Answer
(a) $S$ spans $R^3$.
(b) $S$ is not linearly independent set of vectors.
(c) $S$ is not a basis for $R^3$.
Work Step by Step
Let $S$ be given by $$S=\{(1,0,0),(0,1,0),(0,0,1),(-1,2,-3)\}.$$
(a) For any $u=(x,y,z\in R^3$, one can write it as follows $$u=(x,y,z)=x(1,0,0)+y(0,1,0)+z(0,0,1)+0(-1,2,-3), \quad a,b,c\in R.$$ Then, $S$ spans $R^3$.
(b) Since one have the following combination $$(-1,2,-3)=-(1,0,0)+2(0,1,0)-3(0,0,1)$$ then $S$ is not linearly independent set of vectors.
(c) Since $S$ is not linearly independent set, then it is not a basis for $R^3$.
