Linear Algebra: Subspace Proof

In summary, to prove that the intersection of any collection of subspaces of V is a subspace of V, it must be shown that for all elements in the intersection, their sum and scalar multiple are also in the intersection. This can be demonstrated by considering an arbitrary subspace W in the intersection and using its properties to show that u+v and cu are also in W, thus satisfying the conditions for a subspace.
  • #1
*melinda*
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0
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?
 
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  • #2
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"?
 
  • #3
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?
 
  • #4
Yes, that is one of the essential points of the proof.
 
  • #5
Thanks!
 

1. What is a subspace in linear algebra?

A subspace in linear algebra is a subset of the vector space that satisfies the three main properties: closure under addition, closure under scalar multiplication, and containing the zero vector. In simpler terms, a subspace is a smaller space within a larger vector space that still follows the basic rules of vector addition and scalar multiplication.

2. How do you prove that a set is a subspace?

To prove that a set is a subspace, you must show that it satisfies the three main properties: closure under addition, closure under scalar multiplication, and containing the zero vector. This can be done by showing that the set contains all possible combinations of vectors from the original vector space, and that the zero vector is also included in the set.

3. What is the difference between a subspace and a span?

A subspace is a subset of the vector space that satisfies the three main properties, while a span is the set of all possible linear combinations of a given set of vectors. In other words, a subspace is a subset of the original vector space, while a span is a set of vectors that can be used to create the subspace.

4. Can a subspace contain only one vector?

Yes, a subspace can contain only one vector. This vector would have to satisfy the three main properties of a subspace: closure under addition, closure under scalar multiplication, and containing the zero vector. In this case, the subspace would be a line passing through the origin in the vector space.

5. How does the dimension of a subspace relate to the dimension of the original vector space?

The dimension of a subspace is always equal to or less than the dimension of the original vector space. This is because a subspace is a smaller space within the larger vector space, and therefore cannot have more dimensions than the original space. However, it is possible for a subspace to have the same dimension as the original space.

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