Proof of intersection and sum of vector spaces

lukaszh · #1 $Old$ January 12th, 2009, 01:38 PM

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

NonCommAlg · #2 $Old$ January 12th, 2009, 01:58 PM

Quote:

Originally Posted by lukaszh $View Post$

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 $<x,v+w>=<x,v>+<x,w>=0+0=0.$ 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 $(V+W)^{\bot} \subseteq V^{\bot}\cap W^{\bot}.$

lukaszh · #3 $Old$ January 12th, 2009, 03:25 PM

I dont understand this:

Quote:

Originally Posted by NonCommAlg $View Post$

... then $<x,v+w>=<x,v>+<x,w>=0+0=0.$ thus $V^{\bot}\cap W^{\bot} \subseteq (V+W)^{\bot}....$

How this $V^{\bot}\cap W^{\bot} \subseteq (V+W)^{\bot}$ emerges of $<x,v+w>=<x,v>+<x,w>=0+0=0$
Thanks

NonCommAlg · #4 $Old$ January 12th, 2009, 04:35 PM

Quote:

Originally Posted by lukaszh $View Post$

I dont understand this:

How this $V^{\bot}\cap W^{\bot} \subseteq (V+W)^{\bot}$ emerges of $<x,v+w>=<x,v>+<x,w>=0+0=0$
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. $<x,z>=0, \ \forall z \in V+W.$ so you choose an element of $z \in V+W.$ then $z=v+w,$ for some $v \in V, w \in W.$ we have $<x,z>=<x,v>+<x,w>=0,$ because

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

lukaszh · #5 $Old$ January 13th, 2009, 03:14 AM

Thank you, it's simple :-)