109年 - 109 國立臺北大學_碩士班一般入學考試試題_資訊工程學系：線性代數與離散數學#147254

科目：【阿摩】未分類題庫 | 年份：109年 | 選擇題數：0 | 申論題數：12

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所屬科目：【阿摩】未分類題庫

選擇題 (0)

申論題 (12)

1. (10%) Consider an RSA cryptosystem with p = 11, q = 29, and public-key (e, n) = (3, 319). What is the value of d used in the secret-key? What is the encryption of the message M = 100?
2. (20%) Consider the graph below. Answer the questions and prove your answer. Does it have an Euler circuit? Does it have an Euler path? Does it have a Hamilton circuit? Does it have a Hamilton path?
3. (20%) Answer the following questions.

(a) In how many ways are there to arrange the letters of the word "APPLE"?
(b) How many of these ways start or end with the letter P?
(c) Find the number of solutions to \( x + y + z = 25 \), where \( x, y \), and \( z \) are nonnegative integers.
(d) Answer part (c), but assume that \( x \geq 3 \) and \( y \geq 8 \).
4. Give the matrix A = $$\begin{bmatrix} -6 & -6 & 6 \\ 3 & -3 & -3 \\ 2 & 6 & 2 \end{bmatrix}$$.

(a) \((10\%)\) Compute the LU decomposition of A.
(b) (5%) Determine the nullspace of A.
(c) \((5\%)\) Give a basis for the column space of A.
5. Give the matrix B = $$\begin{bmatrix} 1 & 0 \\ 0 & 2 \\ 0 & 0 \end{bmatrix}$$.

(a) \((10\%)\) Find the eigenvalues and the eigenvectors of \(\mathbf{B}^{\mathrm{T}}\mathbf{B}\)
(b) \((10\%)\) Find the singular value decomposition of \(\mathbf{B}\).
6. (10%) Assume that both C and D are n×n matrices. Determine whether L(D) = CD³ is a linear operator from Rⁿⁿ to Rⁿⁿ.