Orthogonal Projections AND Orthogonal Complements PDF

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ORTHOGONAL PROJECTIONS AND ORTHOGONAL COMPLEMENTS

  In every question below, it is assumed that V, h·, ·i is a vector space with scalar product and W ∈ V is a subspace. 1) Define what it means that the vector y ∈ W is an orthogonal projection of a vector x ∈ V onto the subspace W .   2) Let B = u1 , . . . , un ⊆ W be a basis in W . Let w ∈ V . Prove that if w ⊥ uj , j = 1, . . . , n, then w ⊥ z for every z ∈ W . 3) State the formula for the projection of a vector x ∈ V onto the subspace W = span {u}, where u ∈ V . 4) Let V = Rn , n > 2 and let W = span B, where B = {u1 , u2 } and let B be orthonormal. Let A ∈ Mn,2 (R) such that A = (u1 |u2 ). State the formula for the projection of a vector x ∈ V onto the subspace W . 5) Let V = Rn and let W = col A, where A ∈ Mnm (n > m). Assume that AT A = Im (such matrices are called partial isometries). State the formula for the projection of a vector x ∈ V onto the subspace W . 6) Let PW : V → V be a projection onto a subspace W and let A ∈ Mnn (R) be the matrix of the linear map PW with respect to a basis B = {u1 , u2 , . . . , un } in V . Is it true that A2 = A? Give explanations to your answer. 7) Let S ⊆ V . Define the orthogonal complement S ⊥ . 8) Find ∅⊥ . Here, ∅ is empty set. Use the definition of orthogonal complement to explain your answer. 9) Find V ⊥ . Use the definition of orthogonal complement to explain your answer. 10) Use the Subspace Theorem to prove that the orthogonal complement S ⊥ is a subspace. 11) a) State the Double Orthogonal Complement Theorem. b) Let V = R2 and let S = {e1 }, where e1 is the first standard basis vector. i) Find W = S ⊥ . Give explanations. ii) Find basis in W . iii) Find W ⊥ = S ⊥⊥ . Give explanations. iv) Is it true that S ⊥⊥ = S ? v) Does it contradict to the Double Orthogonal Complement Theorem? Give explanations. 12) Let x ∈ W and x ∈ W ⊥ . Prove that x = 0.

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