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16 1. Preliminaries 19. Suppose the linear transformation T : V → V satisfies T 2 = I. Let U = {x ∈ V | T x = x} and W = {x ∈ V | T x = −x}. (a) Show that U and W are subspaces of V. (b) Show that for any x ∈ V, we have (x + T x) ∈ U and (x − T x) ∈ W. (c) Show that if the field F does not have characteristic 2, then V = U ⊕ W. (d) Show that for V = Z2 2 and the 2×2 matrix A = 1 1 0 1 , we have A2 = I. However, for this example, U = W, and the conclusion of part (c) does not hold. 20. Let A be a square matrix, and suppose A = S + T , where S is symmetric and T is skew-symmetric. Assuming the field does not have characteristic 2, show that S = A+AT 2 and T = A−AT 2 . 21. Let A be a square, complex matrix, and suppose A = H + iK, where H and K are Hermitian. Show that H = A+A∗ 2 and K = A−A∗ 2i . 22. Recall that we say a relation, ∼, is an equivalence relation on a set S if the following three properties hold: (a) For every a ∈ S, we have a ∼ a. (The relation is reflexive.) (b) If a ∼ b, then b ∼ a. (The relation is symmetric.) (c) If a ∼ b and b ∼ c, then a ∼ c. (The relation is transitive.) Show that similarity is an equivalence relation on the set Mn(F). 23. Let ∼ be an equivalence relation on a set S. For each a ∈ S, define the equivalence class of a to be [a] = {x ∈ S|x ∼ a}. Show that for any a, b ∈ S, either [a] = [b] or [a] ∩ [b] = ∅. Hence, each element of S belongs to exactly one equivalence class. We say the equivalence classes partition the set. 24. Verify formula (1.14).