Stoichiometry of chemically reacting species

While linear algebraic methods are present in almost every problem, they also have a number of direct applications. One of them is formulating and solving balance equations for extensive quantities such as mass and energy. A particularly nice application is stoichiometry of chemical systems, where you will discover most of the the basic concepts of linear algebra under different names. 

We consider a closed system with species denoted by Mj,. .., Mk.  

Let n denote the quantity of species expressed in moles. The k-vector 

The first approach is based on explicitly describing chemical reactions. We suppose that there are p reactions taking place in the system. The j-th reaction is described by equation of the form 

The concept of a closed system can also be introduced without considering reactions. Chemical species are built from building blocks called atoms. Define the atom matrix A, where [A] j is the number of the i-th atom in the molecule of the j-th species Mj. If the number of different atoms is denoted by a then the atom matrix is of dimension a k. The quantities of atoms in the system can be calculated by summing up their quantities in each species, i.e., forming the product An. These quantities remain unchanged if the system is closed, so that 

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