Math 215 HW #6 Solutions

2. Problem 3.1.20. Let S be a subspace of Rn. Explain what (S⊥)⊥ = S means and why it is true. Answer: First, (S⊥)⊥ is the orthogonal complement of S⊥, which is itself the orthogonal complement of S, so (S⊥)⊥ = S means that S is the orthogonal complement of its orthogonal complement. To show that it is true, we want to show that S is contained in (S⊥)⊥ and, conversely, that (S⊥)⊥ is contained in S; if we can show both containments, then the only possible conclusion is that (S⊥)⊥ = S. To show the first containment, suppose v ∈ S and w ∈ S⊥. Then