I'll start off easy.

How do you determine the dimension and a basis of the group of all 2x2 matrices over R. That would be $\displaystyle M_{2 \times 2}^R = \left\{\left(\begin{array}{cc}\alpha&\beta\\\lambd a&\delta\end{array}\right) | (\alpha, \beta, \lambda, \delta)\in R\right\} $