Vector space linear independence

In ra-space any n + 1 of these are linearly dependent. But unless the matrix is rather special in form (derogatory), there exist vectors vx for which any n consecutive vectors are linearly independent (in possible contrast to the behavior in the limit). In fact, this is true of almost every vector vx. Hence, if... 

If the set 3N contravariant Cartesian vectors given by the / aj) vectors and K mf vectors are linearly independent, and thus span the full 3N space of Cartesian... 

The following terminology is important The set ft = z,... xt of vectors x, 6 S is linearly dependent, iff there exists a set of scalars a,. ..at, not all zero, such that orixi + —h a = 0. If this is not possible, then the vectors are linearly independent. A vector x, for which a, 0 is one of the linearly dependent vectors. The set of vectors defines a vector subspace S, of S, called span(ft), which consists of all possible vectors z = aix, + —h atzt. This definition also provides a mapping from the array., a ) e Rk to the vector space span(ft). If ft is a linearly independent set, then the dimension of S, is k, and then the vectors constitutes a basis set in Si. If it is linearly dependent, then there is a subset fti 6 ft of size ki = card (ft,) which is linearly independent and spans the same space. Then ki is the dimension of S,. 

We now consider spaces of n dimensions with basis vectors pi, p2, pj,. . . , pn- These vectors are linearly independent if no nontrivial relation like Eq. 2.56 exists ... 

A set of n vectors of dimension n which are linearly independent is called a basis of an -dimensional vector space. There can be several bases of the same vector... 

It has been shown that the p columns of an nxp matrix X generate a pattern of p points in 5" which we call PP. The dimension of this pattern is called rank and is indicated by liPP). It is equal to the number of linearly independent vectors from which all p columns of X can be constructed as linear combinations. Hence, the rank of PP can be at most equal to p. Geometrically, the rank of P can be seen as the minimum number of dimensions that is required to represent the p points in the pattern together with the origin of space. Linear dependences among the p columns of X will cause coplanarity of some of the p vectors and hence reduce the minimum number of dimensions. 

The number of linearly independent columns (or rows) in a matrix is called the rank of that matrix. The rank can be seen as the dimension of the space that is spanned by the columns (rows). In the example of Figure 4-15, there are three vectors but they only span a 2-dimensional plane and thus the rank is only 2. The rank of a matrix is a veiy important property and we will study rank analysis and its interpretation in chemical terms in great detail in Chapter 5, Model-Free Analyses. 

The elementary reactions in Eqs. (1) are not necessarily linearly independent, and, accordingly, let Q denote the maximum number of them in a linearly independent subset. This means that the set of all linear combinations of them defines a 0-dimensional vector space, called the reaction space. In matrix language 0 is the rank of the S x A matrix (2) of stoichiometric coefficients which appear in the elementary reactions (1) ... 

Horiuti calls H the number of independent intermediates. Temkin (10) describes the equation P = S - H as Horiuti s rule, and the equation R = Q — H as expressing the number of basic overall equations. To avoid confusion, let us confine the term basis and the concept of linear independence to sets of vectors, and let numbers such as H, P, Q, R, S be understood as dimensions of vector spaces. This makes it simple to determine their values and the relations among them, as will be done in Section III. 

We face a number of questions concerning the structure of this subspace. Do we need all vectors a, a, . .., to span the subspace or some of them could be dropped Do these vectors span the whole space R0 How to choose a system of coordinates in the subspace The answers to these questions are based on the concept of linear independence. The vectors a, a2>. .., am are said to be linearly independent if the equality... 

Proposition 2.1 Suppose V is a finite-dimensional vector space with basis vi,. .., u . Suppose ,..., Urn is a linearly independent subset of V. Then m < n. 

Whenever we consider a set of equivalence classes, it behooves us to ask what survives the equivalence. Note what does not survive if dim V > 2, the set P(F) is not a complex vector space addition does not descend. For any element v e V [0], there must be a w e V 0 such that the set [u, w] is linearly independent, by the assumption on dimension. By the definition of linear independence, it follows that for every c e C we have... 

It is also true that for a general vector space any set of linearly independent vectors can be combined in analogous fashion to give a set of orthonormal vectors. In this case the scalar product is defined by... 

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 ... 

A curious feature of the space Ms of thermodynamic variables in an equilibrium state S is that its dimensionality varies with the number of phases, p, even though the values of the intensive variables (which might be used to parametrize the state S) do not. The intensive-type ket vectors R/ of (10.8) can actually be defined for all c + 2 intensities (T, —P, fjL, pi2, , pic) arising from the fundamental equation of a c-component system, U(S, V, n, ri2,. .., nc), even if only /of these remain linearly independent when p phases are present. 

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. 

The occupation number vectors are basis vectors in an abstract linear vector space and specify thus only the occupation of the spin orbitals. The occupation number vectors contain no reference to the basis set. The reference to the basis set is built into the operators in the second quantization formalism. Observables are described by expectation values of operators and must be independent of the representation given to the operators and states. The matrix elements of a first quantization operator between two Slater determinants must therefore equal its counterpart of the second quantization formulation. For a given basis set the operators in the Fock space can thus be determined by requiring that the matrix elements between two occupation number vectors of the second quantization operator, must equal the matrix elements between the corresponding two Slater determinants of the corresponding first quantization operators. Operators that are considered in first quantization like the kinetic energy and the coulomb repulsion conserve the number of electrons. In the Fock space these operators must be represented as linear combinations of multipla of the ajaj... 

The dependence of the used orbital basis is opposite in first and second quantization. In first quantization, the Slater determinants depend on the orbital basis and the operators are independent of the orbital basis. In the second quantization formalism, the occupation number vectors are basis vectors in a linear vector space and contain no reference to the orbitals basis. The reference to the orbital basis is made in the operators. The fact that the second quantization operators are projections on the orbital basis means that a second quantization operator times an occupation number vector is a new vector in the Fock space. In first quantization an operator times a Slater determinant can normally not be expanded as a sum of Slater determinants. In first quantization we work directly with matrix elements. The second quantization formalism represents operators and wave functions in a symmetric way both are expressed in terms of elementary operators. This... 

These properties define a linear vector space over the integers. Also since 98x,...,9Bj belong to SlT and are independent, they form a basis for the space which is therefore of dimension t. ... 

Such a definition can, evidently, be extended to any number of routes. It is clear that if A(1), A(2), A<3) are routes of a given reaction, then any linear combination of these routes will also be a route of the reaction (i.e., will produce the cancellation of intermediates). Obviously, any number of such combinations can be formed. Speaking in terms of linear algebra, the reaction routes form a vector space. If, in a set of reaction routes, none can be represented as a linear combination of others, then the routes of this set are linearly independent. A set of linearly independent reaction routes such that any route of the reaction is a linear combination of these routes of the set will be called the basis of routes. It follows from the theorems of linear algebra that although the basis of routes can be chosen in different ways, the number of basis routes for a given reaction mechanism is determined uniquely, being the dimension of the space of the routes. Any set of routes is a basis if the routes of the set are linearly independent and if their number is equal to the dimension of the space of routes. 

Multiplication of the Dirac characters produces a linear combination of Dirac characters (see eq. (4.2.8)), as do the operations of addition and scalar multiplication. The Dirac characters therefore satisfy the requirements of a linear associative algebra in which the elements are linear combinations of Dirac characters. Since the classes are disjoint sets, the Nc Dirac characters in a group G are linearly independent, but any set of N< I 1 vectors made up of sums of group elements is necessarily linearly dependent. We need, therefore, only a satisfactory definition of the inner product for the class algebra to form a vector space. The inner product of two Dirac characters i lj is defined as the coefficient of the identity C in the expansion of the product il[ ilj in eq. (A2.2.8),... 

This classification technique uses a space that is defined by a unique set of vectors called linear discriminants or LDs. Like the PCs obtained from PCA analysis, LDs are linear combinations of the original M-variables in the X-data that are completely independent of (or orthogonal to) one another. However, the criterion for determining LDs is quite different than the criterion for determining PCs each LD is sequentially determined such... 

We would like to emphasize that, due to the closure constraint, there are only (m — 1) linearly independent internal MEC. Thus, the m vectors defined by Eq. (93) in reality span the (m — 1 )-dimensional space of internal MEC. In order to remove this linear dependence one could adopt the relative internal approach of Sect. 2.1.3. Namely, one then selects the electron population of one atom in the system as dependent upon populations of all remaining atoms, and discards the MEC associated with that atom. All remaining MEC can also be constructed directly from the corresponding internal relative softness matrix. Although the sets of independent internal MEC for alternative choices of the dependent atom will differ from one another, they must span the same (m — 1 )-dimensional linear space of independent internal MEC. For example, in the two-AIM system of Fig. 4 there is only one independent internal MEC direction along the P-line. 

A real, symmetric matrix A is called positive definite if x Ax > 0 for every conforming nonzero real vector x. Extend the result of (a) to show that the covariance matrix E in Eq. (4.C-1) is positive definite if the scalar random variables i ,.... Emu are linearly independent, that is, if there is no nonzero m-vector x such that x Eu vanishes over the sample space of the random vector . 

A set of row or column vectors, vi,..., Vp, is called linearly independent if its only vanishing linear combination Yli i i is the trivial one, with coefRcients Ci all zero. Such a set provides the basis vectors Vi,...,Vp of a p-dimensional linear space of vectors CiVi, with the basis variables Cl,..., Cp as coordinates. 

The unlabeled triangle is the simplex in E (2-simplex) and the unlabeled tetrahedron is the simplex in (3-simplex) evidently, whether enantiomorphous -simplexes can be partitioned into homochirality classes depends on the dimension of E". Recall that an /j-simplex is a convex hull of + 1 points that do not lie in any (n - l)-dimensional subspace and that are linearly independent that is, whenever one of the points is fked, the n vectors that link it to the other n points form a basis for an n-dimensional Euclidean space An n-simplex may be visualized as an n-dimensional polytope (a geometrical figure in E" bounded by lines, planes, or hyperplanes) that has n + vertices, n n + )/2 edges, and is bounded by n + 1 (u — l)-dimensional subspaces. It has been shown that the homochirality problem for the simplex in E is shared by all -sim-... 

This linear system usually has an infinite number of solutions. Its solution space is determined by a set of basis vectors. All solutions of the system can be expressed as linear combination of the basis vectors. The dimension of the nullspace (the number of basis vectors) is given by n - rank S ), where rank(S ) is the number of linearly independent rows in S. ... 

Certainly, ARn is the first algebra to be associated with Rn because it is based on the notion of linear independence of vectors, which is in the foundation of the definition of the vector spaces. 

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