I know what a basis is and everything and I've done a lot of questions relating to this but I don't really get this part.
Q) Determine a basis for each of the following subspaces of R3.Give the dimension of each subspace.
b)the set of vectors of the form (a,b,c), where a+b+c = 0.
The basis is $\{(1,0,-1),(0,1,-1)\}$. This basis will generate the set $\{(a,b,c):a+b+c=0\}$
Deduction: Since, $a+b+c=0$ so, $c=-a-b$ and hence, $$(a,b,c)=(a,b,-a-b)=a(1,0,-1)+b(0,1,-1)\text{ }\forall a,b\in F\text{ , field of scalars.}$$