Proof, intersection and sum of vector spaces

[lukaszh]

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

 

[jambaugh]

It is a "simple" matter of proving an element in the left-hand side is in the right-hand 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.

 

[lukaszh]

Could you show me, how to do it?

 

[jambaugh]

Could you show me, how to do it?

Yes but I'd rather you show me some start first. I take it this is an assignment in studying linear algebra. The point of an assignment if for you to puzzle through the problem and thereby learn.

I'll start you by pointing out that if a vector $v$ is in the subspace $U^\perp$ then it must be perpendicular to all elements of the subspace $U$.

 

[lukaszh]

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