# Proof of intersection and sum of vector spaces

• Jan 12th 2009, 11:38 AM
lukaszh
Proof of intersection and sum of vector spaces
Hi,
how to prove this:
$V^{\bot}\cap W^{\bot}=(V+W)^{\bot}$
• Jan 12th 2009, 11:58 AM
NonCommAlg
Quote:

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 $(V+W)^{\bot} \subseteq V^{\bot}\cap W^{\bot}.$
• Jan 12th 2009, 01:25 PM
lukaszh
I dont understand this:
Quote:

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(Worried)
• Jan 12th 2009, 02:35 PM
NonCommAlg
Quote:

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(Worried)

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

$x \in V^{\bot}\cap W^{\bot}.$
• Jan 13th 2009, 01:14 AM
lukaszh
Thank you, it's simple :-)