108 工程數學(D)¶

說明¶

Linear Algebra（50%）+ Probability（50%）
是非題每題正確得 2%，錯誤得 0%
\(\mathbb{R}\)：實數集
\(\mathbb{C}\)：複數集

Linear Algebra（50%）¶

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 \(A\) be an \(n \times n\) matrix with real entries. If \(A\) is symmetric, then all the eigenvalues of \(A\) are real.

(b) Let \(A\) be an \(n \times n\) matrix. If \(A\) is diagonalizable, then \(A\) has \(n\) distinct eigenvalues.

(d) Let \(S\) be a subspace of \(\mathcal{R}^n\) and \(S^\perp\) be the orthogonal complement of \(S\). Then \(\dim(S) + \dim(S^\perp) = n\).

(e) Let \(A\) and \(B\) be \(2 \times 2\) matrices. Then \(\text{rank}(AB) = \text{rank}(BA)\).

(f) Let \(A\) and \(B\) be \(n \times n\) matrices. If \(A\) is nonsingular and \(B\) is singular, then \(A + 2B\) is nonsingular.

(g) Let \(A\) and \(B\) be \(n \times n\) matrices. If \(A\) is similar to \(B\), then \(A\) and \(B\) have the same eigenvalues.

(h) If the vectors \(\mathbf{v}_1\), \(\mathbf{v}_2\), and \(\mathbf{v}_3\) are linearly independent, then the following vectors

\[2\mathbf{v}_1 + \mathbf{v}_2 + 2\mathbf{v}_3, \quad \mathbf{v}_2 + \mathbf{v}_3, \quad \mathbf{v}_2 + 2\mathbf{v}_3\]

are also linearly independent.

(i) Let \(\mathbf{v}_1\) and \(\mathbf{v}_2\) be \(n \times 1\) vectors. Then \(\text{rank}(\mathbf{v}_1\mathbf{v}_1^T + \mathbf{v}_2\mathbf{v}_2^T) = 2\).

(j) Let \(A\) be an \(n \times n\) symmetric matrix. Let \(\mathbf{v}\) be an \(n \times 1\) vector. If \(\mathbf{v}^TA\mathbf{v} = 0\), then \(A\mathbf{v} = \mathbf{0}\).

我的答案¶

(a)

(b)

©

(d)

(e)

(f)

(g)

(h)

(i)

(j)

第 2 題（20%）¶

Let the matrix \(A = \begin{bmatrix} 1 & -3 & 4 \\ -3 & 1 & 0 \\ 4 & 0 & 1 \end{bmatrix}\). Then

(a) (6%) Find all the eigenvalues of \(A\).

(b) (6%) Find an orthonormal basis for \(\mathcal{R}^3\), consisting of the eigenvectors of \(A\).

(d) (2%) Find the nullity of \(A\).

我的答案¶

(a)

(b)

©

(d)

第 3 題（10%）¶

Let \(T\) be a linear operator on \(\mathcal{R}^3\) such that

\[T\left(\begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}\right) = \begin{bmatrix} 10 \\ 4 \\ 2 \end{bmatrix}, \quad T\left(\begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix}\right) = \begin{bmatrix} -4 \\ 0 \\ 0 \end{bmatrix}, \quad T\left(\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}\right) = \begin{bmatrix} 7 \\ 2 \\ 1 \end{bmatrix}.\]

(a) (4%) Find \(T\left(\begin{bmatrix} 0 & 0 & 1 \end{bmatrix}^T\right)\).

(b) (6%) Find \(T\left(T\left(T\left(T\left(\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}^T\right)\right)\right)\right)\).

我的答案¶

(a)

(b)

Probability（50%）¶

第 4 題（11%）¶

The mathematical notion of random variable (RV) is very important in probability theory.

(a) What is the mathematical definition of a RV? (3%)

(b) What is the mathematical definition of a probability function? (3%)

© Consider the experiment of throwing a fair 2-sided dice, where the sample space \(S = \{ \circ, : \}\) and \(\text{Prob}(\{ \circ \}) = \text{Prob}(\{ : \}) = 0.5\). Define a random variable \(X\) for this experiment and write down the probability mass function PMF of \(X\), i.e., \(\text{Prob}(X=x)\) for all real \(x \in R\). (5%)

我的答案¶

(a)

(b)

©

第 5 題（20%）— 機率：等計程車¶

You are waiting for a taxi. The inter-arrival time interval between two taxi arrivals is random, say \(T\), and inter-arrival time intervals are independent. You know that \(T\) has an exponential distribution and that the probability density function of \(T\) is parameterized by \(\mu > 0\) as follows:

\[f_T(t) = \begin{cases} \mu e^{-\mu t}, & \text{if } t \geq 0; \\ 0, & \text{if } t < 0. \end{cases}\]

(a) You know that the previous taxi arrival is s time ago. Let \(T_R\) be the remaining time you have to wait for the next taxi arrival. Derive the probability that you will need to wait for no more than a time duration r, knowing that the previous arrival was s time ago. (You need to give the derivation, not just the answer. 10%)

(b) Now (time 0), you decide not to take a taxi but to count the number of taxi arrivals in a period \([0, \tau]\). Let the number of arrivals in \([0, \tau]\) be \(K\), which is a RV. Derive \(\text{Prob}(K = 0)\). (You need to give the derivation, not just the answer. 5%)

我的答案¶

(a)

(b)

第 6 題（15%）¶

X and Y are two random variables with a joint probability density function as

\[f_{XY}(x, y) = \begin{cases} cxy, & 0 \leq x \leq y \leq 2; \\ 0, & \text{otherwise.} \end{cases}\]

(a) Derive the conditional cumulative distribution function \(F_{X|Y}(x|y)\). (5%)

(b) \(\mathrm{E}[\mathrm{E}[X|Y]] = ?\) (5%)

我的答案¶

(a)

(b)

©

第 7 題（9%）— 機率：雷達訊號估計¶

You are observing a radar signal sequence

\[Y_k = \theta + \omega_k, \quad k = 1, 2, \ldots\]

where \(\theta\) is an unknown constant and \(\omega_k\) is \(N(0, 1)\) and independent and identical over time index \(k\). Given \(m\) observations of \(Y_k\), i.e., given \(\{y_1, y_2, \ldots, y_m\}\),

(a) How do you estimate the value of \(\theta\)? (5%)

(b) Is your estimate unbiased, i.e., does the mean of your estimate equal \(\theta\)? Why? (4%)

我的答案¶

(a)

(b)