[試題] 97下 黃漢水 線性代數二 期末考

[jrpg0618 (L-eight)]

課程名稱︰線性代數二
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︰2009年6月17日
考試時限：180分鐘
是否需發放獎勵金：是

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 ∈ 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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推 like1234567 :第二題的第一小題  α_3和α_4 打錯了?
推 dave1988    :恩 打錯了
※ 編輯: jrpg0618        來自: 140.112.199.240      (06/18 00:21)