課程名稱︰線性代數 課程性質︰數學系必修 課程教師︰黃漢水 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2009/6/17 考試時限（分鐘）：三小時 是否需發放獎勵金：YES （如未明確表示，則不予發放） 試題 : Let R be the field of all real numbers. 一 Let f(x,y) = (7x^2 + 8xy)/(x^2 + 2xy + 2y^2) for (x^2 + 2xy + 2y^2) ≠0 Find the minimum value and the maximum value of f(x,y) 二 Let A = [ 1 -2 -1 -2] ╭─ [-2 5 3 3] ├─ Mat_4x4(R) [-1 3 3 3] ╰─ [-2 3 3 13] (1) Find an invertible matrix K belonging to Mat_4x4(R) and α_1 α_2 α_3 α_4 belonging to R [α_1 0 0 0 ] such that (K^T)AK = [ 0 α_2 0 0 ] [ 0 0 α_3 0 ] [ 0 0 0 α_4 ] (2) Is there an invertible matrix C belonging to Mat_4x4(R) such that A = (C^T)C? Prove yor answer. 三 Let y = y(t) be a function such that y"'(t) - 2y"(t) + y'(t) = 2exp(t), y(0) = 3, y'(0) = 5, y"(0) = 10 Find y(t). 四 Let F be a field , B belonging to Mat_nxn(F), V_i = { (B^i)u │u belonging to Mat_nx1(F) = F^n } and W = nullspace of B. Suppose that B^3 = 0_nxn belonging to Mat_nxn(F) and u_1 u_2 u_3 belonging to F^n → → → such that (B^3)u_1 = 0, (B^3)u_2 = 0, Bu_3 = 0 and (B^2)u_1, (B^2)u_2, u_3 are linearly independent. (1) Prove that u_1, u_2,Bu_1, Bu_2, (B^2)u_1, (B^2)u_2, u_3 are linearly independent. (2) If dim(V_1) = 2, dim(V_2∩W) = 2, dim(W) = 3, then prove that for any u 屬於 F^n, there are α_1 α_2 α_3 α_4 α_5 α_6 α_7 u = α_1(u_1) + α_2(u_2) + α_3(Bu_1) + α_4(Bu_2) + α_5(B^2)u_1 + α_6(B^2)u_2 + α_7(u_3) 五 Let [2 0 2] [0 2 0] [1 2 2] [1 0 0 1 0 1] C_1 = [1 0 0] C_2 = [0 1 0 1 1 0] over F = Z_3 = {0 1 2} [0 1 0] [0 0 1 0 1 1] [0 0 1] B = C_1*C_2 and A = B + 2*I_6 Find an invertible matrix K belonging to Mat_6x6(F) such that K^(-1)AK = J is Jordan form of A --

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