Can two subspaces have vectors in common

In summary, F62 is a 6-dimensional field where each element is an element of F2. The question is whether two 4-dimensional subspaces in F62 can have exactly 9 or 8 vectors in common. Based on the understanding that if two subspaces have any vectors in common, they also share the subspace spanned by those vectors, the answer is that having exactly 9 or 8 vectors in common is not possible. This is because a one-dimensional subspace, which contains just one vector, cannot have exactly 9 or 8 vectors in common with another subspace.
  • #1
brru25
29
0

Homework Statement



Can two 4-dimensional subspaces of F62 have exactly 9 vectors in common? Can they have exactly 8 vectors in common?

F62 is the 6-dimensional field where each (a1, a2, a3, a4, a5, a6) is an element of F2.

The Attempt at a Solution



F62 obviously has 26 = 64 elements. I want to say that if each of the 4 vectors in both subspaces are the same, then there can be up to that are also the same, which would mean that 9 isn't possible, but I don't think that's right at all.

Thank you ahead of time for your help.
 
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  • #2
Well, if two subspace has any vectors in common, then they have the subspace spanned by those vector in common. How many vectors does a one-dimensional subspace contain?
 
  • #3
HallsofIvy said:
Well, if two subspace has any vectors in common, then they have the subspace spanned by those vector in common. How many vectors does a one-dimensional subspace contain?

Just one (I would think)??
 

FAQ: Can two subspaces have vectors in common

Can two subspaces have vectors in common?

Yes, it is possible for two subspaces to have vectors in common. This can happen when the two subspaces are not mutually exclusive, meaning they both contain some of the same vectors.

How can two subspaces have vectors in common?

Two subspaces can have vectors in common if they share the same basis, or if one subspace is a subset of the other. This means that the vectors in the shared subspace will be linear combinations of the vectors in the other subspace.

Is it possible for two subspaces to have no vectors in common?

Yes, it is possible for two subspaces to have no vectors in common. This can happen if the two subspaces are mutually exclusive, meaning they do not share any of the same vectors.

Can two subspaces with no vectors in common still intersect?

No, if two subspaces have no vectors in common, they cannot intersect. Intersection between subspaces means that they share at least one vector, which is not the case if they have no vectors in common.

Can two subspaces with no vectors in common still be considered as subspaces of the same vector space?

Yes, two subspaces with no vectors in common can still be considered as subspaces of the same vector space. This is because subspaces are defined by their properties and not by the specific vectors they contain. As long as they satisfy the requirements of a subspace, they can be considered as such.

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