課程名稱︰線性代數二
課程性質︰必修
課程教師︰林紹雄
開課系所︰數學系
考試時間︰2006/4/14 10:10~13:00
Math 201 14410-01 (Linear Algebra) Midterm Exam (4/14/2006)
There Problems A to E with a total of 130 points. Please write down your
computational, or proof steps clearly on the answer sheets.
A. Let A be the matrix
[ 1 -1 0 -1 ]
A = [ 0 2 0 1 ]
[ -2 1 -1 1 ]
[ 2 -1 2 0 ]
(a) (10 points) Find its charateristic polynomials and eigenvalues. Find a
maximal set of linearly independent eigenvectors of A.
-1
(b) (10 points) Find an invertible matrix P in M_4(R) such that P AP is its
Jordan canonical form.
tA
(c) (10 points) Use (b) compute e for real t.
2 2 2 3
B. Let Q(x,y,z) = 2( x + y + z - xy - xz + yz ) be a quadratic form in R .
(a) (6 points) write down the symmetric matrix A in M_3(R) such that
T
Q(x,y,z) = [x y z] A [x y z] . Determine whether Q is positive definite or
not.
(b) (10 points) Find an orthogonal matrix R in M_3(R) with det(R) = 1 such
T T
that Q is reduced to its normal form in (u,v,w) where [u v w] = R[x y z] .
(c) (4 points) Determine whether the equation Q(x,y,z) = -1 has a real
solution. Does Q(x,y,z) = -1 always have complex solutions (i.e. x,y,z are
complex)?
C. Multinational companies in the U.S., Japan and Europe have assets of 4
trillion US dollars. At the start, 2 trillion US dollars are in the U.S. and
2 trillion US dollars in Europe. Each year 1/2 the U.S. money stays home, 1/4
goes to both Japan and Europe. For Japan and Europe, 1/2 stays home and 1/2 is
sent to the U.S..
(a) (4 points) Find the transition matrix A such that
[ US ] [ US ]
[ J ] = A [ J ]
[ E ]year k+1 [ E ]year k
(b) (12 points) Find the eigenvalues and eigenvectors of A. Determine the
k
steady state of A and compute lim A .
k→∞
(c) (4 points) Find the limiting distribution of the 4 trillion dollars as the
world ends.
D. Prove the following statements. Each has 10 points.
H
(a) Let M, A in M_n(C). Assume that AM + M A = -I_n and A is positive definite.
Prove that M is a stable matrix, i.e. when λ is an eigenvalue of M, then
the real part of λ is negative.
(b) Let A be a n*n nonzero nilpotent matrix of rank r. If the minimal
k n
polynomial of A is m(x) = x , prove that k ≧ ────.
n - r
tA
(c) Let A in M_n(C). Prove that e is unitary iff A is skew-Hermitian, i.e.
H
A = -A
E. Determine which of the following statements is true. Prove your answer (or
give a counterexample). Each has 10 points.
(a) Let A and B be two 4*4 matrices. Suppose that A satisfies a polynomial
equation p(A) = 0 iff B satisfies the same equation p(B) = 0. Then A and B
are similar.
3
(b) Let the vector-valued function u: R → R satisfies the system du/dt = Au
3
and u(0) = u0 in R , where
[ 0 -1 0 ]
A = [ 1 0 -1 ]
[ 0 1 0 ]
Then |u(t)| = |u0| for all t in R, and there exists some time T such that
u(T) = u0.
2 2
(c) If a 5*5 matrix has x (x - 1) as its minimal polynomial, then its Jordan
canonical form must be either
[ 0 0 0 0 0 ] [ 0 0 0 0 0 ]
[ 1 0 0 0 0 ] [ 1 0 0 0 0 ]
[ 0 0 0 0 0 ] or [ 0 0 1 0 0 ]
[ 0 0 0 1 0 ] [ 0 0 1 1 0 ]
[ 0 0 0 1 1 ] [ 0 0 0 0 1 ]
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※ 編輯: Nzing 來自: 140.112.240.198 (04/15 19:41)