GATE | GATE-CS-2014-(Set-3) | Question 15

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

Answer: (B)
Explanation: 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.
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