課程名稱︰線性代數二 課程性質︰數學系大一必修 課程教師︰陳其誠 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2011/03/25 考試時限（分鐘）：110分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : LINEAR ALGEBRA EXAM I 3/25 2011 Write your answer on the answer sheet. In this examination, C([-1,1]) denote the inner product space consisting of all real valued continuous functions on the close interval [-1,1] with the inner product defined by 1 <f,g> = ∫f(x)g(x)dx. -1 Also, P_k(F) denote the vector formed by all polynomials of degree not greater than k with coefficients in F. ┌1 -4 -3┐ (1) (25 points) Let A =│0 -1 -1│. Calculate the characteristic polynomial of └0 0 0┘ A (10 points), find all the eigenvalues of A (5 points) and find a matrix Q so that QAQ^-1 is diagonal (10 points). (2) (20 points) Let S =｛v_1,v_2,v_3｝be the ordered basis of standard inner product space R^3, with v_1 = (2,2,1), v_2 = (1,0,1), v_3 = (0,1,2). Apply the Gram-Schmidt process to S to obtain an orthogonal basis of R^3 (10 points). Then normalize the vectors in this basis to obatin an orthonormal basis β＝｛u_1,u_2,u_3｝and find a_1,a_2,a_3 so that (3,2,5) = (a_1)(u_1) + (a_2)(u_2) + (a_3)(u_3) (10 points). (3) Either give a brief reason or give a counter example for each of the foll- owing assertions (5 points each): (a) We have (S⊥)⊥ = Span(S), for each subset S of a finite dimensional inner product space. (b) Suppose f(x) ∈ C([-1,1]) has the nth derivative f(n)(x) and T_n(x) = n f(k)(0) Σ────x^k is its nth Taylor polynomial. Then ||f-T_n|| ≦ ||f-g|| k=0 k! for all g ∈ P_n(R). (c) An inner product space V is spanned by the subset ｛v∈V | <v-v_0,v-v_0> ≦ ε｝, for any given positive number ε and any given vector v_0 ∈ V. (d) Every matrix A satisfying A^2 = 0 is diagonalizable. (e) If a 2x2 matrix A satisfies A^2 = I, then its characteristic polynomi- al equals x^2 - 1. (4) Give a rigorous proof for each of the following assertions. (a) (15 points) Suppose T and U are diagonalizable linear operators on a finite dimensional vector space V. Show that T and U are simultaneous- ly diagonalizable if and only if T。U = U。T. (b) (10 points) Let V = C([-1,1]). Suppose W_e and W_o denote the subspac- es of V consisting of the even and odd functions respectively. Prove that W_e⊥ = W_o. (c) (5 points) In an inner product space with order basis｛v_1,...,v_n｝, the matrix A = (a_ij), with the ij entry a_ij = <v_i,v_j>, is inverti- ble. --

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