Subspace of $R^3$ If $$ W = \\{(a, b, c) : a+b+c=0; a, b,c \in R\\}$$ is a subspace of $\mathbb{R}^3$ then what would be the dimensions of $W$?. I believe that this signifies a complete space $R^3$ also containing the origin but my teacher says that it must be 2-D subspace of $R^3$. I don't get it why its dimension is 2. Can anybody help me apprehend intuitively?

Notice that

$$u=(a,b,c)\in W\iff u=(a,b,-a-b)=a(1,0,-1)+b(0,1,-1)\in S$$ where

$$S=\operatorname{Span}((1,0,-1),(0,1,-1))$$ so $W=S$ which is a subspace of $\Bbb R^3$ of dimension $2$.

Notice also that using the linear form $\varphi:\Bbb R^3\to\Bbb R,(a,b,c)\mapsto a+b+c$, we see that $W$ is nothing but the kernel of $\varphi$ and its dimension is 2 due to the rank-nullity theorem.