A and B are two subspaces contained in a finite vector space V and dimA = dimB Can we conclude A=B? In that subspaces A and B are really the same subspace and every element in one is in the other? I think yes because if dimA=dimB then their basis will contain the same number of vectors. These vectors span their respective subspaces. Given that each basis set contain an equal number of linearly indepedent vectors, they can be reduced to the standard vectors in V. Hence both spaces can be spanned by a single set of standard vectors in V so space A=B.