问题 问答题

设二次型

,其中二次型矩阵A有特征值4.

试用正交变换将二次型f化为标准形,并写出所用坐标变换;

答案

参考答案:二次型f的矩阵[*],由λ=4是矩阵A的特征值,有
[*]
所以a=3.
由矩阵A的特征多项式
[*]
得到矩阵A的特征值为λ1=1,λ2=4,λ3=-2.
对于λ=1,由(E-A)x=0,[*]
得到矩阵A属于λ=1的特征向量α1=(-1,1,1)T
对于λ=4,由(4E-A)x=0,
[*]
得到矩阵A属于λ=4的特征向量α2=(1,-1,2)T
对于λ=-2,由(-2E-A)x=0,
[*]
得到矩阵A属于λ=-2的特征向量α3=(1,1,0)T
由于α1,α2,α3已两两正交,故只需单位化,有
[*]
那么,令
[*]
在正交变换x=Qy下,有
[*]
二次型[*].

选择题
完形填空

Growing up on her own

WORLD No 2 tennis player Jelena Jankovic from Serbia (塞尔维亚) has won her second title of the year. She became champion of the China Open on September 28.

Jankovic first practiced tennis when she was nine years old. After six months Jankovic played her first match.

It was the national championship for kids up to 10 years old. She came to the semi-finals (半决赛).

When she was 11, she won a national championship for girls up to 12 years old. In order to continue her career, Jelena left Serbia and went to the United States.

At that time, she didn’t speak any English. And her family did not come with her. It was a hard time.

“I did not know what I needed to do when I was going to school. I could not do my homework. When you have a difficulty, there is nobody to help you. So, you learn it the hard way,” Jelena said.

However, it was a good learning experience for Jelena. “It makes me more independent (独立的) and stronger as a person. I know how to do everything by myself so I don’t have to depend on (依赖) my parents or somebody else to do it for me,” she said.

In the US, Jelena received better conditions for practice and development of her game. In 2000, she became a professional (职业选手) and has been shining on the tennis court ever since.

根据上面短文的内容回答问题    

1. Where is Jelena Jankovic from?

2. What is Jelena Jankovic’s job?

3. Why did she leave for the United States?

4. When did she become a professional?

5. What do you think of her hard time in the United States?