Linear Algebra Model of Commerce

This project envolves a practical necessity of creating a tax system exclusively based in the register of purchases and sales of an enterprise, that could be automactly aquired without need to create declarations or to mantain a accountability. On the contrary, this model is possible, there will be and accounting automactly manteined. For this purchases and sales to make sense, there will be the necessity of some type of classification and segregation of the goods and services bought and sold. There are some suggestions, like the HArmonized System for goods. Bar codes or UPC codes are good options to classification, too. Here we will supose this classification to be given and so it is possible to count each class indepently. In our first example we will classificate the goods as apples and oranges and will disregard everything that doesn't fit in the classification.
A set of goods can, this way, be represented by a vector. There will be a number of units of this good and a price involved in the commercial purchase of sale transaction. For representing a set of goods we construct a vector, with each coordinate related to the price of goods in the set. The coordinates will be measured by the square root of the price praticed, in our case, for commodity. If the coordinate is the square root of the price, so that this way the total price of a set of goods is the euclidean norm of the vector. Done so, a commercial unit will transform, in a unit of time (usualy a month) a certain number of each good in another number of goods. Locally, it is a linear transformation, represented by a Matrix. Given a commercial unit's sellings (s) and buying (b) in a given period, expressed as vectors, we call the F matrix of this plant mechanics, any F such that s = F b. This package intends to analyze the mathematical properties of F.
The model begins with a classification of everything that is bought and sold by the enterprise, including work, capital, and rent. The coordinates in this space are defined by the square root of the price practiced in the selling or buying. The square root is used for the Euclidian norm to express the total value spent on the things bought and the total value received selling.
For instance, if someone buys and sells apples and bananas, there will be a diagonal matrix "M", invertible. There will be an eigenvalue for the commerce of apples, measuring the added value and another one to the commerce of bananas. In another enterprise, one buys oranges and sells orange juice, so there will be an industry, with zero diagonal values in F. The singular value decomposition will be needed to examine the added value performance.