Vector linearly dependent

In the case of linearly dependent vectors, each of them can be expressed as a linear combination of the others. For example, the last of the three vectors below can be expressed in the form Zj = z. 

Although hie rank of is two, the linearly dependent vector is not formed by a linear mixture. We have again used MATLAB to perform die computations. 

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,. 

A significant advantage of this procedure is that nearly linearly dependent vectors can be eliminated at this stage, simply reducing the value of m. The resulting unit vectors define an n x in column matrix. Using these unit vectors, the algorithm solves anmxm system of linear equations in the e-space,... 

Result (A.19) proves the necessity of a Lemma for linear independent vectors (and in fact it transforms one basis y-y of p-dimensional vector space to another one by linear transformation Q). To prove this for the remaining linearly dependent vectors y, we express them through those which are independent... 

Since the triples operator (14.4.11) is redundant, we cannot set up a projection basis (/I that is biorthogonal to the linear combination of CSFs ix) in (14.4.13). We shall simply assume that the constitute a linearly independent basis for the space spanned by the linearly dependent vectors /u.3 but we shall not specify their detailed form. 

Any linearly independent set of simultaneous homogeneous equations we can construct has only the zero vector as its solution set. This is not acceptable, for it means that the wave function vanishes, which is contrai y to hypothesis (the electron has to be somewhere). We are driven to the conclusion that the normal equations (6-38) must be linearly dependent. 

The equality holds if, and only if, the vec tors a, b are linearly dependent (i.e., one vec tor is scalar times the other vector). 

Equation (8.91) is singular since it has a zero determinant. Also the column vectors are linearly dependent since the second column is —5 times the first column and therefore the system is unobservable. 

Consider a set of n Af-dimensional vectors and a function (p that assigns a value 1 to each element of Af (i.e. 0 is a dichotomy see above). Baum [baumSSa] showed that if Af consists only of vectors such that no subset of N or fewer of them is linearly dependent, the smallest sized multi-layered perception that can realize an arbitrary dichotomy for Af contains one hidden layer consisting of [(n — 1) /N - -1] neurons. The size of this perception can only be decreased by putting on a more stringent constraint on the set Af. 

The Linear Algebraic Problem.—Familiarity with the basic theory of finite vectors and matrices—the notions of rank and linear dependence, the Cayley-Hamilton theorem, the Jordan normal form, orthogonality, and related principles—will be presupposed. In this section and the next, matrices will generally be represented by capital letters, column vectors by lower case English letters, scalars, except for indices and dimensions, by lower case Greek letters. The vectors a,b,x,y,..., will have elements au f it gt, r) . .. the matrices A, B,...,... 

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... 

Linear Manifolds in Hilbert Space.—Any sequence of m vectors, l/iXl/aV called a linearly dependent sequence if... 

The dependence of the induced field B ", eq.(12), on the applied field B, of course simply implies that we have a new induced field for each mcignitude and direction of B. For the purpose of interpretation and display, the linear dependence on the magnitude of B is circumvented by considering the dimensionless shielding vector which is defined as the negative of the induced field per unit applied field [14,28,30], i.e. 

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. 

In most practical applications, all but one component of ft will be non-negative. One can then choose the vector with the negative component to be the left-hand side of (5.82). However, the fact that fi must be non-negative greatly restricts the types of linear dependencies that can be expressed as a mixture-fraction vector of length... 

Since B depends on the choice of the linearly independent vectors used to form d> , all possible combinations must be explored in order to determine if one of them satisfies (5.96) and (5.97). Any set of linearly independent columns of d> that yields a matrix B satisfying (5.96) and (5.97) will be referred to hereinafter as a mixture-fraction basis. 

No other linear dependency is apparent, and thus we should expect to find a mixture-fraction vector with two components. 

However, care must be taken to avoid the singularity that occurs when C is not full rank. In general, the rank of C will be equal to the number of random variables needed to define the joint PDF. Likewise, its rank deficiency will be equal to the number of random variables that can be expressed as linear functions of other random variables. Thus, the covariance matrix can be used to decompose the composition vector into its linearly independent and linearly dependent components. The joint PDF of the linearly independent components can then be approximated by (5.332). 

In Sections VILA and VII.B, we make use of the fact that the link matrix L allows to account for linear dependencies within the partial derivatives. In particular, the dependence of a (vector) functionf S) = /(Sind, Sdep) is expressed... 

Figure 4-15. Linear dependence, three vectors f ,i, f ,2, f ,3 lie in one plane...

Figure 4-16. Near linear dependence if the base vectors are almost parallel...

The rank of a matrix Y is the number of linearly independent rows or columns in this matrix. The columns of Y are linearly dependent if one of the column vectors y j can be written as a linear combination of the other columns. The same holds for rows. 

Collecting the orthogonal complement to the C ) in the (Vpam x Vg-eej-dimensional matrix L (note that the constraints vectors may be linearly dependent), we can express the requirements in Eq. (26) as a linear transformation to a new set of variables, x ... 

The set of all direct mechanisms in a system contains within it a basis for the vector space of all mechanisms. In general, there are more direct mechanisms than basis elements, which means that there can exist linear dependence relations among direct mechanisms but, even so, they differ chemically. That is, a direct mechanism with a given step omitted cannot be considered to result from a combination of two other mechanisms in which that step is assumed to occur. In the latter case the net velocity of zero for that step would result from a cancellation of equal and opposite net velocities rather than from the complete absence of the step. The set of all direct mechanisms (unlike a basis) is a uniquely defined attribute of a chemical system. In fact what we have called a direct mechanism is what is usually called a mechanism in chemical literature, even though the definition may be implicit. 

Since the covariance matrix is diagonal, a vector denoted by V is used to store the variances. If the number of balances does not exceed ten, the module also computes the tabular value of the chi square distribution at significance level a = 0.05 and degrees of freedom nb. The return value ER = 1 of the status flag indicates that the rows of the matrix W are linearly dependent, and hence you should drop at least one of the balance equations. If the source of linear dependence is not clear then the module M10 can help to uncover it. 

The singular limit (S 10.2-6), in which one of the / independent R-) is a zero vector (implying that the original basis vectors R,) were linearly dependent), occurs at a critical state Sc, where the number of phases p and dimension/ are changing. Critical state limits will be examined in Chapter 11. 

Equation (11.8) expresses the geometrical necessity of linear dependence among any /+ 1 vectors in an /-dimensional space. Such linear dependence (corresponding to a null... 

---