PRO/VS-02002

Prove that the set of matrices
\begin{equation*}
  \begin{pmatrix}
     1 & 0 \\ 1& 1
  \end{pmatrix}
,~
  \begin{pmatrix}
     0& 1 \\ 1& 1
  \end{pmatrix}
,~
  \begin{pmatrix}
     1 & 1 \\ 0& 1
  \end{pmatrix}
,  ~  \begin{pmatrix}
     1 & 1 \\ 1& 0
  \end{pmatrix},
\end{equation*}
is a basis in the real vector space of all \(2\times 2\)  real matrices.