Mathematics Exam > Mathematics Questions > Let V(F) be a vector space of dimension 10 an...
Start Learning for Free
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?
| FREE This question is part of | Download PDF Attempt this Test |
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)
Free Test
FREE
| Start Free Test |
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.
|
Explore Courses for Mathematics exam
|
|
Similar Mathematics Doubts
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?
Question Description
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?.
Solutions 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? in English & in Hindi are available as part of our courses for Mathematics.
Download more important topics, notes, lectures and mock test series for Mathematics Exam by signing up for free.
Here you can find the meaning of 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? defined & explained in the simplest way possible. Besides giving the explanation of
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?, a detailed solution 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? has been provided alongside types of 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? theory, EduRev gives you an
ample number of questions to practice 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? tests, examples and also practice Mathematics tests.
|
|
Explore Courses for Mathematics exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup