Example of two subspaces $W_1$ and $W_2$ of $ V$ such that $W_1∪W_2$ is also a subspace of $V$

Question

So I know the obvious counter example would be to let:

$W_1 = \{(a, 0) | a \in\mathbb{R}\}$ and $W_2 = (0, 0)$. Where $W_1 + W_2 = (a, 0)$ which is an element of $W_1\cup W_2$. But if I wanted $W_1 = \{(a, 0) | a \in\mathbb{R}\}$ and $W_2 = \{(b, 0) | b \in\mathbb{R}\}$ then $W_1 + W_2 = (a+b, 0)$, would that belong to the union of $W_1$ and $W_2$ ? or no?

Also, I think my understanding of this would show that it has to be stable by addition and multiplication so, for example, if $W_1 = \{(a, 0) | a \in\mathbb{R}\}$ and $W_2 = \{(2a, 0) | a \in\mathbb{R}\}$ then $W_1 + W_2 = (3a, 0)$ that would belong to the union because it's like a scalar that is 3?

I feel completely confused as to if the last 2 examples I gave belong to the union or not.

Answer 1

The union of two subspaces is a subspace if and only if one is contained in the other. This is the case for example with your example $W_1=\{\,(a,0)\mid a\in \mathbb R\,\}$, $W_2=\{\,(b,0)\mid b\in \mathbb R\,\}$ because in fact $W_1=W_2$ (the different names of "dummy variables" does not make the sets different).

Note however that $\{\,(2a,0)\mid a\in \mathbb R\,\} + \{\,(a,0)\mid a\in \mathbb R\,\}$ is not $\{\,(3a,0)\mid a\in \mathbb R\,\}$ (well, actually it is, but not for the reason you are thinking of; if we replace $\mathbb R$ with a field where $3=0$, the distinction becomes more relevant).

Answer 2

If you want the union to be a subspace, given any elements $u,v\in W_1\cup W_2$ then $u\in W_1$ or $u\in W_2$ and the same for $v$. The problem arises when $u$ and $v$ are elements of different subspaces (if they are in the same subspace their sum would also be in the union).

Assume $u\in W_1$ and $v\in W_2$ then, for the union to be a subspace, $u+v$ must be an element of either $W_1$ or $W_2$, i.e. $ u+ v\in W_i$ with $i=1$ or $2$. This implies $u$ and $v$ are both elements of the same subspace. So, as @Hagen von Eitzen pointed out, one subspace must be contained in the other.