# 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 !

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It should be $W_1\cap W_2$ instead of $W_1\cup W_2$. – joriki Sep 27 '12 at 10:46
A free book by Jim Hefferon on Linear Algebra at joshua.smcvt.edu/linearalgebra Page 129 has a good explanation – Vikram Sep 27 '12 at 11:13

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

$$W_{1}\cup W_{2}=\{v_{1}|v_{1}\in W_{1}\}\cup\{v_{2}|\in W_{2}\}$$ 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, $(1,1)\not\in W_{1}\cup W_{2}$

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thank you ! would you be so kind to tell me what you mean by $sp_{\mathbb{E}}$ though? :P – foaly Sep 27 '12 at 10:58
@foaly - sorry, it was a typo. is it clear now ? – Belgi Sep 27 '12 at 10:59
no.. I don't know what that $sp$ think means :P probably some kind of notation not known to me (yet) ? – foaly Sep 27 '12 at 11:02
it means span. do you know what this means ? – Belgi Sep 27 '12 at 11:05
oh ya. the notations i know are $\\span(1,0)$ and $<1,0>$ (or similar to that). Why do you write the index $\mathbb{R}$ to it? – foaly Sep 27 '12 at 11:06

$$W_1+W_2=\{w_1+w_2\mid w_1\in W_1\land w_2\in W_2\}\;.$$

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