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Determine intersection dimension of two 4D subspaces

Capital OneData Scientist · Technical Screen
MediumData ScientistTechnical ScreenStatistics & Math

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This question tests the application of the Grassmann Dimension Formula to determine the possible dimensions of the intersection of two subspaces within a larger vector space.

Problem Statement

Let V be a 7-dimensional real vector space. Suppose you have two subspaces, U and W, both of which are 4-dimensional. We are interested in the dimension of their intersection, dim(U ∩ W).

  1. Use the Grassmann Dimension Formula to determine the range of all possible values for dim(U ∩ W).
  2. Identify which values in the set {0, 1, 2, 3, 4} are theoretically impossible for this intersection.
  3. For each possible dimension, provide a brief description of how you would construct basis vectors for U and W to achieve that intersection dimension.
  4. Provide a formal proof for why any dimensions identified as "impossible" cannot occur.

Definitions

  • The Grassmann Dimension Formula states: dim(U + W) = dim(U) + dim(W) - dim(U ∩ W).
  • Recall that for any two subspaces U and W of a vector space V, dim(U + W) <= dim(V).

Hints

Hint 1

Recall the Grassmann Dimension Formula: dim(U + W) = dim(U) + dim(W) - dim(U ∩ W).

Hint 2

Since U + W is a subspace of V, its dimension cannot exceed the dimension of the ambient space V (which is 7).

Hint 3

Use the constraint that dim(U + W) must be at least the dimension of the largest individual subspace (4) and at most the total dimension of the ambient space (7).

Follow-up questions

  • How does the result change if V were only 6-dimensional?
  • If we choose two 4-dimensional subspaces 'randomly' in a 7-dimensional space, what is the most likely dimension of their intersection?
  • How would you represent these subspaces computationally using matrices, and what matrix operation would you perform to find the intersection dimension?