TITLE: Adding two subspaces
QUESTION [6 upvotes]: I have two subspaces: 
$$W_1 = \{(x, 3x) : x\in \Bbb R \}$$ and 
$$W_2 = \{(2x, 0): x\in \Bbb R \}$$
How do I get $W_1 + W_2$? 
I tried simply adding a sample vector from each, i.e. $$ (1, 3) + (2, 0) = (3, 3)$$ but I don't think this makes sense since this new vector doesn't fit it $W_1$ nor $W_2$....

REPLY [3 votes]: Carefully note that for any two sets (not only for subspaces) $S$ & $T$, $S+T=${$s+t:s\in S, t\in T$}. Thus your sample vector viz $(3,3)$ is just a single element of $W_1+W_2$. You need to accommodate all such in $W_1+W_2$. Thus what should be the general form of a vector in $W_1+W_2$? Isn't it $(x,3x)+(2y,0)$ for $x,y\in \mathbb R$?