Notes

Just to mention here, this solution won't show every step of calculation, as this would take ages and is quite hard to do on a computer. Instead it will show the main intermediary results and the way of calculation (As it was in the computer science course in Germany).

As this solution is not written for mathematic experts, I'll try to skip as much special terms as possible. This definitely increases the amount of text needed to describe this solution, and in some cases it might even become slightly unprecise. But you should notice that this solution could be much briefer if you would use words as basis transformation, linear dependent, vector space ... .