Basis stoichiometric subspace

Here, mattix A is the stoichiometric coefficient matrix formed from the reaction stoichiometry, and C is any concentration vector that satisfies Equation 6.7. Determination of the subspace of concenttations orthogonal to the stoichiometric subspace, is in fact equivalent to computing the mil space of A. We shall denote by N the mattix whose columns form a basis for the null space of A. Linear combination of the columns in N hence generates the set of concentrations orthogonal to the stoichiometric subspace. 

Finally, one may compute the set of vectors orthogonal to the stoichiometric subspace. Again computation gives a set of two vectors that form a basis for the null space. Two representative vectors for this space are given below ... 

The columns in A span the stoichiometric subspace S in (Ca-Cb-Cc-Cd-Ce space). To find N—the columns of N that form a basis for the nullspace of S—we simply... 

Matrix N has columns that form a basis for the nullspace of the stoichiometric subspace S, and hence there must he (n-d) columns in N if there are n components participating in d independent reactions (the stoichiometric coefficient matrix A has size nxd). 

From Section 7.2.1.1, the dimension of the AR is three (d=3) and there are four components (n = 4). It is expected that the dimension of the subspace orthogonal to the stoichiometric subspace is (4 - 3) = 1. Therefore, for the three-dimensional Van de Vusse system, the null space is given by a one-dimensional subspace (in other words, the basis for the null space is composed of a single, linearly independent vector). This is confirmed when the null space of A is computed, giving... 

Acyclic auxiliary dynamical system with one attractor have a characteristic property among all auxiliary dynamical systems the stoichiometric vectors of reactions Aj A q form a basis in the subspace of concentration space with c, = 0. Indeed, for such a system there exist n—1 reactions, and their... 

Let us assume that the auxiliary dynamical system is acyclic and has only one attractor, a fixed point. This means that stoichiometric vectors form a basis in a subspace of concentration space with — 0. For every reaction A,- A the following linear operators Qu can be defined ... 

The sets of mole numbers that satisfy (11.2.3) are sets of stoichiometric coefficients, Vy, and the subspace of N that satisfies (11.2.4) is called the nullspace of A. The dimension of the nullspace is the number of independent vectors Vy (basis vectors) that satisfy (11.2.3) that is, the nullspace has dimension H, which is the number of independent chemical reactions. 

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