課程名稱︰線性代數二
課程性質︰數學系必修
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︰2011年05月27日
考試時限：130分鐘
是否需發放獎勵金：是
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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