We begin with a relation of linear dependence using unknown scalars $a$ and $b$. We wish to know if these scalars must both be zero. Recall that the zero vector in $C$ is $(-1,\,-1)$ and that the definitions of vector addition and scalar multiplication in Example CVS are not what we might expect.
\begin{align*}
\zerovector
&=(-1,\,-1)\\
&=a(0,\,2) +b(2,\,8)&&\knowl{./knowls/definition.RLD.knowl}{\text{Definition RLD}}\\
&=(0a+a-1,\,2a+a-1) + (2b+b-1,\,8b+b-1)&&\knowl{./knowls/example.CVS.knowl}{\text{Example CVS}}\\
&=(a-1,\,3a-1) + (3b-1,\,9b-1)\\
&=(a-1+3b-1+1,\,3a-1+9b-1+1)&&\knowl{./knowls/example.CVS.knowl}{\text{Example CVS}}\\
&=(a+3b-1,\,3a+9b-1)\\
\end{align*}
From this we obtain two equalities, which can be converted to a homogeneous system of equations,
\begin{align*}
-1&=a+3b-1&a+3b&=0\\
-1&=3a+9b-1&3a+9b&=0
\end{align*}
This homogeneous system has a singular coefficient matrix, and so has more than just the trivial solution (Definition NM). Any nontrivial solution will give us a nontrivial relation of linear dependence on $S$. So $S$ is linearly dependent (Definition LI).