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Let V(F) be a vector space of dimension 10 and let U and W are two distinct 7 -dimensional vector subspaces of V. Then dimensions of the subspace (U∩W) cannot be?
- a)4
- b)5
- c)6
- d)8
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Let V(F) be a vector space of dimension 10 and let U and W are two dis...
We have dim U = dim W = 7 and U and W are distinct
⇒ U + W subspace properly contains U and W.
⇒ dim(U + W) > 7
But also ( U + W ) be a subspace of V
⇒ dim(U + W) ≤ dim V = 10
⇒ (U + W) can have dimensions 8. 9 or 10.
Now dim (U ∩ W) = dim U + dim W - dim (U + W)
= 14 - dim(U + W)
⇒ U + W subspace properly contains U and W.
⇒ dim(U + W) > 7
But also ( U + W ) be a subspace of V
⇒ dim(U + W) ≤ dim V = 10
⇒ (U + W) can have dimensions 8. 9 or 10.
Now dim (U ∩ W) = dim U + dim W - dim (U + W)
= 14 - dim(U + W)
Most Upvoted Answer
Let V(F) be a vector space of dimension 10 and let U and W are two dis...
The dimensions of the subspace (U ∩ W) can range from 0 to 6.
The dimension of the subspace (U + W) can range from 7 to 13.
To see why, note that the dimension of the intersection (U ∩ W) can be at most 7, since both U and W are 7-dimensional subspaces. If the intersection is non-zero, then U and W are not distinct subspaces.
On the other hand, the dimension of the sum (U + W) can be at most 13, since the sum of two subspaces cannot have a dimension greater than the sum of the dimensions of the individual subspaces.
Therefore, the dimensions of the subspace (U ∩ W) can range from 0 to 6, and the dimensions of the subspace (U + W) can range from 7 to 13.
The dimension of the subspace (U + W) can range from 7 to 13.
To see why, note that the dimension of the intersection (U ∩ W) can be at most 7, since both U and W are 7-dimensional subspaces. If the intersection is non-zero, then U and W are not distinct subspaces.
On the other hand, the dimension of the sum (U + W) can be at most 13, since the sum of two subspaces cannot have a dimension greater than the sum of the dimensions of the individual subspaces.
Therefore, the dimensions of the subspace (U ∩ W) can range from 0 to 6, and the dimensions of the subspace (U + W) can range from 7 to 13.
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Let V(F) be a vector space of dimension 10 and let U and W are two dis...
We have dim U = dim W = 7 and U and W are distinct
⇒ U + W subspace properly contains U and W.
⇒ dim(U + W) > 7
But also ( U + W ) be a subspace of V
⇒ dim(U + W) ≤ dim V = 10
⇒ (U + W) can have dimensions 8. 9 or 10.
Now dim (U ∩ W) = dim U + dim W - dim (U + W)
= 14 - dim(U + W)
⇒ U + W subspace properly contains U and W.
⇒ dim(U + W) > 7
But also ( U + W ) be a subspace of V
⇒ dim(U + W) ≤ dim V = 10
⇒ (U + W) can have dimensions 8. 9 or 10.
Now dim (U ∩ W) = dim U + dim W - dim (U + W)
= 14 - dim(U + W)
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