Linear subspaces

Orthogonal complements

SOLVED: Problem I: Orthogonal Complement of a Subspace: Suppose that V is a subspace of R^n. Let VL = x ∈ R^n : x · y = 0, ∀y ∈ V be the set of vectors orthogonal to every element in V. (a) Verify that VL is a subspace of R^n. (b) Suppose that V = span(V1, V2, ..., Vk) for some V1, V2, ..., Vk ∈ R^n. Express V and VL as subspaces induced by the matrix A = [V1 V2 ... Vk] and its transpose A^T. (c) Show that (VL)^L = V. (d) Show that dim(V) dim(VL) = n. (e) Show that V ⊆ W for another