107 工程數學(D)¶
說明¶
- 第 1 題至第 5 題為單選題，每題 10 分
- 第 6 題至第 8 題為非選擇題
- \(\mathbb{R}\)：實數集
- \(\mathbb{C}\)：複數集
選擇題（50%）¶
第 1 題（10%）— 機率¶
X and Y are independent continuous random variables with probability density functions (PDFs) shown below. What is the probability of \(P(-1 < X + Y < 1)\)?
（圖示：X 的 PDF 在 [-1, 1] 上為均勻分布，Y 的 PDF 在 [-2, 2] 上為均勻分布）
(A) ¼ (B) ⅔ (C) ¾ (D) ⅝ (E) none of the above.
我的答案¶
第 2 題（10%）— 機率¶
X and Y are random variables with a joint PDF: \(P(X=x, Y=y) = e^y\) for \(x \geq 0\) and \(y \geq x\).
What are the probability densities of \(P(X=0.5 | Y=3)\), \(P(X=2 | Y=3)\), and \(P(X=2.5 | Y=3)\), respectively?
(A) ⅓, ⅓, and 0 (B) 0, ⅓, ⅓ (C) ⅓, 0, and ⅓ (D) ⅓, ⅓, and ⅓ (E) none of the above.
我的答案¶
第 3 題（10%）— 機率¶
X is a continuous random variable as given in Problem 1. Let \(Y = -2X^2\). Please derive the cumulative distribution function (CDF) of \(Y\), i.e., \(F(Y)\), and calculate \(F(-3)\), \(F(-1)\), and \(F(1)\), respectively.
(A) 0, \(1-(1/2)^{1/2}\) and 1 (B) 0, \(2^{1/2}-1\), and 1 (C) \(1/2\), \(2^{1/2}/2\), and 1 (D) 0, \(2^{1/2}/2\), and 0 (E) none of the above.
我的答案¶
第 4 題（10%）— 機率：擲筊¶
Bob goes to a temple for some advices from God. Before he can make a request, he needs to get an approval first by tossing a pair of fair crescent-shaped "coins." If both coins are "head", it is called a smile-cup and he can try again. If both are "tail", it is called a no-cup, and he has to stop and leave the temple. If one of the coins is "head" and the other is "tail", it is called an approval. Based on this ritual, what is the probability that he gets an approval from God?
(A) ½ (B) ⅝ (C) ⅔ (D) ¾ (E) none of the above.
我的答案¶
第 5 題（10%）— 機率¶
Based on Problem 4, how many tosses on average Bob made before he leaves the temple (either he is rejected or gets an approval)?
(A) 1 (B) 4/3 (C) 3/2 (D) 2 (E) none of the above.
我的答案¶
非選擇題（50%）¶
第 6 題（21%）— 線性代數：線性變換與對角化¶
Let \(V\) be an \(n\)-dimensional vector space associated with the field of the real number set \(\mathcal{R}\). Consider a linear transformation \(T : V \to V\). Suppose there exist \(n\) vectors \(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n \in V\) that are linearly independent, each of them being an eigenvector of \(T\). In other words, for any \(k \in \{1, 2, \ldots, n\}\),
for some \(\lambda_k \in \mathcal{R}\).
(a) (7%) Let \(\mathbf{u} = \mathbf{x}_1 + 2\mathbf{x}_2 + \cdots + n\mathbf{x}_n\). Find \(T(\mathbf{u})\) and express it in terms of \(\lambda_k\)'s and \(\mathbf{x}_k\)'s. Justify your answer.
(b) (7%) Prove or disprove that \(T\) is diagonalizable, that is, there exist an ordered basis \(\mathcal{B} = \{\mathbf{b}_1, \ldots, \mathbf{b}_n\}\) and a diagonal matrix \(D\) such that for any \(\mathbf{v} \in V\), \([T(\mathbf{v})]_\mathcal{B} = D[\mathbf{v}]_\mathcal{B}\), where \([\mathbf{x}]_\mathcal{B} = [c_1, c_2, \ldots, c_n]^T \in \mathcal{R}^n\) where \(c_k\)'s are the unique set of scalars that satisfies \(\mathbf{x} = c_1\mathbf{x}_1 + c_2\mathbf{x}_2 + \cdots + c_n\mathbf{x}_n\) for any \(\mathbf{x} \in V\).
© (7%) Find the necessary and sufficient condition of the scalars \(\lambda_k\)'s such that \(T\) is an invertible operation (i.e., there exists \(U : V \to V\) such that \(T(U(\mathbf{x})) = \mathbf{x}\) and \(U(T(\mathbf{x})) = \mathbf{x}\) for any \(\mathbf{x} \in V\)).
我的答案¶
(a)
(b)
©
第 7 題（9%）— 線性代數：內積空間¶
Consider the vector space \(\mathcal{R}^4\) over the field \(\mathcal{R}\) with standard addition
and standard scalar multiplication
for any \(\mathbf{u} = \begin{bmatrix} u_1 & u_2 & u_3 & u_4 \end{bmatrix}^T\), \(\mathbf{v} = \begin{bmatrix} v_1 & v_2 & v_3 & v_4 \end{bmatrix}^T \in \mathcal{R}^4\) and \(c \in \mathcal{R}\). Suppose we define
Is \(\langle \cdot, \cdot \rangle\) an inner product of the vector space \(\mathcal{R}^4\)? Why?
我的答案¶
第 8 題（20%）— 線性代數：行簡化與線性獨立¶
Consider an \(m \times n\) matrix \(A = \begin{bmatrix} \mathbf{a}_1 & \mathbf{a}_2 & \cdots & \mathbf{a}_n \end{bmatrix}\) and let its reduced row echelon form be \(R = \begin{bmatrix} \mathbf{r}_1 & \mathbf{r}_2 & \cdots & \mathbf{r}_n \end{bmatrix}\). Then, there exists an invertible matrix \(P\) of size \(m \times m\) such that \(PA = R\).
(a) (10%) Suppose a subset of column vectors of \(A\), \(\{\mathbf{a}_{p_1}, \mathbf{a}_{p_2}, \cdots, \mathbf{a}_{p_k}\}\), is linearly independent. Here \(k\) is an integer, \(k \leq n\), and \(1 \leq p_1 < p_2 < \cdots < p_k \leq n\). Show that \(\{\mathbf{r}_{p_1}, \mathbf{r}_{p_2}, \cdots, \mathbf{r}_{p_k}\}\) is also linearly independent.
(b) (10%) Show that \(A\mathbf{x} = \mathbf{b}\) is consistent for all \(\mathbf{b} \in \mathcal{R}^m\) if and only if \(\text{rank } A = m\) ("if": 5%; "only if": 5%).
我的答案¶
(a)
(b)