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Fundamental Subspaces of S in Java Creation 3 of 9 in Java Fundamental Subspaces of S




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Fundamental Subspaces of S using jar toproduce 3 of 9 in asp.net web,windows application data matrix dx2/dt v2 Null space Left null space Row space (--1,1). (1,1). -- 1Vdyn dx1/dt Column space Vdyn Figure 8.6: A gra phical depiction of the null, row, and column spaces for x1 x2 . Since the fluxes v1 and v2 are finite, all these three spaces are finite.

The singular vectors shown are multiplied by 2 to make the figure more simplistic.. by the (1, 1) jvm Code39 / 2 and (1, 1)/ 2 vectors. They are orthogonal and span the null and row spaces respectively; vss is mapped into the origin by S, whereas vdyn is mapped onto u1 = ( 1, 1)/ 2 and stretched by the singular value (see equation 8.10).

Note that the bounded range of the uxes also set the bounds of the column space. The extreme points of the row space (the open triangle and square) correspond to the maximum allowable values on the time derivatives of x1 and x2 . Thus, the extreme points of the row space lead to extreme points in the column space.

The implication of nonnegativity and nite size of the ux and concentration values will be discussed in subsequent chapters. Numerical example We can trace these mappings using a speci c numerical example. If we pick v = (2 2, 2)T , then VT v = 1 3 (8.

11). which corresponds to the projection of v onto the two right singular vectors. In other words, this ux vector is decomposed into one unit of v1 and three of v2 . Then VT v = 2 0 (8.

12). 8.3 SVD of S for the Elementary Reactions which nally maps onto the left singular vectors as x = U VT v = U 2 0 = 2 u1 + 0 u2 = 2u1 = 2 2 (8.13). Bilinear associat Code 3/9 for Java ion Consider the reaction of equation 6.13 written as v1 x1 + x2 v2 where x1 = C, x2 = P and x3 = C P. The SVD matrix is 1/ 3 1 1 2/ 6 0 6 1 = 1/ 3 1/ 6 1/ 2 0 1 1 1 0 1/ 3 1/ 6 1/ 2 of the stoichiometric 0 0 0 1/ 2 1/ 2 1/ 2 1/ 2 x3 (8.

14). There is only one singular value, and thus the column space is spanned by one left singular vector u1 . It is the reaction vector si normalized to unit length. The row space is spanned by vT = (1, 1)/ 2.

The row and 1 column spaces are related by Sv1 = 1 u1 or 1 1 1 1/ 3 = 6 1/ 3 1/ 3 1 = 2 1 1 (8.15). 1 1 1 or 1/ 2 1/ 2. (8.16). 1 1 1 1 1 1. 1 1. (8.17). Note that the row jsp Code 39 and null spaces are spanned by the same right singular vectors, vT and vT , respectively, as the reversible conversion. This same 1 2 decomposition is true of the ux vector, leading to an analogous interpretation. The column space is simply spanned by the normalized form of the reaction vector.

The left null space is now two dimensional. The orthonormal basis vectors for the column and left null space are shown in Figure 8.7.

The second and third left singular vectors u2 and u3 that span the left null space are hard to interpret chemically. We will address this issue in what follows..

Fundamental Subspaces of S Figure 8.7: The t awt Code39 hree-dimensional depiction of the orthonormal basis for the column and left null spaces for a simple bilinear association. The plane is the left null space and the line is the column space.

The vectors shown are an orthonormal set. If the flux vector is on the left-hand side of the plane as indicated, then the reaction is proceeding in the forward direction, and vice versa..

Linear combinatio ns of fluxes and concentrations We can begin to familiarize ourselves with the details of these transformations. The ux vector in the dynamic equations can be transformed using VT as dx = SVVT v dt or x1 1 0 d x2 = 1 (v1 v2 ) 0 (v1 + v2 ) dt 1 0 x3 (8.18).

(8.19). forming two group ings of the uxes. The second term corresponds to the null space, and the combination v1 + v2 is a type III extreme pathway that we will discuss in 9. The rst corresponds to the row space and the grouping v1 v2 is the net ux through the reaction, and it is orthogonal to v1 + v2 (see Figure 8.

5). Multiplying equation 8.19 by UT leads to ( x1 x2 + x3 )/ 3 0 6 d (v1 v2 ) (v1 + v2 ) 0 (2x1 x2 + x3 )/ 6 = 0 dt 2 2 0 0 (x2 + x3 )/ 2 Note that the singular value of 6 shows up and that the two column vectors on the right-hand side of the equation are the two columns of .

This system is now fully decomposed, showing how independent groupings of concentrations are moved by independent groupings of the uxes. As noted earlier, the two left singular vectors that span the left null.
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