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

(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^TB)

(c) A triangular matrix is nonsingular if and only if its diagonal entries are all nonzero.

(d) If x₁, x₂, ..., x_k span a vector space V, then they are linearly independent.

(e) Let {u₁, u₂}, {v₁, v₂}, and {w₁, w₂} be bases for \( \mathbb{R}^2 \). If X is the transition matrix corresponding to a change of basis from {u₁, u₂} to {v₁, v₂} and Y is the transition matrix corresponding to a change of basis from {v₁, v₂} to {w₁, w₂}, then Z = XY is the transition matrix corresponding to a change of basis from {u₁, u₂} to {w₁, w₂}.

(f) If L: V → V is a linear transformation and L(v₁) = L(v₂), then v₁ - v₂ ∈ ker(L).

(g) Any two matrices with the same trace are similar.

(h) If {u₁, u₂, ..., u_k} is an orthonormal set of vectors in R^n and U = (u₁, u₂, ..., 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 λ₁, λ₂, ..., λ_n be the eigenvalues of matrix A, then det(A) = λ₁ + λ₂ + ... + λ_n.

2. (10%) Let A = [1 1; 1 1; 0 0]
Compute the singular values and singular value decomposition of A.

(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^-1b.

(e) Solve x1 of the linear system using Cramer's Rule.

(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.

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 $y \equiv 4 \pmod{15}$, $y \equiv 7 \pmod{21}$, $y \equiv 14 \pmod{35}$.