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 -- 「我們愛星星至深無懼於黑暗。」 "We have loved the stars too fondly to be fearful of the night." --

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