Can two subspaces have vectors in common

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SUMMARY

Two 4-dimensional subspaces of the 6-dimensional field F2 cannot have exactly 9 vectors in common. Given that F2 has 64 elements, the maximum number of vectors that can be shared is determined by the dimensionality of the subspaces. If two subspaces share any vectors, they must also share the subspace spanned by those vectors, which restricts the number of common vectors to a maximum of 8.

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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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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?
 
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)??
 

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