Where is the difference between the union and sum of sets? My book writes: 

Definition. A vector space $V$ is called the direct sum of $W_1$ and $W_2$ if $W_1$ and $W_2$ are subspaces of $V$ such that $W_1 \cap W_2=\{0\}$ and $W_1 + W_2 = V$. We denote that $V$ is the direct sum of $W_1$ and $W_2$ by writing $V=W_1\oplus W_2$.

I'm not sure what I should imagine $W_1 + W_2 = V$ as.
Thank you for any help !

Take this example to clarify the difference: $$V=\mathbb{R}^{2}$$ $$W_{1}=sp_{\mathbb{R}}\{(1,0)\}=\{(a,0)|a\in\mathbb{R}\}$$ $$W_{2}=sp_{\mathbb{R}}\{(0,1)\}=\{(0,b)|b\in\mathbb{R}\}$$
Then,
$$W_{1}+W_{2}=\{w_{1}+w_{2}|w_{i}\in W_{i}\}=\{(a,0)+(0,b)|a,b\in\mathbb{R}\}=\{(a,b)|a,b\in\mathbb{R}\}$$
but,
$$W_{1}\cup W_{2}=\{v_{1}|v_{1}\in W_{1}\}\cup\{v_{2}|v_{2}\in W_{2}\}$$
and this set is consistent of all elements of the form $(a,0)$ and
$(0,b)$ (where $a,b\in\mathbb{R})$ but, for example, $(1,1)\not\in W_{1}\cup W_{2}$.
