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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 !

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    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

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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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    okay. thanks :)2012-09-27
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$W_1+W_2=\{w_1+w_2\mid w_1\in W_1\land w_2\in W_2\}\;.$

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    Elegant and simple.2018-01-07