Answer
The set $S=\{(2,3),(6,9)\}$ is not a basis for $R^2$.
Work Step by Step
The set $S=\{(2,3),(6,9)\}$ is not a basis for $R^2$ because the vectors in $S$ are not linearly independent. Because one can see that $(6,9)=3(2,3)$, and hence we have the non trivial combination
$$3(2,3)-(6,9)=(0,0)$$
Which means that the vectors $(2,3),(6,9)$ are not Linearly independent. Hence the set $S=\{(2,3),(6,9)\}$ is not a basis for $R^2$ .
