Linearly independent equations

A proper set of chemical equations for a system is made up of R linearly independent equations. 

Nf = 0 The problem is exactly determined. If NF = 0, then the number of independent equations is equal to the number of process variables and the set of equations may have a unique solution, in which case the problem is not an optimization problem. For a set of linear independent equations, a unique solution exists. If the equations are nonlinear, there may be no real solution or there may be multiple solutions. 

Since the individual variations of the orbitals are linearly independent, equation (A.36) can only be true if the quantity inside the large square brackets is zero for every value of a, namely... 

In addition, if each calibration sample contains only one analyte, then R contains already the spectra of the pure components and C K x K) is a diagonal matrix (i.e. non-zero values only in the main diagonal). Hence eqn (3.20) simply calculates S by dividing the spectrum of each pure calibration sample by its analyte concentration. To obtain a better estimate of S, the number of calibration samples is usually larger than the number of components, so eqn (3.19) is used. We still have two further requirements in order to use the previous equations. First, the relative amounts of constituents in at least K calibration samples must change from one sample to another. This means that, unlike the common practice in laboratories when the calibration standards are prepared, the dilutions of one concentrated calibration sample carmot be used alone. Second, in order to obtain the necessary number of linearly independent equations, the number of wavelengths must be equal to or larger than the number of constituents in the mixtures (7>A). Usually the entire spectrum is used. 

From eqns. (86) one must choose a set of linearly independent equations and by using known methods find the reaction rate constants. 

Matrix inversion is not widely used in practice, but from a theoretical point of view is extremely useful, because it allows us to calculate the minimum number of projections that are required for a complete reconstruction. If we have p projections of a structure, and each projection contains r rays, a reconstruction procedure amounts to solving a system of p-r equations in n2 unknowns, and algebra tells us that a solution exists only if the number of linearly independent equations is equal to the number of the unknowns. 

Next, suppose that you are interested in solving n linear independent equations in n unknown variables ... 

Figure 3.3 Graphical representation of the two consistent and linearly independent equations of Example 3.15.

Since the operator set described in (24) is not in general known, different operator basis sets must be chosen. Any new set of basis operators can always be expressed as a linear combination of the (N+l) operators given in (24). It is obvious that the new operator basis set can also yield only 2N linearly independent equations when inserted into (19). To study this problem of linear dependence among the EOM equations in more detail, consider the following simple analysis. Let (19) be expressed in terms of some set of 2Af basis operators, (flj), as... 

It is to be emphasized that at most 2N basis operators can yield linearly independent equations when substituted into (19). A set of 2 N basis operators for which < ) 0 forms what is often called a complete set of basis operators for EOM calculations in that they give the desired excitation energies. However, when expanding one set of operators in terms of a different operator basis as in (26), a basis of (iV-H) operators is in general needed. Thus there are two entirely different senses in which an operator can be complete. EOM completeness involves 2N operators, but (Y-t-1) are necessary for general operator completeness. 

Assuming that the concentrations of the intermediate and of the final products equal zero at the beginning of the reaction (bo = Cq = do = eg = 0), the following three linear independent equations with mass-balance result ... 

Analysis of these two equations shows that at least two more linear independent equations must be foimd in order to be able to describe the system completely. 

The maximum number of linear independent equations for one knot with n fluxes is [Pg.157]

The number of stoichiometrically independent reactions is given by the rank of the matrix Rp, which can be determined with e.g. the aid of the Gaussian method of elimination. As a result, the stoichiometrical coefficients of linearly independent equations for the reaction system are necessary and sufficient for, for example, calculation of the conversion of the key variables and therefore also for all other components. Thus... 

Example for Determination of Number of Linearly Independent Equations 409... 

We now consider the rectangular reservoir domain defined by the index ranges 1 < i < 11 and 1 < j < 11, and specifically, a Dirichlet formulation where pressures of 10, 20, 30, and 40 are specified in clockwise fashion along the four edges of the box. This no-well formulation, as discussed earlier, is associated with a unique solution. If Equation 7-15 is written for each and every node (ij) internal to the computational box, and the assigned boundary values are included into the set of linear equations, we obtain 11 x 11, or 121 unknowns that are fully determined by 121 linearly independent equations. Over one hundred coupled equations are required for this very coarse mesh.t... 

The last O Eq. 26.4 constitutes a system of three linear independent equations for the three... 

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