Concentrations Orthogonal to the Stoichiometric Subspace

1 Null Space For many purposes in AR theory, it is useful to understand the set of concentrations that lie perpendicular (orthogonal) to S, which are spanned by the stoichiometric coefficient matrix A. For instance, the computation of critical DSR solution trajectories and CSTR effluent compositions that form part of the AR boundary require the computation of this space. It is therefore important that we briefly provide details of this topic here. It is simple to show from linear algebra that all points orthogonal to the space spanned by the columns of A are those that obey the following relation  

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. 

Assume that A contains n rows (there are n components in the system). If A has d linearly independent columns (indicating d linearly independent reactions), then rank(A) = d. If the rank of A is d, then the null space of A must have rank (n - d), or rank(N) = (n - d) (Strang, 2003). Thus, N is a matrix having n rows and (n-d) columns. 

Vectors Uj, U2,. .., form a basis for the null space of A — linear combination of the set of vectors uj,. .., 

Calculate the null space of matrix A—null(A)—for the following matrices  

---

Since there is only one column in A (corresponding to a single reaction), rank(A) = 1. To compute the set of concentrations orthogonal to the stoichiometric subspace, we compute the null space of A . Hence, since the rank of A is one, we expect the rank of the null space to be (3 - 1) = 2. We may compute the null space using standard methods such as elementary row operations. It... 

---