Answer
See below
Work Step by Step
Let $S$ be the subspace of $R^3$ spanned by the vectors $v_1 = (1, 0, 1) \\ v_2 = (0, 1, 1)\\v_3 = (2, 0, 2)$.
We can see that $2v_1=2(1,0,1)=(2,0,2)=v_2$
then $v=av_1+bv_2+cv_3 \in S\\
=av_1+bv_2+2cv_1\\
=(a+2c)av_1+bv_2\\
\rightarrow v \in span \{v_1,v_2\}\\
\rightarrow span\{v_1,v_2,v_3\} \subset span \{v_1,v_2\}$
Since $\{v_1,v_2\} \subset \{v_1,v_2,v_3\}\\
\rightarrow span\{v_1,v_2\} \subset \{v_1,v_2,v_3\}\\
\rightarrow S=span\{v_1,v_2\}$
Notice that $v_1$ and $v_2$ are not proportional and then $v_1,v_2$ are linearly independent in $R^3$
Hence, $\{v_1,v_2\}$ is a basic for $S \rightarrow \dim[S]=2 $
