0 1 0
1 1 0 0
Solve
(A 2I ) p3 p2 ,
( A 2I ) p3


0
0
0

p3


0

we
obtain
that
0 1 0 1
1 0 1
P


0
0
1 ,
0 1 0
1 1 0
(1) -----------------------------------------------------------------------------------------------------------------
(2)
1 0 1
T


0
1
0
ห้องสมุดไป่ตู้
(The column vectors of T are the eigenvectors of A)
1
p, q p(t)q(t)dt . 1
(1) Construct an orthonormal basis for P[1,1] from the basis 1, x, x2 by using the Gram-Schmidt orthogonalization process.
Suppose that AR33 and A2 5A 6I O . (1) What are the possible minimal polynomials of A? Explain. (2) In each case of part (1), what are the possible characteristic polynomials of A? Explain.
0
1 2
Determinant divisor of order D1() 1 , D2 () 1 , D3 () p() ( 1)( 2)2
Elementary divisors are ( 1) and ( 2)2 .
---------------------------------------------------------------------------------------------------------------------(2) The Jordan canonical form is --------------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------is the subspace consisting of all constant polynomials.
--
第 3 页（共 3 页）
Part II (选做题, 每题 10 分) 请在以下题目中（第 6 至第 9 题）选择三题解答. 如果你做了四题，请在题号上画圈标明需要批改的三题. 否 则，阅卷者会随意挑选三题批改，这可能影响你的成绩. 第 6 题 Let P4 be the vector space consisting of all real polynomials of degree less than 4 with usual addition and
(1) Find the matrix A representing with respect to the ordered basis [1, x, x2 ] for P3 .
(2) Find a basis for P3 such that with respect to this basis, the matrix B representing is diagonal. (3) Find the kernel（核） and range （值域）of this transformation. Solution:
(2) Let f (x) x2 1 P[1,1] . Find the projection of f onto the subspace spanned by{1, x }. Solution:
(1)
1
1 1 , 1 dx 1
,2 u1
1, 2
p1 x,
Case x 1, the characteristic polynomial is (x 1)3
Case x2 5x 6 , the characteristic polynomial is (x 1)2 (x 6) or (x 6)2 (x 1)
------------------------------------------------------------------------------------------------------------------第 5 题（10 分） 得分
第 2 题（15 分） 得分
Let be the linear transformation on P3 (the vector space of real polynomials of degree less than 3) defined by ( p(x)) xp '(x) p ''(x) .
(3) Compute eAt . (Give the details of your computations.)
Solution:
第 2 页（共 3 页）
(1)
1 1 0

I

A


0
2
0

,(特征多项式
p() ( 1)( 2)2 . Eigenvalues are 1, 2, 2.)
The range is the subspace spanned by the vectors x, x2 1
----------------------------------------------------------------------------------------------------------------------第 3 题（20 分） 得分
Solution:
(1) An annihilating polynomial of A is x2 5x 6 .
The minimal polynomial of A divides any annihilating polynomial of A. The possible minimal polynomials are x 6 , x 1, and x2 5x 6 . --------------------------------------------------------------------------------------------------------------(2) The minimal polynomial of A divides the characteristic polynomial of A. Since A is a matrix of order 3, the characteristic polynomial of A is of degree 3. The minimal polynomial of A and the characteristic polynomial of A have the same linear factors. Case x 6 , the characteristic polynomial is (x 6)3
(2)
proj x2 1,u1 u1 x2 1,u2 u2 x2 1,
1 2
1 x2 1, 2
3x 2
3x 2
----------------------------------------------------------------------------------------------------------------
P 1


0
0
1

0 1 0
--------------------------------------------------------------------------------------------------------------------
第 4 题（10 分） 得分
Let
A


1 0
2 0
0 0

.
Find
the
Moore-Penrose
inverse
A of
A.
Solution:
1
P (PT P )1 PT
(1,
,0
)
G

GT
(GGT
)1

1 5

2 0

也可以用 SVD 求.
----------------------------------------------------------------------------------------------------------------
0 0 1
The corresponding eigenvectors in P3 are
0 0 0
T 1 AT