Answer
The set $S=\{(0,0,0),(1,0,0),(0,1,0)\}$ is not a basis for $R^3$.
Work Step by Step
The set $S=\{(0,0,0),(1,0,0),(0,1,0)\}$ is not a basis for $R^3$ because the vectors in $S$ are not linearly independent. Indeed, one can see that the following linear combination $$a(0,0,0)+b(1,0,0)+c(0,1,0)=(0,0,0), \quad a,b,c\in R$$ is valid for any non-zero value of $a$ and $b=c=0$, hence the vectors of $S$ are not linearly independent. Hence, the set $S=\{(0,0,0),(1,0,0),(0,1,0)\}$ is not a basis for $R^3$.
