How to solve the following vector space problem?

Question

I am stuck with the following problem :

Let $S=\{(x,y,z)\in \Bbb R^3: 2x+y-z=0\}$. Show that the basis for this sub space is $(0,1,1)$.

Clearly, $(0,1,1)\in S$. To prove that certain set of vectors form a basis of $S$, we have to prove they are linearly independent and they span S.

Can someone elaborate and explain in detail to tackle this problem?

Answer 1

For any $(x,y,z)\in S$, we can write $z=2x+y$, so all elements of $S$ are of the form $$(x,y,2x+y)=x\cdot(1,0,2)+y\cdot(0,1,1).$$ Therefore, $\{(1,0,2), (0,1,1)\}$ is a basis for $S$.