112 工程數學(D)¶

說明¶

$\mathbb{R}$：實數集
$\mathbb{C}$：複數集
$\mathrm{i}$：虛數單位
$(a + bi)^\dagger = a - bi$：共軛複數
$A^\dagger$：共軛轉置（conjugate transpose）
$I_n$：$n \times n$ 單位矩陣
$\mathbf{0}$：零向量或零矩陣

第 1 題（Basis and Dimension）¶

This problem concerns the basic definition of vector space, and its basis and dimension.

(a) (5%) Let $\mathbb{Z}_2^n := \{0, 1\}^n$ the set of all $n$-bit strings for any integer $n$. The set $\mathbb{Z}_2^n$ forms a vector space over $\mathbb{Z}_2$. For example, for vectors $101, 001 \in \mathbb{Z}_2^3$ and scalars $0, 1 \in \mathbb{Z}_2$, we have

\[101 + 001 \pmod{2} := (1 \oplus 0)(0 \oplus 0)(1 \oplus 1) = 100 \in \mathbb{Z}_2^3;$$ $$101 \cdot 1 = (1 \cdot 1)(0 \cdot 1)(1 \cdot 1) = 101 \in \mathbb{Z}_2^3;$$ $$101 \cdot 0 = (1 \cdot 0)(0 \cdot 0)(1 \cdot 0) = 000 \in \mathbb{Z}_2^3.\]

What is the dimension of the vector space $\mathbb{Z}_2^n$ over $\mathbb{Z}_2$? Give a basis of the vector space $\mathbb{Z}_2^n$ over $\mathbb{Z}_2$.

(b) (5%) We denote '$\dagger$' by a conjugate transpose. For example:

\[A = \begin{bmatrix} 1 & -2 - \mathrm{i} & 5 \\ 1 + \mathrm{i} & \mathrm{i} & 4 - 2\mathrm{i} \end{bmatrix} \in \mathbb{C}^{2 \times 3};\]

\[A^\dagger = \begin{bmatrix} 1 & 1 - \mathrm{i} \\ -2 + \mathrm{i} & -\mathrm{i} \\ 5 & 4 + 2\mathrm{i} \end{bmatrix}.\]

A (possibly complex) matrix $A$ is Hermitian if and only if $A^\dagger = A$. Consider a vector space $\{A \in \mathbb{C}^{n \times n} : A^\dagger = A\}$ over field of real numbers. What is its dimension?

我的答案¶

(a)

(b)

第 2 題（Matrix Inversion）¶

(a) (5%) Let $A$ be an $n \times n$ non-singular matrix that satisfies $A^3 - 4A^2 + 3A - 5I_n = \mathbf{0}$. Calculate the inverse of $A$ in terms of a polynomial of $A$.

(b) (5%) Let $\omega := e^{\frac{2\pi \mathrm{i}}{n}}$ for some integer $n$ and let

\[B := \begin{bmatrix} 1 & 1 & 1 & 1 & \cdots & 1 \\ 1 & \omega & \omega^2 & \omega^3 & \cdots & \omega^{n-1} \\ 1 & \omega^2 & \omega^4 & \omega^6 & \cdots & \omega^{2(n-1)} \\ 1 & \omega^3 & \omega^6 & \omega^9 & \cdots & \omega^{3(n-1)} \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \omega^{n-1} & \omega^{2(n-1)} & \omega^{3(n-1)} & \cdots & \omega^{(n-1)(n-1)} \end{bmatrix}.\]

Calculate the inverse of $B$. Express your answer in the most simplified form.

我的答案¶

(a)

(b)

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第 3 題（5%）¶

Suppose columns of a matrix $A$ are $n$ vectors in $\mathbb{R}^m$. Answer the following questions.

(a) (True or False) $A$ is an $n \times m$ matrix.

(b) If the columns are linear independent, what is the rank of $A$?

(d) If the columns form a basis for $\mathbb{R}^m$, what can you say about the rank, $n$, and $m$?

(e) Suppose $A$ has rank $r$, it means that $A$ has $r$ ______ columns.

(f) (True or False) The map $T : \mathbb{R}^n \to \mathbb{R}^m$ defined by $T(\mathbf{x}) := A\mathbf{x}$ is a linear transform.

(Getting 5 points if all answers are correct. Otherwise, 0 point.)

我的答案¶

(a)

(b)

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(d)

(e)

(f)

第 4 題（Eigenvalues and Eigenvectors）¶

(a) (5%) Let

\[A := \begin{bmatrix} 3 & 1 & -1 \\ 1 & 3 & -1 \\ -1 & -1 & 3 \end{bmatrix}.\]

Write down the 3 eigenvalues of $A$ with multiplicity in the decreasing order.

(b) (10%) Suppose that an $n \times n$ matrix $B$ satisfies

\[B := \begin{bmatrix} a & b & b & \cdots & b \\ b & a & b & \cdots & b \\ b & b & a & \cdots & b \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ b & b & b & \cdots & a \end{bmatrix},\]

where $a > 0$ and $b > 0$. Write down all the eigenvalues of $B$ with multiplicity in the decreasing order in terms of $a$, $b$, and $n$.

我的答案¶

(a)

(b)

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第 5 題¶

Consider a random variable $X$ and $\mathrm{E}[|X|] < \infty$. Hence, its expectation $\mathrm{E}[X]$ exists. Let us denote $\mathrm{E}[X]$ as $\mu_X$ for notational simplicity. The absolute deviation from the mean is $|X - \mu_X|$, and its expectation is denoted as

\[d_X := \mathrm{E}[|X - \mu_X|].\]

Let $\sigma_X$ denote the standard deviation of $X$ if it exists.

(a) (5%) Suppose the probability density function of $X$, $f_X(t)$, is proportional to $e^{-\lambda|t|}$, for some $\lambda > 0$. Derive $d_X$ in terms of $\sigma_X$.

(b) (5%) Let $X$ be a normal random variable. Derive $d_X$ in terms of $\sigma_X$.

我的答案¶

(a)

(b)

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第 6 題¶

Let $X$ be continuous random variable with cumulative distribution function $F_X(t)$, $t \in \mathbb{R}$ and probability density function $f_X(t)$, $t \in \mathbb{R}$. Furthermore, $f_X(t) = f_X(-t)$ for any $t \in \mathbb{R}$, and $\mathrm{E}[X^2] < \infty$. Let $Y$ be another random variable, independent of $X$, that takes values at $1$ or $-1$ with equal probability, that is,

\[Y = \begin{cases} 1, & \text{with probability } 1/2 \\ -1, & \text{with probability } 1/2 \end{cases}\]

Let $Z = XY$, the product of $X$ and $Y$.

(a) (5%) Are $X$ and $Z$ correlated? Justify your answer rigorously by deriving the covariance between $X$ and $Z$.

(b) (5%) Are $X$ and $Z$ independent? Justify your answer rigorously by deriving the joint cumulative distribution function of $X$ and $Z$.

我的答案¶

(a)

(b)

第 7 題¶

Let $U$ be an uniform random variable over the interval $(0, 1)$. Given $U = u$, $X_1, X_2, \ldots$ are independent and identically distributed Bernoulli $u$ random variables. Let $W_n$ denote the number of "1"s in the length-$n$ sequence $(X_1, X_2, \ldots, X_n)$.

(a) (5%) Derive the conditional probability mass function of $(X_1, X_2, \ldots, X_n, W_n)$ given $U$:

\[P_{X_1, X_2, \ldots, X_n, W_n | U}(x_1, x_2, \ldots, x_n, w | u).\]

(b) (5%) Derive the conditional probability mass function of $(X_1, X_2, \ldots, X_n)$ given $W_n$:

\[P_{X_1, X_2, \ldots, X_n | W_n}(x_1, x_2, \ldots, x_n | w).\]

(d) (5%) Derive the probability mass function of $W_n$.

(e) (5%) Derive the joint probability mass function of $(X_1, X_2, \ldots, X_n)$:

\[P_{X_1, X_2, \ldots, X_n}(x_1, x_2, \ldots, x_n).\]

我的答案¶

(a)

(b)

©

(d)

(e)