Question

If S₁ and S₂ are subspaces of Rⁿ with the same dimension, then.. Please continue with your question or any statements or reasoning you'd like to provide regarding S₁ and S₂ in this context.

Answer 1

If S₁ and S₂ are subspaces of Rⁿ with the same dimension, then the intersection of S₁ and S₂ will also have the same dimension.

Step-by-step explanation:

If S₁ and S₂ are subspaces of Rⁿ with the same dimension, then the intersection of S₁ and S₂ will also have the same dimension as S₁ and S₂. In other words, if both subspaces have dimension k, the intersection will also have dimension k. This is because the dimension of a subspace is the number of linearly independent vectors that span the subspace. Since the intersection of two subspaces is a subset of both subspaces, it cannot have more linearly independent vectors than the subspaces themselves, and therefore will have the same dimension.