix that performs a transformation of the data into an arbitrarily high-dimensional feature-space, where linear classification relates to nonlinear classifiers in the original space the input data lives in. SVMs are quite a recent Machine Learning method that received a lot of attention because of their superiority on a number of hard problems [150]. 

Candidate mineral compositions or test vectors are tested by linearly rotating the NC eigenvectors towards a test vector by using a least squares procedure ( ) and determining if the test vector could possibly lie in the vector space defined by the NC eigenvectors. In this way, suspected minerals are kept or rejected from further consideration. (From this step of the analysis, TTFA derives its name.) When NC mineral compositions have been determined that adequately reproduce the original data and are consistent with other information, such as XRD or Infrared analysis, this aspect of TTFA is finished. At this point, we have successfully determined the matrix C of Equation 5. 

Definition 2.7 Suppose V and W are vector spaces and T V —> W is a linear transformation. IfT is invertible and T W V is a linear transformation, then we say that T is an isomorphism of vector spaces (or isomor-phism/t r short) and that V and W are isomorphic vector spaces. 

Exercise 2.14 (Used in Section 5.5) Let V denote a complex vector space. Let y denote the set of complex linear transformations from V to C. Show that y is a complex vector space. Show that ifV is finite dimensional then dim y = dim V. The vector space y is called the dual vector space or, more simply, the dual space. 

Remark. Apart from the question whether the set of all eigenfunctions is complete, one is in practice often faced with the following problem. Suppose for a certain operator W one has been able to determine a set of solutions of (7.1). Are they all solutions For a finite matrix W this question can be answered by counting the number of linearly independent vectors one has found. For some problems with a Hilbert space of infinite dimensions it is possible to show directly that the solutions are a complete set, see, e.g., VI.8. Ordinarily one assumes that any reasonably systematic method for calculating the eigenfunctions will give all of them, but some problems have one or more unsuspected exceptional eigenfunctions. 

This shows that there is still some linearity. In particular, there is a subset of knots that form a vector space and is therefore a linear sector of the model. It is the set of the knots with zero helicity or with unlinked lines. Note also that the theory is fully linear from the local point of view, as a consequence of the local equivalence with Maxwell s theory shown in Section V.A. By this we mean that the set of the electromagentic knots contains all the linear combinations of standard solutions around any point. 

Linear algebra deals with finite dimensional real or complex spaces, called R" or Cn for any positive integer n. A typical n-vector i e K" or Cn has the form of a row... 

A matrix M is an ordered array of numbers, usually describing a linear transformation of one vector P to another R. The components of M relate the components of R to those of P. If the dimensionalities of the vector spaces of P and R are equal, then the matrix is square with equal numbers of rows and columns.We deal only with square matrices except for the NX 1 matrices that describe vectors as row or column matrices, for example ... 

The determinant can be regarded as a measure of the volume which is spanned by the column vectors (or row vectors) of the matrix in the vector space. For example, in a two dimensional space, two vectors can span a surface area, provided that the vectors are not parallel. If they should be parallel, the surface area would be zero. In the three-dimensional space, three vectors can span a volume, provided that they do not lie in the same plane. If they should do so, the volume would be zero. In that case, any of the three vectors can be expressed as a linear combination of the two others the vectors are said to be linear dependent. 

The maximum number of linearly independent vectors in a linear space or subspace is said to be the dimension of the space. Such a linearly independent set of vectors is said to be a basis for that space, by which it is meant that any arbitrary vector in the space can be expressed as a linear combination of the basis set. 

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