推 like1234567 :第二題的第一小題 α_3和α_4 打錯了? 06/17 22:35
推 dave1988 :恩 打錯了 06/17 23:51
※ 編輯: jrpg0618 來自: 140.112.199.240 (06/18 00:21)
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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