課程名稱︰代數導論二
課程性質︰必修
課程教師︰康明昌
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2008/6/18
考試時限（分鐘）：2小時
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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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