Four matrices of order two

Determine whether the matrices

$$A_1=\left(\begin{array}{cc}1& 1\\ 0& 0\end{array}\right),A_2=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),A_3=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),A_4=\left(\begin{array}{cc}0& 0\\ 1& 1\end{array}\right)$$

form a basis of $M_2(\mathbb{R})$, and find the dimension of the subspace they span.