How to calculate with residue classes in $\mathbb{Z}{/5\mathbb{Z}}$?
- $- \overline x \neq \overline x$ but $- \overline x = \overline{5-x}$
- $\overline x + \overline y = \overline{x+y}$
- $\overline x \cdot \overline y = \overline{x\cdot y}$
So the following calculation should be right?
$ \overline 2 \cdot \overline 1 + \overline 2 \cdot \overline 1 - \overline 3 \cdot \overline 4 = \overline{2\cdot1} + \overline{2\cdot 1} - \overline{3\cdot4} = \overline{2\cdot1 + 2\cdot 1 - 3 \cdot 4} = \overline{-8} = \overline{-3} = \overline{5-3} = \overline{2} $
How to use this, when solving a LES like the following with coefficients in $\mathbb{Z}_{5}$?
$ \left( \begin{array}{cccc|c} 3 & -1 & 0 & 2 & -4\\ 1 & 0 & -3 & 2 & 2\\ 2 & 2 & -3 & 0 & 1\\ \end{array} \right) \rightsquigarrow \left( \begin{array}{cccc|c} 0 & -1 & 9 & -4 & -10\\ 1 & 0 & -3 & 2 & 2\\ 0 & 0 & 3 & -4 & -1\\ \end{array} \right) $
Can I run the Gaussian algorithm without taking to much care of the residue classes and convert the integers after I've finished the algorithm?
Don't multiply a row by 5 during the Gaussian algorithm