Dear All,
I hope you can help me solving the following problem.
Given several subspaces of a linear -dimensional vector space, each subspace being determined by a set of linearly independent vectors, find a set of vectors spanning the intersection of the subspaces.
It suffices to solve the problem just for two subspaces, as the rest can be done iteratively. Is there any simple method for determining the dimension of the intersection in question without finding the vectors?
Thank you for any hint.
I meant that the dimension of any subspace in the form , where the sum is running over any subset of subspace indices is known. This can be easily performed by computing the rank of the sum (union).
Your expression for 3 subspaces implies that one really needs to know , so there is no possibility to avoid the solution of the main problem. I see no other way to determine .
Concerning the main problem it can be of course easily solved in the 3D case. Consider two subspaces of dimension 2 (if one of the subspaces has a smaller dimension the case is trivial). Than either the subspaces are equivalent and the intersection can be represented by any pair or the intersection has dimension 1 and is represented by the vector
.
Can it be somehow generalized to higher dimensions? I can imagine the following. Let be an operation, which produces vectors, orthogonal to the vectors of . Than the problem were solved by:
But what is the explicit construction of the operation ? I would guess that the Gram-Schmidt orthogonalization should work... Any other suggestions?