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Proof, intersection and sum of vector spaces 
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#1
Jan1109, 01:36 PM

P: 32

Hello,
how to prove this: [tex]V^{\bot}\cap W^{\bot}=(V+W)^{\bot}[/tex] Thanks 


#2
Jan1109, 02:58 PM

Sci Advisor
PF Gold
P: 1,776

It is a "simple" matter of proving an element in the lefthand side is in the righthand side and vis versa by parsing the definitions. But you'll learn little by seeing it done. You need to go through the steps of discovering the tricky details and resolving them so you appreciate the implications.



#3
Jan1109, 04:00 PM

P: 32

Could you show me, how to do it?



#4
Jan1109, 07:15 PM

Sci Advisor
PF Gold
P: 1,776

Proof, intersection and sum of vector spaces
I'll start you by pointing out that if a vector [itex]v[/itex] is in the subspace [itex]U^\perp[/itex] then it must be perpendicular to all elements of the subspace [itex]U[/itex]. 


#5
Jan1209, 04:43 AM

P: 32

I know this:
[tex]\left(v\in V^{\bot}\wedge v\in W^{\bot}\right)\Rightarrow\left(v\in V^{\bot}\cap W^{\bot}\right)[/tex] [tex]\left(x\in V\wedge x\in W\right)\Rightarrow\left(x\in V\cap W\right)[/tex] I can also write that [tex]v^Tx=0\,;\; x\in V, v\in V^{\bot}[/tex] [tex]w^Ty=0\,;\; y\in W, w\in W^{\bot}[/tex] 


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