![GATE 2014 [Set-3]](https://solutionsadda.in/wp-content/uploads/2019/05/green-new-logo.png){kind=logo}
GATE 2014 [Set-3]
April 4, 2025GATE 2014 [Set-3]
April 4, 2025GATE 2014 [Set-3]
| Question 15 |
If V1 and V2 are 4-dimensional subspace of a 6-dimensional vector space V, then the smallest possible dimension of V1∩V2 is ______.
| 2 | |
| 3 | |
| 4 | |
| 5 |
Question 15 Explanation:
In a 6 dimensional vector space, sub space of 4 dimensional subspace V1, V2 are provided. Then the V1∩V2?
For eg: a two dimensional vector space have x, y axis. For dimensional vector space, it have x, y, z axis.
In the same manner, 6 dimensional vector space has x, y, z, p, q, r (assume).
Any subspace of it, with 4 dimensional subspace consists any 4 of the above. Then their intersection will be atmost 2.
[{x,y,z,p} ∩ {r,q,p,z}] = #2
V1 ∩ V2 = V1 + V2 – V1 ∪ V2 = 4 + 4 + (-6) = 2
For eg: a two dimensional vector space have x, y axis. For dimensional vector space, it have x, y, z axis.
In the same manner, 6 dimensional vector space has x, y, z, p, q, r (assume).
Any subspace of it, with 4 dimensional subspace consists any 4 of the above. Then their intersection will be atmost 2.
[{x,y,z,p} ∩ {r,q,p,z}] = #2
V1 ∩ V2 = V1 + V2 – V1 ∪ V2 = 4 + 4 + (-6) = 2
Correct Answer: A
Question 15 Explanation:
In a 6 dimensional vector space, sub space of 4 dimensional subspace V1, V2 are provided. Then the V1∩V2?
For eg: a two dimensional vector space have x, y axis. For dimensional vector space, it have x, y, z axis.
In the same manner, 6 dimensional vector space has x, y, z, p, q, r (assume).
Any subspace of it, with 4 dimensional subspace consists any 4 of the above. Then their intersection will be atmost 2.
[{x,y,z,p} ∩ {r,q,p,z}] = #2
V1 ∩ V2 = V1 + V2 – V1 ∪ V2 = 4 + 4 + (-6) = 2
For eg: a two dimensional vector space have x, y axis. For dimensional vector space, it have x, y, z axis.
In the same manner, 6 dimensional vector space has x, y, z, p, q, r (assume).
Any subspace of it, with 4 dimensional subspace consists any 4 of the above. Then their intersection will be atmost 2.
[{x,y,z,p} ∩ {r,q,p,z}] = #2
V1 ∩ V2 = V1 + V2 – V1 ∪ V2 = 4 + 4 + (-6) = 2