課程名稱︰線性代數二 課程性質︰數學系大一必修 課程教師︰陳其誠 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2011/05/27 考試時限（分鐘）：130分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : LINEAR ALGEBRA EXAM Ⅲ 5/27 2011 You should give all details in order to gain the corresponding points. Let ┌0 0 1┐ ┌0 0 0 -1┐ A =│1 0 -3│, B =│0 0 1 0│, └0 1 3┘ │0 1 0 0│ └1 0 0 -1┘ ┌0 0 … -a_0┐ ┌ 3 0 1┐ │1 0 … -a_1│ C =│ 2 2 2│, D =│0 1 … -a_2│. └-1 0 1┘ │: : : │ └0 0 … -a_100┘ (1) (20 points) Find a Jordan canonical form J and an invertible matrix Q so that A = QJQ^-1 (2) (20 points) Let T: R^6 → R^6 be a linear operator with the characteristic polynomial f(t) = t(t-1)^2(t+1)^3 and the minimal polynomial p(t) = t(t-1)(t+1)^2. Find a Jordan canonical form of T. (3) (15 points) Let T=L_B: R^4 → R^4. It is known that (T-I)(T+I)(T^2+T+I)=0. Determine the space K_t^2+t+1:=｛x∈R^4│[(T^2+T+I)^p](x)=0, for some non-negative integer p｝ (4) (15 points) Find the minimal polynomial of C. (5) Either give a brief reason or give a counter example for each of the foll- owing assertions (5 points each): Let T be a linear operator on a finite dimensional vector space V. (a) Suppose f(t) is a polynomial with f(0)≠0 and f(T) = 0. Then T must be invertible. (b) If T is invertible, then there exist a polynomial g(t) so that T^-1 = g(T). (c) If J is a Jordan cononical form of T, then J^t and J are similar. (d) The characteristic polynomial as well as minimal polynomial of T is determined by the Jordan canonical form (if exists). (e) The Jordan canonical form (if exists) of T is determined by the chara- cteristic polynomial and minimal polynomial. (f) Suppose D is a rational form of T with t^101 + a_100‧t^100 + … + a_1‧t + a_0 = t(t-100)^100. Then T is not diagonalizable. --

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