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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 !

  • 1
    It should be $W_1\cap W_2$ instead of $W_1\cup W_2$.2012-09-27
  • 0
    A free book by Jim Hefferon on Linear Algebra at http://joshua.smcvt.edu/linearalgebra/ Page 129 has a good explanation2012-09-27

2 Answers 2

6

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

  • 0
    thank you ! would you be so kind to tell me what you mean by $sp_{\mathbb{E}}$ though? :P2012-09-27
  • 0
    @foaly - sorry, it was a typo. is it clear now ?2012-09-27
  • 0
    no.. I don't know what that $sp$ think means :P probably some kind of notation not known to me (yet) ?2012-09-27
  • 0
    it means span. do you know what this means ?2012-09-27
  • 0
    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?2012-09-27
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    to say where the scalars can be taken from. when this is clear people omit the field in the notation2012-09-27
  • 0
    okay. thanks :)2012-09-27
7

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

  • 0
    Elegant and simple.2018-01-07