Is the Intersection of Subspaces a Subspace in a Vector Space?

In summary, the question is asking to prove that the intersection of two subspaces in a vector space is also a subspace. The question also states that the vector space is over GF(2). The first step in solving this question is to understand the definition of a subspace and show that the intersection satisfies the three rules of a subspace. This is a common theorem and can also be found in any textbook. It is important to show your work and attempt the question before seeking help.
  • #1
zhenghao1
2
0
Hi to all,

I really need help fast.

How do I solve this question? A solution would be much appreciated. THANKS A MILLION!

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Let S1 and S2 be the two subspaces in a vector space V. Show that the intersection of S1 and S2 is also a subspace in V.
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  • #2
You need to show some work. What is the definition of a subspace?
 
  • #3
Question

Ok sorry, there was one piece of information that I left out.

The question is:

Let S1 and S2 be the two subspaces in a vector space V. Show that the intersection of S1 and S2 is also a subspace in V.

Assume vector space over GF(2).

Can someone please help me? Thanks!
 
  • #4
That doesn't change anything. You need to show us what you have attempted already? What is the definition of a subspace?
 
  • #5
Let a, b any two elements in the intersection and show that they satisfy the 3 rules of subspace. Any textbook should also have the proof already because this is a common theorem.
 

FAQ: Is the Intersection of Subspaces a Subspace in a Vector Space?

What is the definition of "intersection of subspaces" in linear algebra?

The intersection of subspaces in linear algebra refers to the set of all elements that are common to two or more subspaces. It is denoted by ∩ and can also be thought of as the largest subspace that is contained in all of the given subspaces.

How is the intersection of subspaces calculated?

The intersection of subspaces can be calculated by finding the common elements or vectors between the given subspaces. This can be done by solving a system of linear equations, where the equations represent the conditions for the vectors to be in both subspaces.

What is the significance of the intersection of subspaces?

The intersection of subspaces is significant in linear algebra as it helps in understanding the relationships between different subspaces. It also plays a crucial role in solving systems of linear equations and determining the dimension of a vector space.

Can the intersection of subspaces be empty?

Yes, the intersection of subspaces can be empty. This means that there are no common elements or vectors between the given subspaces. It is also possible for the intersection of subspaces to be a single element or a subspace of lower dimension.

How does the intersection of subspaces relate to linear independence?

If the intersection of subspaces is non-empty, then the vectors in the intersection are linearly dependent. This is because they can be expressed as a linear combination of the same set of vectors. On the other hand, if the intersection is empty, then the vectors are linearly independent.

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