課程名稱︰線性代數二 課程性質︰數學系大一必修 課程教師︰余正道 開課學院：理學院 開課系所︰數學系 考試日期︰2020年04月24日(五) 考試時限：10:00-12:15，共130分鐘 試題 : 1. [20%] Let V = {continuous function [0, 1] → R} with the pairing 1 k (f | g) = ∫ f(t)g(t)dt. Let h (t) = t for k≧0. 0 k 1 (a) Show that ( | ) defines an inner product on V and (h | h ) = -------. p q p+q+1 (b) Apply the Gram-Schmidt process to the independent subset {h1,h2,h3,h4} of V to obtain an orthogonal subset. 1 (c) Consider the n ×n matrix H = (-------) for any positive n p+q+1 1≦i,j≦n integer n. Prove that det(Hn) > 0. (d) Let W = {f∈V|f(1/2 + x) = f(1/2 - x), 0≦x≦1/2}. Find the orthogonal complement of W. Justify your answer. 2. Let V be an inner product space and W⊂V a finite dimensional subspace with ⊥ the orthogonal complement W. ⊥ (a) [5%] Show that V = W⊕W. (b) [10%] Let P:V→W be the orthogonal projection associated with the decomposition in (a). Show that for v∈V, Pv is the best approximation of v in W (i.e., Pv is the unique vector among w∈W such that the length ∥v-w∥ is minimal. 2 (c) [10%] Suppose E∈L(V) is a projection (i.e., E = E) with image W. Suppose ∥Ev∥≦∥v∥ for all v∈V. Prove that E = P defined in (b). 3. Let V be a finite dimensional inner product space. ⊥ (a) [7%] Let W⊂V be a subspace. Then V = W ⊕ W and every v∈V decomposes ⊥ uniquely as v = v' + v'' with v'∈W, v''∈W. Define a map f ∈L(V) by W f(v) = v' - v''. Show that f is both self-adjoint and unitary. W W (b) [8%] Suppose f∈L(V) is self-adjoint and unitary. Show that f = f for W some W as in (a). 4. [15%] Let A,B∈M(C). Assume that A,B are normal and AB = BA. Show that there n * * exists a unitary P∈M(C) such that P AP and P BP are both diagonal. n 5. [10%] Let f be a non-degenerate symmetric bilinear form on a finite ⊥ dimensional V. For a subspace W, let W = {v∈V | f(v,w) = 0 for all w∈W}. ⊥ Show that dim W + dim W = dim V. 6. Let f be a non-degenerate symmetric bilinear form on a finite dimensional V. (a) [8%] Show that for any basis B = {v1,...,vn} of V there exists a unique basis B' = {v1',...,vn'} such that f(vi,vj') = δ . ij (b) [7%] Suppose V is complex. Show that there exists a basis B such that B' = B where B' is defined in (a). 7. [20%] Let f be a non-degenerate symmetric bilinear form on a finite dimensional V. A subspace U is called isotropic if f(u,u') = 0 for all u,u'∈U. (a) Suppose U is isotropic subspace of dimension r. Show that there exists a basis {u1,...,ur} of U and an isotropic subspace W with a basis {w1,...,wr} such that f(ui,wj) = δ . ij (b) Suppose V is real, f has signature (p,q). Show that any isotropic U has dim U ≦ min{p,q} and there exists an isotropic U with dim U = min{p,q}. --

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