Linear Algebra: Subspace Proof

[*melinda*]

Prove that the intersection of any collection of subspaces of V is a subspace of V.

Ok, I know I need to show that:

1. For all u and v in the intersection, it must imply that u+v is in the intersection, and

2. For all u in the intersection and c in some field, cu must be in the intersection.

I can show both 1. and 2. for the trivial case where the intersection is zero, but I'm not sure what I need to do for the arbitrary case.

Any suggestions?

 

[Hurkyl]

Well, I'm not sure where you're stuck, so I'll throw out two random hints.


(a) Let W be one of the spaces in the intersection...

(b) What is the definition of "x is in the intersection"?

 

[*melinda*]

Since u and v are elements of the intersection, u and v will also be elements of any subspace W that is in the intersection. And since u and v are in W and W is a subspace, this guarantees that u+v will also be in W. This same argument would apply to scalar multiplication.

Is that the right idea?

 

[Hurkyl]

Yes, that is one of the essential points of the proof.

 

[*melinda*]

Thanks!