Linear Combinations of Vectors

The idea of a linear combination is an important idea that will be encountered when we discuss how a matrix operator affects a linear combination of vectors. 

Of course, we need to check that the p g) we defined is actually a linear transformation. Here the physics helps us. Recall from Section 1.2 that linear combinations of vectors can be interpreted physically if a beam of particles contains a mixture of orthogonal states, then the probabiUties goverrung experiments with that beam can be predicted from a linear combination of those states. Thus observer A s and observer B s linear combinations must be compatible. In other words, if observer A takes a linear combination ci/i -+ c fi, while observer B takes the same linear combination of the corresponding states cip(g)/i C2p(g )h. the answers should be compatible, i.e.,... 

An important concept is the matrix rank. The highest order of the nonzero determinant generated by a given matrix is designated as rgA. With a zero determinant the matrix is singular. The matrix whose elements along the main diagonal are equal to unity and all the rest are zero is called a unit or identity matrix. If some vector Y is a linear combination of vectors, it is called linear-dependent... 

The first equation as in the nondegenerate case, can be satisfied by taking the correction to be orthogonal to the unperturbed ground state vector which is satisfied by any linear combination of vectors from the manifold related to the /-th degenerate eigenvalue. Inserting the required expansion we get ... 

Vectors The collection of all vectors containing n elements satisfying basic axioms (see, for example, Ramkrishna and Amundson (1985)) is called a linear space denoted by 91 . The basic axioms define scalar multiplication, vector addition (from which evolves the concept of linear combination), and a null vector that has all elements zero. Thus a linear combination of vectors Xj, j = is expressed as the vector E ia Xj, where the a/s are numbers. If the a/s are all... 

C is therefore a linear combination of vectors Cj and C2, which are the concentrations of beakers 1 and 2, respectively. 

Mixture concentrations are commonly viewed as a linear combination of vectors. In AR theory, the vector difference C2 - Cl is frequently referred to as the mixing vector... 

A subspace of a vector space is a nonempty subset that is also a vector space. That is, vector subspaces also obey the laws of vector addition and scalar multiplication. If x and y are two vectors that lie in a vector subspace, then linear combinations of x and y will produce vectors that also lie in the subspace. For example, linear combinations of vectors [0, 0, 1] and [2, 0, 0] produce vectors that lie in a two-dimensional subspace (a plane) in R linear combinations of vector [5, 0.2, -3, 1, 8] lie in a one-dimensional subspace (a line) in R. ... 

The two operations introduced by the axioms can occur in combinations. Using the associativity, we define uniquely a linear combination of vectors v , v , with coefficients (real numbers), , ... 

Numerical Methods for Chemical Engineers Using Excel , VBA, and MATEAB 3.4.3 Linear Combinations of Vectors... 

Alternatively, the electron can exchange parallel momentum with the lattice, but only in well defined amounts given by vectors that belong to the reciprocal lattice of the surface. That is, the vector is a linear combination of two reciprocal lattice vectors a and b, with integer coefficients. Thus, g = ha + kb, with arbitrary integers h and k (note that all the vectors a,b, a, b and g are parallel to the surface). The reciprocal lattice vectors a and are related to tire direct-space lattice vectors a and b through the following non-transparent definitions, which also use a vector n that is perpendicular to the surface plane, as well as vectorial dot and cross products ... 

To derive the DIIS equations, let us consider a linear combination of coordmate vectors q ... 

For states of different symmetry, to first order the terms AW and W[2 are independent. When they both go to zero, there is a conical intersection. To connect this to Section III.C, take Qq to be at the conical intersection. The gradient difference vector in Eq. f75) is then a linear combination of the symmetric modes, while the non-adiabatic coupling vector inEq. (76) is a linear combination of the appropriate nonsymmetric modes. States of the same symmetry may also foiiti a conical intersection. In this case it is, however, not possible to say a priori which modes are responsible for the coupling. All totally symmetric modes may couple on- or off-diagonal, and the magnitudes of the coupling determine the topology. 

Plane waves are often considered the most obvious basis set to use for calculations on periodic sy stems, not least because this representation is equivalent to a Fourier series, which itself is the natural language of periodic fimctions. Each orbital wavefimction is expressed as a linear combination of plane waves which differ by reciprocal lattice vectors ... 

The above equation for x provides an example of expressing a vector as a linear combination of other vectors (in this case, the basis vectors). The vector x is expressed as... 

Any other modes we can think of are a linear combination of these. For example the double antisite, which is formed by exchanging a pair of A and B atoms, is equivalent to n + ni - 2 T. This would be an equally good choice as a basis vector instead of one of the three above. 

Just as a known root of an algebraic equation can be divided out, and the equation reduced to one of lower order, so a known root and the vector belonging to it can be used to reduce the matrix to one of lower order whose roots are the yet unknown roots. In principle this can be continued until the matrix reduces to a scalar, which is the last remaining root. The process is known as deflation. Quite generally, in fact, let P be a matrix of, say, p linearly independent columns such that each column of AP is a linear combination of columns of P itself. In particular, this will be true if the columns of P are characteristic vectors. Then... 

Step 3.—Choose an arbitrary basis for this set of vectors. Let Px and P3 form such a basis. Express all the vectors as linear combinations of Px and P2 (this can be done in only one way). 

Step 6.—Again expressing the vectors as a linear combination of P2 and P3, obtain ... 

Now z — cy < 0 must hold for all j in order to have obtained a solution x° whose components are given by the coefficients expressing P0 as a linear combination of Pi and P2. To impose the condition zy — cy < 0 on the parameter t, is to solve a set of simultaneous—not necessarily linear—inequalities in. Then Pi and P2 would be an optimal basis for this interval of values of. By fixing a value of immediately outside the interval and in the neighborhood of a boundary point, the vector to be eliminated and that to be introduced into the basis are produced in the usual manner, and the process is then repeated. If no value of t satisfies the set of inequalities, then by fixing at a given 0, the usual procedure is used to eliminate a vector and introduce another into the basis. 

To decide on a change of basis in this case we put = 11 in order to determine the solution for vaiuesof 10. This violates s3 — c3 < 0, as can be seen fr< the above analysis. Hence P3 must come into the basis. The vector to eliminated is obtained as usual by expressing P0 as a linear combination of and P2 at 6 = 11, which gives -... 

If there is an interval of values of t common to these inequalities, one fixes a value of t at 10 + e where > 0 is arbitrarily small and t0 is a value on the boundary of the interval, and proceeds in the usual way to obtain a change of basis and then determine a neighboring interval of values of t. The solution vector in each case is given by the weights obtained in expressing P0—the column vector whose coefficients are the 6,(0—as a linear combination of the basis. The process terminates in a finite number of steps since the number of vectors in the problem is finite. 

On the strength of these results we can now express any arbitrary vector > in as a linear combination of the basis vectors ... 

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