## Elementary Linear Algebra 7th Edition

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