# Discrete Cartesian Coordinate Transformations: Using Algebraic Extension Methods
## 4 Results
### Advantages of Representations of Linear Operators Using Idempotent Elements  
Let us show that for each specific Galois field that meets the criterion formulated above, there exists a distinguished system of discrete coordinates on the plane in which the action of the operator on the coordinate vectors is factorized. The element of the algebraic ring obtained by the combined method of algebraic expansions and represented in the form (60) corresponds to the representation of the matrix describing, in particular, the coordinate transformation in the following form:  
\[\mathbf{W}=\mathbf{AE}_{1}+\mathbf{BE}_{2} \tag{61}\]  
The detailed mapping between matrices and elements obtained by algebraic expansions is given by the formulas  
\[\mathbf{A}=A_{1}\mathbf{E}_{0}+A_{2}\mathbf{G}\leftrightarrow A_{1}+A_{2}\theta \tag{62}\]  
\[\mathbf{B}=B_{1}\mathbf{E}_{0}+B_{2}\mathbf{G}\leftrightarrow B_{1}+B_{2}\theta \tag{63}\]  
We will show that such a representation does provide a simplification of the description of coordinate transformations for the case of the considered Galois fields. The use of Formula (1) based on the classical method of algebraic expansions is not the only possible one. Along with it is admissible to use the representation of two-dimensional coordinates using the element \(i\), which corresponds to the nonstandard method of algebraic expansions. In terms of ordinary vector expressions, this corresponds to the representation of a radius vector on a two-dimensional discrete plane as  
\[\mathbf{r}=r_{1}\mathbf{e}_{1}+r_{2}\mathbf{e}_{2} \tag{64}\]  
where  
\[\mathbf{e}_{1}= \begin{pmatrix}\beta\\ \alpha\end{pmatrix} ;\ \mathbf{e}_{2}= \begin{pmatrix}\beta\\ -\alpha\end{pmatrix}  \tag{65}\]  
The following relations are valid:  
\[\mathbf{E}_{1} \begin{pmatrix}\beta\\ \alpha\end{pmatrix}  = \begin{pmatrix}\beta&-\alpha\\ \alpha&\beta\end{pmatrix}\begin{pmatrix}\beta\\ \alpha\end{pmatrix} = \begin{pmatrix}\beta^{2}-\alpha^{2}\\ 2\alpha\beta\end{pmatrix} = \begin{pmatrix}\beta\\ \alpha\end{pmatrix}  \tag{66}\] \[\mathbf{E}_{1} \begin{pmatrix}\beta\\ -\alpha\end{pmatrix}  = \begin{pmatrix}\beta&-\alpha\\ \alpha&\beta\end{pmatrix}\begin{pmatrix}\beta\\ -\alpha\end{pmatrix} = \begin{pmatrix}\beta^{2}+\alpha^{2}\\ 0\end{pmatrix} = \begin{pmatrix}0\\ 0\end{pmatrix} \] (67) \[\mathbf{E}_{2} \begin{pmatrix}\beta\\ \alpha\end{pmatrix}  = \begin{pmatrix}\beta&\alpha\\ -\alpha&\beta\end{pmatrix}\begin{pmatrix}\beta\\ \alpha\end{pmatrix} = \begin{pmatrix}\beta^{2}+\alpha^{2}\\ 0\end{pmatrix} = \begin{pmatrix}0\\ 0\end{pmatrix} \] (68) \[\mathbf{E}_{2} \begin{pmatrix}\beta\\ -\alpha\end{pmatrix}  = \begin{pmatrix}\beta&\alpha\\ -\alpha&\beta\end{pmatrix}\begin{pmatrix}\beta\\ -\alpha\end{pmatrix} = \begin{pmatrix}\beta^{2}-\alpha^{2}\\ -2\alpha\beta\end{pmatrix} = \begin{pmatrix}\beta\\ -\alpha\end{pmatrix}  \tag{69}\]  
These relations mean that the action of an operator represented in the form (47) on a vector represented in the form (64) is factorized. \[\mathbf{Wr}=(\mathbf{A}\mathbf{E}_{1}+\mathbf{B}\mathbf{E}_{2})(r_{1}\mathbf{e}_{1}+r_{2}\mathbf{e}_{2})=r _{1}\mathbf{A}\mathbf{e}_{1}+r_{2}\mathbf{B}\mathbf{e}_{2} \tag{70}\]  
Ratio (70), which can also be written in terms of elements obtained by the combined method of algebraic expansions, shows that each component of the vector describing the coordinates of a pixel in terms of the corresponding Galois field is subject to its <<WORD> operator. Moreover, this operator is isomorphic to a separate element of the Galois field obtained by the method of classical algebraic expansions of the basic field. The next task that arises when using the proposed approach is to bring the vectors \(\mathbf{A}\mathbf{e}_{1}\) and \(\mathbf{B}\mathbf{e}_{2}\) (or their corresponding algebraic elements) to the form of the form (64). This problem is obviously solvable since it reduces to the solution of linear equations.