# Thread: Proof of intersection and sum of vector spaces

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

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

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

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

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

4. Originally Posted by lukaszh
I dont understand this:

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

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

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

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