作者TassTW (为文载道尊於势)
看板NTU-Exam
标题[试题] 96下 康明昌 代数导论二 期末考
时间Wed Jun 18 14:03:55 2008
课程名称︰代数导论二
课程性质︰必修
课程教师︰康明昌
开课学院:理学院
开课系所︰数学系
考试日期(年月日)︰2008/6/18
考试时限(分钟):2小时
是否需发放奖励金:是
(如未明确表示,则不予发放)
试题 :
1. Let K be a field, T belongs to Mn(K)
(a) Give the definitions of the minimum polynomial and the characteristic
polynomial of T.
(b) State the Cayley-Hamilton Theoerm for the matrix T.
2. Let K be a field, V be an n-dimensional vector space over K, and
R = End_K(V). Define V as an R-module by fv = f(v) for any f in R,
and v in V.
Show that V is a simple R-module.
3. Let N1, N2, N' be submodules of an R-module M. Assume that:
(i) N1 belongs to N2
(ii) N1 intersects N' = N2 intersects N'
(iii) N1 + N' = N2 + N'.
Show that N1 = N2
4. Let f: N -> M be a homomorphism of R-modules N and M. Suppose that
g: M -> N is also a homomorphism of R-modules satisfying that:
g o f: N -> N is an isomorphism.
Show that M contains a submodule L so that:
(i) M = f(N) + L
(ii) F(N) intersects L = {0}
5. Let K be a field and A,B belongs to Mn(K) be defined by
┌a1 0 ... 0┐
│a2 0 0│
A = │ . . .│
│ . . .│
└an 0 ... 0┘
┌b1 0 ... 0┐
│b2 0 0│
B = │ . . .│
│ . . .│
└bn 0 ... 0┘
Assume a1 != 0.
Show that there exists C in Mn(K) with B = CA
6. Define
┌ 1 1 2 2┐
T = │ 1 -2 -1 -1│in M4(C)
│-2 1 -1 -1│
└ 1 1 2 2┘
(a) Find minimum polynomial of T
(b) Find explicitly a vector v in C^4 so that
v, T(v), T^2(v) are linearly independent over C
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