Answer
The set $S=\{(1,0,0,1),(0,2,0,2),(1,0,1,0),(0,2,2,0)\}$ is not a basis for $R^4$.
Work Step by Step
The set $S=\{(1,0,0,1),(0,2,0,2),(1,0,1,0),(0,2,2,0)\}$ is not a basis for $R^4$ because $S$ is linearly dependent set of vectors. Indeed, assume that $$a(1,0,0,1)+b(0,2,0,2)+c(1,0,1,0)+d (0,2,2,0)=(0,0,0,0), \quad a,b,c,d\in R.$$
Which yields the following system of equations
\begin{align*} a+c&=0\\ 2b+2d&=0\\ c+2d&=0\\ a+2b&=0. \end{align*}
The augmented matrix is given by $$\left[ \begin {array}{cccc} 1&0&1&0\\ 0&2&0&2 \\ 0&0&1&2\\ 1&2&0&0\end {array} \right] $$ and its determinant is zero. This means that the system has non trivial solutions and hence $S$ is linearly dependent set of vectors. Hence, the set $S=\{(1,0,0,1),(0,2,0,2),(1,0,1,0),(0,2,2,0)\}$ is not a basis for $R^4$.
