If V1 and V2 are 4-dimensional subspaces of a 6-dimensional vector space V, then the smallest possible dimension of V1 ∩ V2 is _____.

GATE 2014 · Engineering Mathematics · Vector Space · medium

Answer: 2

1. Apply the subspace dimension formula: dim(V1 ∩ V2) = 4 + 4 - dim(V1+V2) = 8 - dim(V1+V2).
2. Apply the upper bound on V1+V2: The maximum possible value of dim(V1+V2) is 6 (when V1+V2 = V). This is achievable: let V1 = span{e1,e2,e3,e4} and V2 = span{e3,e4,e5,e6} in R^6; then V1+V2 = R^6.
3. Compute minimum intersection dimension: When dim(V1+V2) = 6, dim(V1 ∩ V2) = 8-6 = 2. With the example above, V1 ∩ V2 = span{e3,e4}, which indeed has dimension 2.