Proving that the interesection of subspaces is a subspace

In summary, the conversation discusses proving that the intersection of two subspaces, U1 and U2, is also a subspace of the vector space V. The definition of a subspace is mentioned, along with the need to show that the intersection is closed under addition and scalar multiplication. The conversation concludes with a suggestion for a direct proof using arbitrary elements a and b from the intersection.
  • #1
Freye
28
0

Homework Statement



Let U1; U2 be subspaces of the vector space V . Prove that their intersection U1 \ U2 is
also a subspace of V

Homework Equations



I see how any equations could be used here

The Attempt at a Solution



Well intuitively this seems obvious from the get go. If U1 and U2 are subspaces, then their intersection, which can at most contain all of U1 if U1=U2, and at the very least the 0 vector if U1 and U2 share no common vectors other than the 0 vector. But I don't know how to prove this. It seems like from what I've said, I've neglected all of the intermediate possibilities.
 
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  • #2
What is the definition of a subspace? (Post it here)

Pick arbitrary elements in the intersection and show that all the properties of a subspace hold, using the premise.
 
  • #3
A subspace must be closed under addition and multiplication, use the same addition and scalar multiplication as it's parent vector space, and have the same additive identity as it's parent vector space.

But how can I pick arbitrary elements of the intersection if I don't even know what V is. It could be a vector space of functions, or of complex numbers, or probably of something else that I don't yet know. So how can I write out V so that it doesn't exclude any possibilities, yet I can work with it to prove the conditions for it's subspaces?
 
  • #4
Like this:

Let a and b be elements of U1 \ U2.

How do we know that a+b is in U1 \ U2?

Use the properties of the premise that U1 and U2 are subspaces of V.
 
  • #5
I'm sorry, I'm really not giving you much to go on.

Try a direct proof. It won't take more than 4 lines.

What does it mean that a is in the intersection of U1 and U2? Literally - what does that mean? Same for b. Now put that together to say something about a+b in relation to U1, and similarly with relation to U2.
 
  • #6
Ok thank you, actually that "let a and b be elemnts of U1\U2" was actually quite helpful
 
  • #7
No problem, glad to be of help.
 

Related to Proving that the interesection of subspaces is a subspace

1. What is a subspace?

A subspace is a subset of a vector space that satisfies the three properties of closure under addition, closure under scalar multiplication, and contains the zero vector.

2. How do you prove that the intersection of subspaces is also a subspace?

To prove that the intersection of subspaces is a subspace, you need to show that it satisfies the three properties of closure under addition, closure under scalar multiplication, and contains the zero vector. This can be done by showing that any two vectors in the intersection will also be in the subspace, and that any scalar multiple of a vector in the intersection will also be in the subspace.

3. Can the intersection of two subspaces be empty?

Yes, it is possible for the intersection of two subspaces to be empty. This would occur if the two subspaces do not have any common vectors.

4. Are there any special cases where the intersection of subspaces is not a subspace?

Yes, there are two special cases where the intersection of subspaces is not a subspace. The first is when one subspace is a subset of the other, in which case the intersection will be equal to the smaller subspace. The second is when the two subspaces are parallel, meaning they have the same direction but different magnitudes.

5. Can the intersection of more than two subspaces be a subspace?

Yes, the intersection of any number of subspaces can be a subspace as long as it satisfies the three properties of closure under addition, closure under scalar multiplication, and contains the zero vector. This means that the intersection of any finite number of subspaces will also be a subspace.

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