武汉大学2014年基础数学复试笔试题回忆 - Eufisky - The lost book
武汉大学2015年基础数学复试笔试试题

武汉大学2014年基础数学复试笔试题回忆

Eufisky posted @ 2015年7月19日 23:28 in 考研 with tags 考研 武汉大学 , 827 阅读

试题来自陈洪葛的博客.

 

问题1.(10分)函数$f(x)$在$(-1,1)$上连续,除了$0$这一点外可导。

  1. 若$f(x)$的导函数当$x\to 0$时极限存在,证明$f(x)$在$0$点的导数存在。
  2. 上述命题的逆命题是否成立?就是说$f(x)$在$0$点的导数存在是不是一定有$f(x)$在$x\to 0$的极限存在?成立请证明,否则给出反例。

 

问题2(10分)证明函数$f(x)$在$(a,b)$上一致连续的充分必要条件是对$(a,b)$上的收敛数列$\{x_{n}\}$,数列$\{f(x_{n})\}$也收敛。

 

 

问题3(10分)

证明含参变量积分

\[\int_{0}^{+\infty}\frac{\sin{xy}}{y(1+x)}dy\]

关于$x$在$0<\delta\leq x<+\infty$上一致收敛,在$0<x<+\infty$上非一致收敛。

 

 

问题4(10分)设$X$是带有度量空间上的紧集,$E\subset X,\varphi(x)$是$E$上的变换,且满足

\[d(\varphi(x),\varphi(y))<d(x,y) \qquad (x\neq y,x,y\in E).\]

证明$\varphi(x)$在E中存在唯一的不动点。

 

 

问题5(10分)(Tauber定理)

设在$-1<x<1$上有

\[f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}\]

并且

\[\lim_{n\to\infty}na_{n}=0.\]

若$\displaystyle \lim_{x\to 1^{-}}f(x)=S$,则$\displaystyle \sum_{n=0}^{\infty}a_{n}$收敛且其和为$S$.

 

 

问题6 (15分)

 

讨论微分方程过点y=0的解的存在性和唯一性,其中$\alpha>0$.

\[\frac{dy}{dx}=|y|^{\alpha}.\]

 

 

问题7 (15分)设$A$是$n$阶可逆复方阵,证明存在分解

\[A=UT,\]

其中$U$是酉矩阵,$T$是主对角线上都是正数的上三角型矩阵,并证明这种分解的唯一性。

 

 

问题8 (20分)

已知$A = \left( {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right) \in \mathbb{C}^{2\times 2}$,定义$\mathbb{C}^{2\times 2}$的变换$f:f(X)=XA,\forall X\in \mathbb{C}^{2\times 2}$.

  1. 证明$f$是$\mathbb{C}^{2\times 2}$的线性变换;
  2. 求$f$在$\mathbb{C}^{2\times 2}$的基

    \[{E_{11}} = \left( {\begin{array}{*{20}{c}}1&0\\0&0\end{array}} \right),{E_{12}} = \left( {\begin{array}{*{20}{c}}0&1\\0&0\end{array}} \right),{E_{21}} = \left( {\begin{array}{*{20}{c}}0&0\\1&0\end{array}} \right),{E_{22}}= \left( {\begin{array}{*{20}{c}}0&0\\0&1\end{array}} \right)\]

    下的矩阵$M$.

  3. 给出$\mathbb{C}^{2\times 2}$的两个非零的$f$不变子空间$V_1$和$V_2$,使得$\mathbb{C}^{2\times 2}=V_1\oplus V_2$,请阐述理由.
  4. 证明:存在$\mathbb{C}^{2\times 2}$的一个基,使得$f$在这一基下的矩阵为对角矩阵当且仅当矩阵$A$与对角矩阵相似.

 

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JDC Result Barisal 说:
2022年9月02日 06:06

In the Bangladesh Education System, Barisal board has a good record and the Barisal Division also successfully completed JSC and JDC terminal examination tests 2022 as per schedules along with all other educational boards of the country, JDC Result Barisal and there are a huge number of general and mass education students have appeared to the Grade 8 final exams from the division.The Bangladesh Secondary and Higher Secondary Education, Barisal Board has successfully completed the Junior Certificate & Junior Dakhil Terminal exams on November like as previous years, and the school education department has to conduct evaluation process through answer sheet corrections for both general and mass education JSC & JDC exam answer sheet to calculate subject wise marks of the student, once the evaluation is completed the JSC Result 2022 Barisal Board is announced with full mark sheet with total CGPA of the student.


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