# Thread: Proof of intersection and sum of vector spaces

1. ## Proof of intersection and sum of vector spaces

Hi,
how to prove this:
$V^{\bot}\cap W^{\bot}=(V+W)^{\bot}$

2. Originally Posted by lukaszh
Hi,
how to prove this:
$V^{\bot}\cap W^{\bot}=(V+W)^{\bot}$
if $x \in V^{\bot}\cap W^{\bot},$ and $v \in V, \ w \in W,$ then $=+=0+0=0.$ thus $V^{\bot}\cap W^{\bot} \subseteq (V+W)^{\bot}.$

conversely, we have $V \subseteq V+W,$ and hence $(V+W)^{\bot} \subseteq V^{\bot}.$ similarly $(V+W)^{\bot} \subseteq W^{\bot}.$ thus $(V+W)^{\bot} \subseteq V^{\bot}\cap W^{\bot}.$

3. I dont understand this:
Originally Posted by NonCommAlg
... then $=+=0+0=0.$ thus $V^{\bot}\cap W^{\bot} \subseteq (V+W)^{\bot}....$
How this $V^{\bot}\cap W^{\bot} \subseteq (V+W)^{\bot}$ emerges of $=+=0+0=0$
Thanks

4. Originally Posted by lukaszh
I dont understand this:

How this $V^{\bot}\cap W^{\bot} \subseteq (V+W)^{\bot}$ emerges of $=+=0+0=0$
Thanks
in order to prove that $V^{\bot}\cap W^{\bot} \subseteq (V+W)^{\bot},$ you need to show that if $x \in V^{\bot}\cap W^{\bot},$ then $x \in (V+W)^{\bot}.$ to prove that $x \in (V+W)^{\bot},$ you need to show that $x$ is orthogonal to every

element of $V+W,$ i.e. $=0, \ \forall z \in V+W.$ so you choose an element of $z \in V+W.$ then $z=v+w,$ for some $v \in V, w \in W.$ we have $=+=0,$ because

$x \in V^{\bot}\cap W^{\bot}.$

5. Thank you, it's simple :-)