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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?
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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...
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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Community 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 distinct 7 -dimensional vector subspaces of V. Then dimensions of the subspace (U∩W) cannot be?a)4b)5c)6d)8Correct answer is option 'D'. Can you explain this answer?
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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)4b)5c)6d)8Correct answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about 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)4b)5c)6d)8Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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)4b)5c)6d)8Correct answer is option 'D'. Can you explain this answer?.
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)4b)5c)6d)8Correct answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about 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)4b)5c)6d)8Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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)4b)5c)6d)8Correct answer is option 'D'. Can you explain this answer?.
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