106 工程數學(D)¶

說明¶

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

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) Similar matrices have the same characteristic polynomial.

(b) If \(A^T = -A\), then \(\det A = 0\).

© Let \(A = [\mathbf{a}_1 \ \mathbf{a}_2 \ \mathbf{a}_3 \ \mathbf{a}_4]\) and \(A' = [\mathbf{a}_1 \ \mathbf{a}_4 \ \mathbf{a}_2 \ \mathbf{a}_3]\). If \(A\mathbf{x} = \mathbf{b}\) is consistent, then \(A'\mathbf{x} = -2\mathbf{b} + 5\mathbf{a}_2\) is also consistent.

(d) If \(\lambda\) is an eigenvalue of \(A^2\), then \(\lambda\) is also an eigenvalue of \(A\).

(e) Let \(V\) and \(W\) be subspaces of \(\mathcal{R}^n\). If \(V^\perp = W^\perp\), then \(V = W\).

(f) Let \(R\) be the reduced row echelon form of \(A\). Then the reduced row echelon form of \([A \ A]\) is \([R \ 0]\).

(g) If \(A\) is a nonzero symmetric matrix, then \(A^2\) is also a nonzero symmetric matrix.

(h) Let \(A\) be an \(m \times n\) matrix and \(\mathbf{b}\) be a vector in \(\mathcal{R}^m\). If \(A\mathbf{x} = \mathbf{b}\) has a unique solution, then \(n \geq m\).

(i) If two vectors spaces are isomorphic, then they have the same dimension.

(j) Let \(\mathcal{S} = \{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}\) be a linearly independent subset of \(\mathcal{R}^n\). Let \(\mathbf{v}\) and \(\mathbf{w}\) be \(n \times 1\) vectors. If \(\mathbf{v} \cdot \mathbf{v}_i = \mathbf{w} \cdot \mathbf{v}_i\) for \(i = 1, 2, \ldots, n\), then \(\mathbf{v} = \mathbf{w}\).

我的答案¶

(a)

(b)

©

(d)

(e)

(f)

(g)

(h)

(i)

(j)

第 2 題（15%）— 正交補與最小距離¶

Let \(W = \text{Span } \{\mathbf{v}_1, \mathbf{v}_2\}\) where

\[\mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \\ -1 \\ 1 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 2 \\ 1 \\ -1 \\ 0 \end{bmatrix}.\]

(a) (5%) Find a basis for \(W^\perp\).

(b) (5%) Find an orthonormal basis for \(W^\perp\).

我的答案¶

(a)

(b)

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第 3 題（15%）— 線性變換與對角化¶

Let \(T\) be a linear operator on \(\mathcal{P}_2\) defined by

\[T(p(x)) = (p(0) + p(1)) + (p'(0) + p(1))x + (p(0) + p'(1))x^2,\]

where \(p'(x)\) is the derivative of \(p(x)\).

(a) (6%) Let \(\mathcal{B} = \{1, x, x^2\}\) be a basis for \(\mathcal{P}_2\). Find \([T]_\mathcal{B}\), the matrix representation of \(T\) with respect to \(\mathcal{B}\).

(b) (9%) Find a basis \(\mathcal{B}'\) for \(\mathcal{P}_2\) such that \([T]_{\mathcal{B}'}\) is a diagonal matrix.

我的答案¶

(a)

(b)

第 4 題（20%）— 機率¶

Provide answers to the following 4 sub-problems. Mathematical derivation is optional.

(a) (5%) Consider the trials of testing each IC for being accepted or rejected. Let \(X\) be the random variable that represents the minimum number of trials needed until there are \(k > 0\) accepted ICs and \(F_X(x)\) be its cumulative distribution function (CDF). Let \(Y\) be the random variable that represents the number of accepted ICs in \(n > 0\) trials and \(F_Y(y)\) be its CDF. Write \(F_X(x)\) in terms of \(F_Y(y)\).

(b) (5%) Assume that packet arrivals in a time duration of \(\tau\) follow the Poisson distribution with expected value \(\lambda\). It is known that there is one packet arrival in a duration of \(\tau\). What is the probability distribution of its arrival time \(T\)?

\[W = \lim_{n \to \infty} n \min(X_1, X_2, \ldots, X_n)\]

be another random variable. Find the probability density function (PDF) of \(W\).

(d) (5%) Let \(X_1, X_2, \ldots, X_N\) be iid Gaussian random variables each with expected value 10 and standard deviation 2, where \(N\) is a Poisson random variable with expected value 5. What is the variance of \(X_1 + X_2 + \cdots + X_N\)?

我的答案¶

(a)

(b)

©

(d)

第 5 題（16%）— 機率：連續性定理¶

The theorem of Continuity of Probability Measure states that for any increasing or decreasing sequence of events, \(\{E_n, n \geq 1\}\),

\[\lim_{n \to \infty} P[E_n] = P\left[\lim_{n \to \infty} E_n\right],\]

where \(P[\cdot]\) is a probability measure that satisfies the axioms of probability.

(a) (6%) Prove the theorem of Continuity of Probability Measure.

(b) (10%) What is the consequence of the Continuity of Probability Measure? State a theorem whose proof directly relies on the theorem of Continuity of Probability Measure. Then, prove the theorem that you stated.

我的答案¶

(a)

(b)

第 6 題（14%）— 機率：最佳估計¶

Let \(X\) and \(Y\) be two jointly distributed random variables with the joint PDF as follows:

\[f_{X,Y}(x, y) = \begin{cases} 2, & 0 \leq x \leq 1, 0 \leq y \leq x, \\ 0, & \text{otherwise.} \end{cases}\]

For some reason, samples of random variable \(X\) can be observed, but not for random variable \(Y\). It is desired to estimate \(Y\) based on the observation of \(X\).

(a) (7%) Find the best linear estimate (i.e. \(aX + b\) with \(a\), \(b\) being constants to be determined) of \(Y\) given \(X\) such that the mean square estimation error is minimized. What is the minimum mean square error thus found?

(b) (7%) Find the best estimate (that is not necessarily linear) of \(Y\) given \(X\) such that the mean square estimation error is minimized. What is the minimum mean square error thus found?

我的答案¶

(a)

(b)