课程名称︰线性代数二
课程性质︰数学系必修
课程教师︰朱桦
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2015/04/29
考试时限（分钟）：10:30-12:40
试题 :
4
(1) (15%) Let T be a linear operator on V = C defined by
T(a1,a2,a3,a4) = (4a1-2a2+a3+4a4, 6a1-2a2+6a4, -2a2-a3+4a4, -3a1+3a3+a4).
(a) Find an ordered basis β for V such that [T] is a diagonal matrix.
β
(b) Can you find an orthonormal basis γ (under standard inner product)
such that [T] is a diagonal matrix?
γ
m
(c) Find lim A .
m→∞
┌ ┐ 5 4 3 2
(2) (10%) Let A = │ 0 1 0 │. Find A + 2A + 3A + 4A + 5A + 6I .
│-2 -2 1 │ 3
│-1 0 1 │
└ ┘
(3) (15%) Which of the following transition matrices are regular? Explain
your answer.
┌ ┐ ┌ ┐
A = │0.7 0 0.5 0 │, B = │0.7 0 1 0 │
│ 0 0.4 0 1 │ │0.3 0 0 0 │
│0.3 0 0.5 0 │ │ 0 0 0 1 │
│ 0 0.6 0 0 │ │ 0 1 0 0 │
└ ┘ └ ┘
(4) (15%) (a) Let V = M (C) with the inner product <A,B> = tr(B*A). Let P
n ×n
be an invertible matrix in V, and T be the linear operator on
-1 P
V defined by T (A) = P AP. Find the adjoint T* .
P P
1
(b) Let V = P (R) with the inner product <f,g> = ∫ f(t)g(t)dt. Let
2 -1
D be the differentiation operator on V. Find the adjoint D*.
t
(5) (10%) Let A∈M (F). For any eigenvalue λ of A and A , let E and E'
n ×n t
denote the corresponding eigenspaces for A and A , respectively.
Prove that dim(E) = dim(E').
(6) (10%) Let A, B ∈M (F).
n ×n
(a) Suppose that (I - AB) is invertible, prove that (I - BA) is invertible and
n -1 -1 n
(I - BA) = I + B(I - AB) A.
n n n
(b) Prove that AB and BA have the same eigenvalues in F.
(7) (15%) Let T be a linear operator on an n-dimensional vector space, and suppose
that T has n distinct eigenvalues. Prove that any linear operator which
commutes with T is a polynomial in T.
(8) (10%) (a) Let T be a linear operator on a vector space V, let v be a nonzero vector
in V, and let W be the T-cyclic subspace of V generated by v. Prove that,
for any w∈W, there exists a polynomial g(t) such that w=g(T)v.
(b) Let T be a linear operator on V and suppose that V is a T-cyclic subspace
of itself. Prove that, if U is a linear operator such that UT = TU, then U=g(T)
for some polynomial g(t).
(9) (10%) Let W be a finite-dimensional subspace of an inner product space V, and let T
be the orthogoal projection of V on W. Prove that <Tx,y> = <x,Ty> for all
x,y∈V.
(10) (10%) Let T be a linear operator on a finite dimensional inner product space. Prove
that N(T*T) = N(T), where N(T) is the null space of T.