# 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 !

• It should be $W_1\cap W_2$ instead of $W_1\cup W_2$. 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 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,

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

• thank you ! would you be so kind to tell me what you mean by $sp_{\mathbb{E}}$ though? :P Sep 27 '12 at 10:58
• @foaly - sorry, it was a typo. is it clear now ? 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) ? Sep 27 '12 at 11:02
• it means span. do you know what this means ? 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? Sep 27 '12 at 11:06

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

• Elegant and simple. Jan 7 '18 at 15:06