2018

Problem 1

Consider to solve the following simultaneous linear equation:

$$\mathbf{Ax}=\mathbf{b}$$

where $\mathbf{A}\in {\mathbb{R}}^{m\times n}$, $\mathbf{b}\in {\mathbb{R}}^m$ are a constant matrix and a vector, and $\mathbf{x}\in {\mathbb{R}}^n$ is an unknown vector. Answer the following questions.

(1) An $m\times (n+1)$ matrix $\bar{\mathbf{A}}=(\mathbf{A}|\mathbf{b})$ is made by adding a column vector after the last column of matrix $\mathbf{A}$. In the case of $\mathbf{A}=\left(\begin{array}{ccc}1& 0& -1\\ 1& 1& 0\\ 0& 1& 1\end{array}\right)$ and $\mathbf{b}=\left(\begin{array}{c}2\\ 4\\ 2\end{array}\right)$,

$$\bar{\mathbf{A}}=\left(\begin{array}{cccc}1& 0& -1& 2\\ 1& 1& 0& 4\\ 0& 1& 1& 2\end{array}\right)$$

is obtained. Let the $i$-th column vector of the matrix $\bar{\mathbf{A}}$ be ${\mathbf{a}}_i$ $(i=1,2,3,4)$.

(i) Find the maximum number of linearly independent vectors among ${\mathbf{a}}_1,{\mathbf{a}}_2$ and ${\mathbf{a}}_3$.

(ii) Show that ${\mathbf{a}}_4$ can be represented as a linear sum of ${\mathbf{a}}_1,{\mathbf{a}}_2$ and ${\mathbf{a}}_3$, by obtaining scalars $x_1$ and $x_2$ that satisfy ${\mathbf{a}}_4=x_1{\mathbf{a}}_1+x_2{\mathbf{a}}_2+{\mathbf{a}}_3$.

(iii) Find the maximum number of linearly independent vectors among ${\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{a}}_3$ and ${\mathbf{a}}_4$.

(2) Show that the solution of the simultaneous linear equation exists when $\mathrm{rank}\bar{\mathbf{A}}=\mathrm{rank}\mathbf{A}$ for arbitrary $m,n,\mathbf{A}$ and $\mathbf{b}$.

(3) There is no solution when $\mathrm{rank}\bar{\mathbf{A}}>\mathrm{rank}\mathbf{A}$. When $m>n$, $\mathrm{rank}\mathbf{A}=n$ and $\mathrm{rank}\bar{\mathbf{A}}>\mathrm{rank}\mathbf{A}$, obtain $\mathbf{x}$ that minimizes the squared norm of the difference between the left hand side and the right hand side of the simultaneous linear equation, namely $\Vert \mathbf{b}-\mathbf{Ax}{\Vert }^2$.

(4) When $m<n$ and $\mathrm{rank}\mathbf{A}=m$, there exist multiple solutions for the simultaneous linear equation for arbitrary $\mathbf{b}$. Obtain $\mathbf{x}$ that minimizes $\Vert \mathbf{x}{\Vert }^2$ among them, by adopting the method of Lagrange multipliers and using the simultaneous linear equation as the constraint condition.

(5) Show that there exists a unique $\mathbf{P}\in {\mathbb{R}}^{n\times m}$ that satisfies the following four equations for arbitrary $m,n$ and $\mathbf{A}$.

$$\mathbf{APA}=\mathbf{A}$$ $$\mathbf{PAP}=\mathbf{P}$$ $$(\mathbf{AP}{)}^T=\mathbf{AP}$$ $$(\mathbf{PA}{)}^T=\mathbf{PA}$$

(6) Show that both $\mathbf{x}$ obtained in (3) and $\mathbf{x}$ obtained in (4) are represented in the form of $\mathbf{x}=\mathbf{Pb}$.

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考虑解以下线性方程组：

$$\mathbf{Ax}=\mathbf{b}$$

其中 $\mathbf{A}\in {\mathbb{R}}^{m\times n}$，$\mathbf{b}\in {\mathbb{R}}^m$ 是常数矩阵和向量，$\mathbf{x}\in {\mathbb{R}}^n$ 是未知向量。回答以下问题。

(1) $m\times (n+1)$ 矩阵 $\bar{\mathbf{A}}=(\mathbf{A}|\mathbf{b})$ 是在矩阵 $\mathbf{A}$ 的最后一列后添加一个列向量得到的。对于 $\mathbf{A}=\left(\begin{array}{ccc}1& 0& -1\\ 1& 1& 0\\ 0& 1& 1\end{array}\right)$ 和 $\mathbf{b}=\left(\begin{array}{c}2\\ 4\\ 2\end{array}\right)$,

$$\bar{\mathbf{A}}=\left(\begin{array}{cccc}1& 0& -1& 2\\ 1& 1& 0& 4\\ 0& 1& 1& 2\end{array}\right)$$

得到。设矩阵 $\bar{\mathbf{A}}$ 的第 $i$ 列向量为 ${\mathbf{a}}_i$ $(i=1,2,3,4)$。

(i) 求 ${\mathbf{a}}_1,{\mathbf{a}}_2$ 和 ${\mathbf{a}}_3$ 中线性无关向量的最大数量。

(ii) 证明 ${\mathbf{a}}_4$ 可以表示为 ${\mathbf{a}}_1,{\mathbf{a}}_2$ 和 ${\mathbf{a}}_3$ 的线性组合，获得满足 ${\mathbf{a}}_4=x_1{\mathbf{a}}_1+x_2{\mathbf{a}}_2+{\mathbf{a}}_3$ 的标量 $x_1$ 和 $x_2$。

(iii) 求 ${\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{a}}_3$ 和 ${\mathbf{a}}_4$ 中线性无关向量的最大数量。

(2) 证明当 $\mathrm{rank}\bar{\mathbf{A}}=\mathrm{rank}\mathbf{A}$ 时，线性方程组的解存在，对于任意 $m,n,\mathbf{A}$ 和 $\mathbf{b}$。

(3) 当 $\mathrm{rank}\bar{\mathbf{A}}>\mathrm{rank}\mathbf{A}$ 时，没有解。当 $m>n$，$\mathrm{rank}\mathbf{A}=n$ 且 $\mathrm{rank}\bar{\mathbf{A}}>\mathrm{rank}\mathbf{A}$ 时，求使左边和右边之差的平方范数 $\Vert \mathbf{b}-\mathbf{Ax}{\Vert }^2$ 最小的 $\mathbf{x}$。

(4) 当 $m<n$ 且 $\mathrm{rank}\mathbf{A}=m$ 时，对于任意 $\mathbf{b}$，线性方程组存在多个解。采用拉格朗日乘子法，使用线性方程组作为约束条件，求使 $\Vert \mathbf{x}{\Vert }^2$ 最小的 $\mathbf{x}$。

(5) 证明存在唯一的 $\mathbf{P}\in {\mathbb{R}}^{n\times m}$，它满足以下四个方程，对于任意 $m,n$ 和 $\mathbf{A}$。

$$\mathbf{APA}=\mathbf{A}$$ $$\mathbf{PAP}=\mathbf{P}$$ $$(\mathbf{AP}{)}^T=\mathbf{AP}$$ $$(\mathbf{PA}{)}^T=\mathbf{PA}$$

(6) 证明在 (3) 中得到的 $\mathbf{x}$ 和在 (4) 中得到的 $\mathbf{x}$ 都可以表示为 $\mathbf{x}=\mathbf{Pb}$ 的形式。

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Problem 2

Let $f_1$ be a positive constant function on $[0,1]$ with $f_1(x)=c$, and let $p$ and $q$ be positive real numbers with $1/p+1/q=1$. Moreover, let $\{f_n\}$ be the sequence of functions on $[0,1]$ defined by

$$f_{n+1}(x)=p{\int }_0^x(f_n(t){)}^{1/q}\mathrm{d}t(n=1,2,\dots )$$

Answer the following questions.

1. Let $\{a_n\}$ and $\{c_n\}$ be the sequences of real numbers defined by $a_1=0$, $c_1=c$ and

$$\begin{array}{rl}{a}_{n+1}& =q^{-1}{a}_n+1(n=1,2,\dots ),\\ c_{n+1}& =\frac{p(c_n{)}^{1/q}}{a_{n+1}}(n=1,2,\dots ).\end{array}$$

Show that $f_n(x)=c_n{x}^{a_n}$.

2. Let $g_n$ be the function on $[0,1]$ defined by $g_n(x)=x^{a_n}-x^p$ for $n\ge 2$. Noting that $a_n\ge 1$ holds true for $n\ge 2$, show that $g_n$ attains its maximum at a point $x=x_n$, and find the value of $x_n$.

3. Show that ${\lim }_{n\to \infty }{g}_n(x)=0$ for any $x\in [0,1]$.

4. Let $d_n$ be defined by $d_n=(c_n{)}^{g_n}$. Show that $d_{n+1}/d_n$ converges to a finite positive value as $n\to \infty$. You may use the fact that ${\lim }_{t\to \infty }(1-1/t{)}^t=1/e$.

5. Find the value of ${\lim }_{n\to \infty }{c}_n$.

6. Show that ${\lim }_{n\to \infty }{f}_n(x)=x^p$ for any $x\in [0,1]$.

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设 $f_1$ 是定义在 $[0,1]$ 上的正常数函数，其满足 $f_1(x)=c$，且 $p$ 和 $q$ 为正实数，满足 $1/p+1/q=1$。此外，设 $\{f_n\}$ 为定义在 $[0,1]$ 上的函数序列，其定义为

$$f_{n+1}(x)=p{\int }_0^x(f_n(t){)}^{1/q}\mathrm{d}t(n=1,2,\dots )$$

回答下列问题。

1. 设 $\{a_n\}$ 和 $\{c_n\}$ 为实数序列，其定义为 $a_1=0$, $c_1=c$ 并且

$$\begin{array}{rl}{a}_{n+1}& =q^{-1}{a}_n+1(n=1,2,\dots ),\\ c_{n+1}& =\frac{p(c_n{)}^{1/q}}{a_{n+1}}(n=1,2,\dots ).\end{array}$$

证明 $f_n(x)=c_n{x}^{a_n}$。

2. 设 $g_n$ 为定义在 $[0,1]$ 上的函数，其定义为 $g_n(x)=x^{a_n}-x^p$ 对于 $n\ge 2$。注意 $a_n\ge 1$ 对于 $n\ge 2$ 成立，证明 $g_n$ 在点 $x=x_n$ 取得最大值，并求出 $x_n$ 的值。

3. 证明 ${\lim }_{n\to \infty }{g}_n(x)=0$ 对于任何 $x\in [0,1]$ 成立。

4. 设 $d_n$ 定义为 $d_n=(c_n{)}^{g_n}$。证明 $d_{n+1}/d_n$ 收敛于一个有限正值，当 $n\to \infty$ 时。你可以使用以下事实：${\lim }_{t\to \infty }(1-1/t{)}^t=1/e$。

5. 求 ${\lim }_{n\to \infty }{c}_n$ 的值。

6. 证明 ${\lim }_{n\to \infty }{f}_n(x)=x^p$ 对于任何 $x\in [0,1]$ 成立。

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Problem 3

Let $z_n$ and $w_n$ $(n=0,1,2,\dots )$ be complex numbers. Consider a bag that contains two red cards and one white card. First, take one card from the bag and return it to the bag. $z_{k+1}$ $(k=0,1,2,\dots )$ is generated in the following manner based on the color of the card taken.

$$z_{k+1}=\{\begin{array}{ll}i{z}_k& \text{if a red card was taken,}\\ -i{z}_k& \text{if a white card was taken.}\end{array}$$

Next, take one card from the bag again and return it to the bag. $w_{k+1}$ is generated in the following manner based on the color of the card taken.

$$w_{k+1}=\{\begin{array}{ll}-i{w}_k& \text{if a red card was taken,}\\ i{w}_k& \text{if a white card was taken.}\end{array}$$

Here, each card is independently taken with equal probability. The initial state is $z_0=1$ and $w_0=1$. Thus, $z_n,w_n$ are the values after repeating the series of the above two operations $n$ times starting from the state of $z_0=1$ and $w_0=1$. Here, $i$ is the imaginary unit.

Answer the following questions.

1. Show that $\mathrm{Re}(z_n)=0$ if $n$ is odd, and that $\mathrm{Im}(z_n)=0$ if $n$ is even. Here, $\mathrm{Re}(z)$ and $\mathrm{Im}(z)$ represent the real part and the imaginary part of $z$ respectively.

2. Let $P_n$ be the probability of $z_n=1$, and $Q_n$ be the probability of $z_n=i$. Find recurrence equations for $P_n$ and $Q_n$.

3. Find the probabilities of $z_n=1$, $z_n=i$, $z_n=-1$, and $z_n=-i$ respectively.

4. Show that the expected value of $z_n$ is $(i/3{)}^n$.

5. Find the probability of $z_n=w_n$.

6. Find the expected value of $z_n+w_n$.

7. Find the expected value of $z_n{w}_n$.

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设 $z_n$ 和 $w_n$ $(n=0,1,2,\dots )$ 为复数。考虑一个包含两张红牌和一张白牌的袋子。首先，从袋子中取出一张牌并将其放回袋子。$z_{k+1}$ $(k=0,1,2,\dots )$ 的生成方式如下，基于取出的牌的颜色。

$$z_{k+1}=\{\begin{array}{ll}i{z}_k& \text{如果取出的是红牌,}\\ -i{z}_k& \text{如果取出的是白牌.}\end{array}$$

接下来，再次从袋子中取出一张牌并将其放回袋子。$w_{k+1}$ 的生成方式如下，基于取出的牌的颜色。

$$w_{k+1}=\{\begin{array}{ll}-i{w}_k& \text{如果取出的是红牌,}\\ i{w}_k& \text{如果取出的是白牌.}\end{array}$$

这里，每张牌是独立且以相等概率取出的。初始状态为 $z_0=1$ 和 $w_0=1$。因此，$z_n,w_n$ 是从 $z_0=1$ 和 $w_0=1$ 状态开始重复上述两个操作 $n$ 次后的值。这里，$i$ 是虚数单位。

回答下列问题。

1. 证明 $\mathrm{Re}(z_n)=0$ 当 $n$ 为奇数时，并且 $\mathrm{Im}(z_n)=0$ 当 $n$ 为偶数时。这里，$\mathrm{Re}(z)$ 和 $\mathrm{Im}(z)$ 分别表示 $z$ 的实部和虚部。

2. 设 $P_n$ 为 $z_n=1$ 的概率，$Q_n$ 为 $z_n=i$ 的概率。找出 $P_n$ 和 $Q_n$ 的递推方程。

3. 求 $z_n=1$, $z_n=i$, $z_n=-1$, 和 $z_n=-i$ 的概率。

4. 证明 $z_n$ 的期望值为 $(i/3{)}^n$。

5. 求 $z_n=w_n$ 的概率。

6. 求 $z_n+w_n$ 的期望值。

7. 求 $z_n{w}_n$ 的期望值。