110 工程數學(D)¶

說明¶

\(\mathbb{R}\)：實數集
\(\mathbb{C}\)：複數集
\(T\)：轉置
b：行向量
Column space：\(A\) 的所有線性組合
Nullspace：\(A\mathbf{x} = \mathbf{0}\) 的所有解
Row space = \(A^T\) 的 column space
Left nullspace = \(A^T\) 的 nullspace

第 1 題（20%）— 是非題¶

Label the following statements as being true or false. (No explanation is needed. Each correct answer gets 2% and each wrong answer gets 0%):

(a) Let \(S\) be a nonempty subset of \(\mathcal{R}^n\). Then \((S^\perp)^\perp = \text{Span } S\).

(b) Let \(S_1\) be a linearly independent subset of \(\mathcal{R}^n\) and \(S_2\) be a generating set for \(\mathcal{R}^n\). Then \(S_1\) cannot have more vectors than \(S_2\).

(d) Let \(A_1\) and \(A_2\) be \(m \times n\) matrices. If \(A_1\mathbf{x} = \mathbf{b}_1\) and \(A_2\mathbf{x} = \mathbf{b}_2\) are consistent, then \(\begin{bmatrix} A_1 \\ A_2 \end{bmatrix}\mathbf{x} = \begin{bmatrix} \mathbf{b}_1 \\ \mathbf{b}_2 \end{bmatrix}\) is consistent.

(e) Let \(V\) be a finite dimensional inner product space and let \(\mathcal{B}\) be a basis for \(V\). Then \(\langle f, g \rangle = [f]_\mathcal{B} \cdot [g]_\mathcal{B}\), for any \(f, g \in V\).

(f) There exists a \(3 \times 3\) diagonalizable matrix whose characteristic polynomial is given by \(t^3 - t^2 - 2t\).

(g) Let \(\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}\) be a basis for \(\mathcal{R}^n\). Let \(A\) be \(n \times n\). If \(\|A\mathbf{v}_i\| = \|\mathbf{v}_i\|\), for \(i = 1, 2, \ldots, n\), then \(A\) is orthogonal.

(h) Let \(T : \mathcal{R}^{99} \to \mathcal{R}^{100}\) be linear. Then there exist a pair of distinct vectors \(\mathbf{v}_1, \mathbf{v}_2 \in \mathcal{R}^{99}\) such that \(T(\mathbf{v}_1) = T(\mathbf{v}_2)\).

(i) Let \(Q\) and \(A\) be \(m \times m\) and \(m \times n\) matrices respectively, and \(\mathbf{b} \in \mathcal{R}^m\). Let \(\mathcal{S}_1\) and \(\mathcal{S}_2\) be solution sets to \(A\mathbf{x} = \mathbf{b}\) and \(QA\mathbf{x} = Q\mathbf{b}\) respectively. Then \(\mathcal{S}_1\) is a subspace of \(\mathcal{S}_2\).

(j) Let \(W_1\) and \(W_2\) be subspaces of \(\mathcal{R}^n\). If \(\dim W_1 + \dim W_2 > n\), then \(W_1 \cap W_2\) is a nonzero subspace of \(\mathcal{R}^n\).

我的答案¶

(a)

(b)

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(d)

(e)

(f)

(g)

(h)

(i)

(j)

第 2 題（15%）¶

Let \(V\) be the set of all \(2 \times 2\) symmetric matrices and \(B = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}\). Let \(T\) be a linear operator on \(V\) defined by \(T(A) = B^TAB\), for each \(A \in V\).

(a) (3%) Find a basis for \(V\).

(b) (12%) Find the eigenvalues of \(T\).

我的答案¶

(a)

(b)

第 3 題（15%）¶

Let \(\mathcal{P}_2\) be the set of all polynomials with degree less than equal to 2. For any \(p_1(x), p_2(x) \in \mathcal{P}_2\), their inner product is defined by

\[\langle p_1(x), p_2(x) \rangle = \int_{-1}^{1} p_1(x)p_2(x) \, dx.\]

Let \(W = \{1, x\}\) and \(p(x) = x^2\). Find the unique polynomials \(q(x) \in W\) and \(r(x) \in W^\perp\) such that \(p(x) = q(x) + r(x)\).

我的答案¶

第 4 題（8%）— 機率：硬幣選擇¶

You have two coins. Coin A comes up heads with probability ¼. Coin B comes up heads with probability ½. You choose one of these coins randomly and then you flip it 3 times.

(a) (4%) What is the probability that you observe at least 2 heads?

(b) (4%) If you observe 3 tails, what is the probability that you have chosen Coin A?

我的答案¶

(a)

(b)

第 5 題（24%）— 機率：聯合分布¶

The random variables \(X\) and \(Y\) have the joint probability density function

\[f_{X,Y}(x, y) = C \times \exp\left(-\frac{4x^2 + y^2}{8}\right),\]

where \(-\infty < x < \infty\) and \(-\infty < y < \infty\).

(a) (4%) Find the constant \(C\).

(b) (4%) Find the conditional probability density function \(f_{X|Y}(x|y)\).

(d) (8%) Let \(Z = 3X + 2Y\). Derive the moment generating function \(\phi_Z(s) = \mathrm{E}[e^{sZ}]\). You have to give the derivation, not just the answer.

我的答案¶

(a)

(b)

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(d)

第 6 題（18%）— 機率：電話費計費¶

You make \(N\) phone calls. The length of the \(i\)th phone call, denoted by \(T_i\) in minutes, is an exponential random variable with expected value \(1/\lambda\), where the parameter \(\lambda > 0\). The random variables \(T_1, T_2, \ldots, T_N\) are independent and identically distributed. For any fraction of a minute at the end of a call, the phone company charges for a full minute. In other words, the phone company calculates its charge based on \(W_i = \lceil T_i \rceil\) for the \(i\)th phone call. Here the ceiling function of a real number \(x\), denoted by \(\lceil x \rceil\), is the least integer number greater than or equal to \(x\). For instance, \(\lceil 3.1 \rceil = 4\) and \(\lceil 3 \rceil = 3\).

(a) (3%) What is the probability density function of \(T_1\)?

(b) (7%) Derive the probability mass function of \(W_1\). You have to give the derivation, not just the answer.

我的答案¶

(a)

(b)

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