Basis vectors orthogonal

The unit vectors Ci, whose exact definition, meaning and interpretation depend on the particular application at hand, are called basis vectors and form the elements of a basis. They are particularly simple to work with because they are orthogonal. This means that... 

The n basis vectors which define the basis of a coordinate space 5" are n mutually orthogonal and normalized vectors. Together they form a frame of reference axes for that space. 

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. 

It should be clear that the set of all real orthogonal matrices of order n with determinants +1 constitutes a group. This group is denoted by 0(n) and is a continuous, connected, compact, n(n — l)/2 parameter9 Lie group. It can be thought of as the set of all proper rotations in a real n-dimensional vector space. If xux2,. ..,xn are the orthonormal basis vectors in this space, a transformation of 0(n) leaves the quadratic form =1 x invariant. 

In the theory of optics this phenomenon is accounted for in terms of geometrical construction, but the physical picture is less convincing. Double refraction is a well-documented property of most crystals, at its most spectacular in Iceland spar. The double image of an object viewed through the crystal indicates the existence of two independent rays and not the components of a single ray. In mathematical terms the two rays are linearly independent and therefore orthogonal. Any intermediate situation represents a linear combination of the two orthogonal basis vectors and can be resolved into two components. What happens to an individual photon is however, not clear. 

In words, the first h basis vectors are obtainable by applying pw to other vectors, while the other (n—h) are orthogonal to all of these. h=0 correspond to />< ) =0. In this basis, we see from (21), (22), that... 

The above equations are valid for orthonormal sets of basis vectors. They can be written in very similar ways for general non-orthogonal bases (e.g. F in Figure 4-12). The only difference is the computation of the pseudo-inverse, which can be numerically demanding, but is trivial for orthonormal bases. 

For each coordinate 2 in the full space, we may define a covariant basis vector 0R /02 and a contravariant basis vector 02 /0R, which obey orthogonality and completeness relations... 

By using the chain rule for differentiation and the orthogonality of the basis vectors, we then obtain... 

The unit vectors ei, whose exact definition, meaning and interpretation depend on the particular application at hand, are called basis vectors and form the elements of a basis. They are particularly simple to work with because they are orthogonal. This means that their dot products vanish epej = 0, unless i = j. If i = j, then the scalar or dot product is unity (it is usually convenient, but not necessary, to use bases that are normalized so epei = 1). The shorthand way of representing this information is to write... 

Exercise 6.14 Use the Gram-Schmidt technique of orthogonalization to find a recursive formula for an orthogonal basis ofC[—l, 1] with the property that the kth basis vector is a polynomial of degree n (for n = 0, 1, 2,. . J. Show (from general principles) that the nth basis element is precisely the character of the representation ofSU(T) on P". Use the recursive formula to calculate and /4. 

Linear operators in finite-dimensional spaces. It is supposed that an n-dimensional vector space Rn is equipped with an inner product (, ) and associated norm a = / x, x). By the definition of finite-dimensional space, any vector x 6 Rn can uniquely be represented as a linear combination x = Cj + c of linearly independent vectors, ..., which constitute a basis for the space Rn. The numbers ck are called the coordinates of the vector x. One can always choose as a basis an orthogonal and normed system of vectors. .. , n ... 

Figure 3.2. Examples, in 2-D space, of (a) an LI set of orthogonal basis vectors ei e2, (b) an LI non-orthogonal basis, and (c) a set of three basis vectors in 2-D space that are not LI because...

Example 3.2-2 Consider a basis of three orthogonal unit vectors with e3 (along OZ) normal to the plane of the paper, and consider the proper rotation of this basis about OZ through an angle o by the operator R ( z) (see Figure 3.3). Any vector v may be expressed as the sum of its projections along the basis vectors ... 

The columns of Tk are called scores and are mutually orthogonal but not normalized. They can be used to form a set of column basis vectors for A. 

When the first direction, v has been found, the data are reflected such that the first eigenvector is mapped onto the first basis vector. Then the data are projected onto the orthogonal complement of the first eigenvector. This is simply done by omitting the first component of each (reflected) point. Doing so, the dimension of the projected data points can be reduced by 1 and, consequently, all the computations do not need to be done in the full r-dimensional space. 

To conduct SFA in practice, the singular-value decomposition (SVD, see Chapter 4) of the two subwindows yields a basis of orthogonal vectors spanning the (A,B) subspace, called ej, and another basis for the (B,C) subspace, called fj. The spectrum of B, sB, can be obtained from these two sets of basis vectors as shown in Equation 11.9,... 

The AIM electron-population displacements, d/V, are strongly coupled through the olf-diagonal hardness matrix elements //y y>,i Thus, a given displacement d/Vk strongly affects the chemical potentials of all AIM. This representation considers all AIM populational parameters as independent variables, which can be interpreted as projections of the populational vector (/V, d, + N2 2 + dm) onto the orthogonal system of populational axes associated with the constituent atoms, i.e., the AIM populational basis vectors ... 

It should be observed that, contrary to the orthonormal character of the AIM basis vectors, the MEC/REC vectors f M are not orthogonal, giving rise to the metric... 

The columns of the resulting overall intra-reactant decoupling-transformation matrix (see Eq. (137)) define the rigid IDM. The orthogonal axes Omx = uUlnt represent the new basis vectors in this representation. The transformed hardness matrix assumes the partly decoupled form ... 

A vector P is a quantity with both magnitude and direction, usually designated in boldface type (e.g., B, to), and represented pictorially as an arrow /. A vector in a space of N dimensions may be expressed in terms of N orthogonal unit basis vectors as a linear ordered array of numbers that describe the contributions from... 

The method is easily generalized to a continuous description, where the basis vectors of the orthogonal complement are C distributions a ix) the analog of Eq. (64) is simply... 

Again, I is a vector to which all reaction vectors mj — m2 are orthogonal. However, in this case the total mass of odd-valued x-olefins is also constant, and hence one has also a second basis vector in the orthogonal complement ... 

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