Is $S=\{(a+b,a+c,2c)\mid a,b,c\in \mathbb{R}\}$ a subspace of $\mathbb{R}^3$? 
Is $S=\{(a+b,a+c,2c)\mid a,b,c\in \mathbb{R}\}$ a subspace of $\mathbb{R}^3$?

I just made this question up to practice determining if sets are vector subspaces or not.
From what I can tell, the zero vector, $\vec{0}=(0,0,0)$ exists, when $a=b=c=0$ and is in $S$. Now to confirm if scalar multiplication and addition still hold and exist in $S$. $(a+b,a+c,2c)=a(1,1,0)+b(1,0,0)+c(0,1,2)$, which is a general linear combination of the vectors $(1,1,0),(1,0,0),(0,1,2)$. Since the sum of any such two linear combinations is again a linear combination and the same goes for scalar multiples, the set $S$ is a subspace of $\mathbb{R}^3$.
Is the above reasoning valid?
 A: yup, that's correct. actually you can deduce it's the whole $R^3$ because the vectors you wrote out form a basis
A: Another way in which you see the points @mm noted is to establish a $3\times 3$ matrix as follows: $$\begin{pmatrix}
  1 & 1 & 0\\
  1 & 0 & 0\\
  0 &1 &2\\
  \end{pmatrix}$$ Noe try to find the Reduced Row Echelon Form of it. It'll tell you that you have a basis here.
A: Just a comment about a few "issues" with your proof:
You are not checking whether $0$ exists. A subspace is a set which satisfies certain properties and is a subset of a vector space $V$. In your problem, $V = \mathbb{R}^3$ is a vector space so it contains zero and so zero exists. The question is whether $S$ contains the zero vector. So you should say it like this: $0 \in S$ as when $a = b = c = 0$ we have $(0 + 0, 0 + 0, 2\cdot 0) = 0$.
A similar issue with your wording with addition and scalar multiplication. What you're checking is whether the operations defined on $V$ are closed in $S$ (i.e. if $x$ and $y$ are in $S$ is $x + y$ and $cx$ still in $S$?). More precisely, what you're checking is whether the two binary operations defined on V are still binary operations when restricted to $S$. Mathematically I'm checking that if $\star: V \times V \rightarrow V$ is a binary operation on the vector space $V$ over the field $F$, then whether $\star_S : S \times S \rightarrow S$ is a binary operation on $S$ where $S \subset V$. So it's not really a matter of existence as you're not checking whether there exists a binary operation on $S$ but whether the ones defined on $V$ still work when restricted to $S$. 
I hope this helps!
