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If V1 and V2 are 4-dimensional subspaces of a 6-dimensional vector space V, then the smallest possible dimension of V1 ∩ V2 is ______.
- a)1
- b)2
- c)3
- d)4
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
If V1 and V2 are 4-dimensional subspaces of a 6-dimensional vector spa...
First, note that V1+V2 is still in V, so dim(V1+V2)≤ 6. We know that dim(V1+V2)=dimV1+dimV2−dim(V1∩V2).
So 6≥dim(V1+V2)=dimV1+dimV2−dim(V1∩V2) dim(V1∩V2)≥4+4−6=2. The answer is B.
So 6≥dim(V1+V2)=dimV1+dimV2−dim(V1∩V2) dim(V1∩V2)≥4+4−6=2. The answer is B.
Most Upvoted Answer
If V1 and V2 are 4-dimensional subspaces of a 6-dimensional vector spa...
To determine the smallest possible dimension of the subspace V1 ∩ V2, we need to consider the relationship between the dimensions of V1, V2, and V.
- Given that V is a 6-dimensional vector space, it means that the dimension of V is 6.
- Now, let's consider the possible scenarios for V1 and V2:
1. If dim(V1) = dim(V2) = 4:
In this case, the sum of the dimensions of V1 and V2 would be 4 + 4 = 8, which is greater than the dimension of V (6). Therefore, it is not possible for both V1 and V2 to have a dimension of 4.
2. If dim(V1) = 4 and dim(V2) = 3:
In this scenario, the sum of the dimensions of V1 and V2 would be 4 + 3 = 7, which is still greater than the dimension of V (6). So, this is also not possible.
3. If dim(V1) = 3 and dim(V2) = 4:
Now, the sum of the dimensions of V1 and V2 is 3 + 4 = 7. Since 7 is greater than 6 (the dimension of V), we can conclude that the intersection of V1 and V2 must have a dimension less than or equal to 6 - 1 = 5.
4. If dim(V1) = 2 and dim(V2) = 4:
In this case, the sum of the dimensions of V1 and V2 is 2 + 4 = 6, which is equal to the dimension of V. This means that the intersection of V1 and V2 can have a dimension of at most 6 - 2 = 4.
5. If dim(V1) = 1 and dim(V2) = 4:
Here, the sum of the dimensions of V1 and V2 is 1 + 4 = 5, which is less than the dimension of V (6). Therefore, the smallest possible dimension of V1 ∩ V2 is 5.
From the above analysis, we can see that option B, which states the smallest possible dimension is 2, is incorrect. The correct answer is option D, which states the smallest possible dimension is 4.
- Given that V is a 6-dimensional vector space, it means that the dimension of V is 6.
- Now, let's consider the possible scenarios for V1 and V2:
1. If dim(V1) = dim(V2) = 4:
In this case, the sum of the dimensions of V1 and V2 would be 4 + 4 = 8, which is greater than the dimension of V (6). Therefore, it is not possible for both V1 and V2 to have a dimension of 4.
2. If dim(V1) = 4 and dim(V2) = 3:
In this scenario, the sum of the dimensions of V1 and V2 would be 4 + 3 = 7, which is still greater than the dimension of V (6). So, this is also not possible.
3. If dim(V1) = 3 and dim(V2) = 4:
Now, the sum of the dimensions of V1 and V2 is 3 + 4 = 7. Since 7 is greater than 6 (the dimension of V), we can conclude that the intersection of V1 and V2 must have a dimension less than or equal to 6 - 1 = 5.
4. If dim(V1) = 2 and dim(V2) = 4:
In this case, the sum of the dimensions of V1 and V2 is 2 + 4 = 6, which is equal to the dimension of V. This means that the intersection of V1 and V2 can have a dimension of at most 6 - 2 = 4.
5. If dim(V1) = 1 and dim(V2) = 4:
Here, the sum of the dimensions of V1 and V2 is 1 + 4 = 5, which is less than the dimension of V (6). Therefore, the smallest possible dimension of V1 ∩ V2 is 5.
From the above analysis, we can see that option B, which states the smallest possible dimension is 2, is incorrect. The correct answer is option D, which states the smallest possible dimension is 4.
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