課程名稱︰線性代數二 課程性質︰數學系必修 課程教師︰朱樺 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2015/04/29 考試時限（分鐘）：10:30-12:40 試題 : 4 (1) (15%) Let T be a linear operator on V = C defined by T(a1,a2,a3,a4) = (4a1-2a2+a3+4a4, 6a1-2a2+6a4, -2a2-a3+4a4, -3a1+3a3+a4). (a) Find an ordered basis β for V such that [T] is a diagonal matrix. β (b) Can you find an orthonormal basis γ (under standard inner product) such that [T] is a diagonal matrix? γ m (c) Find lim A . m→∞ ┌ ┐ 5 4 3 2 (2) (10%) Let A = │ 0 1 0 │. Find A + 2A + 3A + 4A + 5A + 6I . │-2 -2 1 │ 3 │-1 0 1 │ └ ┘ (3) (15%) Which of the following transition matrices are regular? Explain your answer. ┌ ┐ ┌ ┐ A = │0.7 0 0.5 0 │, B = │0.7 0 1 0 │ │ 0 0.4 0 1 │ │0.3 0 0 0 │ │0.3 0 0.5 0 │ │ 0 0 0 1 │ │ 0 0.6 0 0 │ │ 0 1 0 0 │ └ ┘ └ ┘ (4) (15%) (a) Let V = M (C) with the inner product <A,B> = tr(B*A). Let P n ×n be an invertible matrix in V, and T be the linear operator on -1 P V defined by T (A) = P AP. Find the adjoint T* . P P 1 (b) Let V = P (R) with the inner product <f,g> = ∫ f(t)g(t)dt. Let 2 -1 D be the differentiation operator on V. Find the adjoint D*. t (5) (10%) Let A∈M (F). For any eigenvalue λ of A and A , let E and E' n ×n t denote the corresponding eigenspaces for A and A , respectively. Prove that dim(E) = dim(E'). (6) (10%) Let A, B ∈M (F). n ×n (a) Suppose that (I - AB) is invertible, prove that (I - BA) is invertible and n -1 -1 n (I - BA) = I + B(I - AB) A. n n n (b) Prove that AB and BA have the same eigenvalues in F. (7) (15%) Let T be a linear operator on an n-dimensional vector space, and suppose that T has n distinct eigenvalues. Prove that any linear operator which commutes with T is a polynomial in T. (8) (10%) (a) Let T be a linear operator on a vector space V, let v be a nonzero vector in V, and let W be the T-cyclic subspace of V generated by v. Prove that, for any w∈W, there exists a polynomial g(t) such that w=g(T)v. (b) Let T be a linear operator on V and suppose that V is a T-cyclic subspace of itself. Prove that, if U is a linear operator such that UT = TU, then U=g(T) for some polynomial g(t). (9) (10%) Let W be a finite-dimensional subspace of an inner product space V, and let T be the orthogoal projection of V on W. Prove that <Tx,y> = <x,Ty> for all x,y∈V. (10) (10%) Let T be a linear operator on a finite dimensional inner product space. Prove that N(T*T) = N(T), where N(T) is the null space of T. --

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