admin 管理员组文章数量: 1087139
2024年4月16日发(作者：源码搭建详细教程)
NUAA
第1页 （共6页）
Matrix Theory, Final, Test Date： 2015年12月28日
矩阵论班号: 学号 姓名
必做题（70分）
题号
得分
1
2
3
4
5
选做题（30分）
总分
Part I (必做题，共5题，70分)
第1题（15分）
得分
Let
P
[1,1]
denote the set of all real polynomials of degree less than 3 with domain
（定义域）
[1,1]
. The addition and scalar multiplication are defined in the usual way.
Define an inner product on
P
[1,1]
by
p,q
p(t)q(t)dt
.
1
1
(1) Construct an orthonormal basis for
P
[1,1]
from the basis
1,x,x
2
by using the
Gram-Schmidt orthogonalization process.
(2) Let
f(x)x
2
1
P
[1,1]
. Find the projection of
f
onto the subspace spanned by{
1,x
}.
Solution:
(1)
11,1
p
1
x,
1
2
1
1
1
dx2
,
u
1
1
2
,
1
2
1
[
1
1
2
1
xdx]
1
2
0
,
u
2
xp
1
xp
1
3
x
,
2
x
2
p
2
10
331
(3x
2
1)
p
2
x,x,xx
,
u
3
2
4
223
xp
2
22
22
------------------------------------------------------------------------------------
-------
(2)
projx
2
1,u
1
u
1
x
2
1,u
2
u
2
x
2
1,
1
2
1
2
x
2
1,
33
xx
22
2212
0
33
2
------------------------------------------------------------------------------------
----------------------------
精品文档
第2题（15分） 得分
Let
be the linear transformation on
P
3
(the vector space of real polynomials of
degree less than 3) defined by
(p(x))xp'(x)p''(x)
.
(1) Find the matrix
A
representing
with respect to the ordered basis [
1,x,x
2
] for
P
3
.
(2) Find a basis for
P
3
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:
(1)
（）10
002
010
A
（x）x
002
（x
2
）22x
2
------------------------------------------------------------------------------------
-----------------------------
(2)
101
T
010
(The column vectors of T are the eigenvectors of A)
001
The corresponding eigenvectors in
P
3
are
000
010
T
1
AT
(
T
diagonalizes
A
)
002
2
[1,x,x
2
1][1,x,x
2
]T
. With respect to this new basis
[1,x,x1]
, the representing
matrix of is diagonal.
---------------------------------------------------------------------------------
---------------------------------- (3) The kernel is the subspace consisting of
all constant polynomials.
The range is the subspace spanned by the vectors
x,x
2
1
------------------------------------------------------------------------------------
-----------------------------------
。
2
欢迎下载
精品文档
第3题（20分） 得分
110
Let
A
020
.
012
(1) Find all determinant divisors and elementary divisors of
A
.
(2) Find a Jordan canonical form of
A
.
(3) Compute
e
At
. (Give the details of your computations.)
Solution:
(1)
10
1
IA
20
,(特征多项式
p(
)(
1)(
2)
2
. Eigenvalues are 1, 2,
0
01
2
2.)
Determinant divisor of order
D
1
(
)1
,
D
2
(
)1
,
D
3
(
)p(
)(
1)(
2)
2
Elementary divisors are
(
1) and (
2)
2
.
------------------------------------------------------------------------------------
----------------------------------
(2) The Jordan canonical form is
100
J
021
002
------------------------------------------------------------------------------------
--------------------------------------
0
(3) For eigenvalue 1,
IA
0
0
1
For eigenvalue 2,
2IA
0
0
1
1
1
0
0
， An eigenvector is
p
1
(1,0,0)
T
1
0
0
, An eigenvector is
p
2
(0,0,1)
T
0
。
3
欢迎下载
1
0
1
版权声明：本文标题：南京航空航天大学Matrix-Theory双语矩阵论期末考试2015 内容由网友自发贡献，该文观点仅代表作者本人， 转载请联系作者并注明出处：http://roclinux.cn/p/1713237228a625352.html， 本站仅提供信息存储空间服务，不拥有所有权，不承担相关法律责任。如发现本站有涉嫌抄袭侵权/违法违规的内容，一经查实，本站将立刻删除。
发表评论