114年 - 114 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#147271

科目：【阿摩】未分類題庫 | 年份：114年 | 選擇題數：0 | 申論題數：22

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所屬科目：【阿摩】未分類題庫

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申論題 (22)

1. (20%) True or False. In each case, answer T if the statement is always true and F otherwise. In the case of a false statement, explain why or give an example to show that the statement is not always true.

(a) If the row reduced echelon form of matrix A involves free variables, then the system Ax=b will have infinitely many solutions.
(b) det(AB^T) = det(A^T B)
(c) A triangular matrix is nonsingular if and only if its diagonal entries are all nonzero.
(d) If x_1, x_2, ..., x_k span a vector space V, then they are linearly independent.
(e) Let {u_1, u_2}, {v_1, v_2}, and {w_1, w_2} be bases for R^2. If X is the transition matrix corresponding to a change of basis from {u_1, u_2} to {v_1, v_2} and Y is the transition matrix corresponding to a change of basis from {v_1, v_2} to {w_1, w_2}, then Z = XY is the transition matrix corresponding to a change of basis from {u_1, u_2} to {w_1, w_2}.
(f) If L: V → V is a linear transformation and L(v_1) = L(v_2), then v_1 - v_2 ∈ ker(L).
(g) Any two matrices with the same trace are similar.
(h) If {u_1, u_2, ..., u_k} is an orthonormal set of vectors in R^n and U = (u_1, u_2, ..., u_k), then UU^T = I_n (the n × n identity matrix).
(i) If A is symmetric and det(A) > 0, then A is positive definite.
(j) Let λ_1, λ_2, ..., λ_n be the eigenvalues of matrix A, then det(A) = λ_1 + λ_2 + ... + λ_n.
2. (10%) Let A = [1 1; 1 1; 0 0]

Compute the singular values and singular value decomposition of A.
3. (10%) Given the following linear system Ax = b:

x_1 + 4x_2 + 3x_3 = 12
-x_1 - 2x_2 = -12
2x_1 + 2x_2 + 3x_3 = 8

(a) Compute A^{-1}.
(b) Compute the LU factorization of A.
(c) Use the LU factorization to determine the value of det(A).
(d) Solve the linear system using x = A^{-1}b.
(e) Solve x1 of the linear system using Cramer's Rule.
4. (10%) Let $L$ be a linear operator on $\mathbb{R}^2$, and let $v_1 = \begin{bmatrix} 1 \\ 3 \end{bmatrix}$, $v_2 = \begin{bmatrix} -3 \\ 5 \end{bmatrix}$, $v_3 = \begin{bmatrix} -9 \\ 1 \end{bmatrix}$.

(a) Are \(\pmb{v}_1, \pmb{v}_2\), and \(\pmb{v}_3\) linearly independent?
(b) If \( L(\pmb{v}_1) = \begin{bmatrix} 2 \\ 4 \end{bmatrix} \) and \( L(\pmb{v}_2) = \begin{bmatrix} 3 \\ 5 \end{bmatrix} \), find the value of \( L(\pmb{v}_3) \).
5. (10%) Solve the recurrence relation \( a_{n} = a_{n - 1} + 2n + 3 \) where \( a_0 = 5 \).
6. (10%) Simplify the following compound proposition to minimize the number of "and", "or", and "not" operations.

$$(p \wedge q \wedge r) \vee (\neg p \wedge \neg r \wedge \neg s) \vee (\neg p \wedge \neg q \wedge s) \vee (\neg p \wedge q \wedge s) \vee (\neg p \wedge r \wedge \neg s)$$
7. (10%) Consider the following program segment. Let $n$ be a positive integer. Find the final values of $a$ and $b$ as functions of $n$.

$a \leftarrow 1$
$b \leftarrow 2$
while $(b \le n)$ do
{
$a \leftarrow a + 3$
$b \leftarrow b \times b \times b$
}
8. (20%) Find all integer solutions for $x$ and $y$ such that $x \equiv 2 \pmod{3}$, $x \equiv 4 \pmod{5}$, $x \equiv 6 \pmod{7}$ and $y \equiv 4 \pmod{15}$, $y \equiv 7 \pmod{21}$, $y \equiv 14 \pmod{35}$.