101年 - 101 國立交通大學_碩士班考試入學試題_資訊聯招：線性代數與機率#113290

科目：研究所、轉學考(插大)◆線性代數與機率 | 年份：101年 | 選擇題數：10 | 申論題數：7

試卷資訊

所屬科目：研究所、轉學考(插大)◆線性代數與機率

選擇題 (10)

複選題
1. Consider a 3x3 matrix A with non-zero elements. Assume A^TAx=0 has non-trivial solutions with two of them being x=[1,1,0]^T and [1,0,0]^T. Which of the following statements are true for b,y ∈ R³? (A) Ay =b must have no solution for some b and exactly one solution for other b. (B) A^Ty=[0,0,5]^T must have at least one exact solution. (C) The least-squares approximation error for the system A^Ty =[0,2,2]^T is 4. (D) Rank of A =2. (E) A must have zero rows.

複選題

2. Suppose has a reduced row echelon form Which of the following statements are true for x,b ∈ R³? (A) Ax=[3,1,0]^T has infinitely many solutions. (B) Ax=[0,1,0]^T has no solution. (C) In this case, Column Space of A = Column Space of U. (D) The matrix A^TA is invertible. (E) The projection of x=[2,1,11]^T onto the Column Space of A is the vector [2,1,0]^T .

複選題

3. Suppose D=AB with Which of the true? (A) Null Space of D = Null Space of B if and only if a≠5. (B) Column Space of D = Column Space of A if and only if b≠2. (C) There exists values of a, b such that Rank of D=1. (D) Rank of D=2 only if a=5 and b=2. (E) Rank of D=3 only if a≠5 and b≠2.

4.Which set is not a vector space? (A) The set of polynomials of degree exactly 3. (B) The set of polynomials having'degree 0, 1, 2, or 3, together with the 0-polynomial. (C) The set of all vectors (x₁, x₂, x₃, x₄) in R⁴ such that x₂ = 5x₃ - 7x₄. (D) The set of all vectors of the form [a+b³, a-b³, 2a] in R³. (E) The set of all vectors [a+b³, a-b³] in R².

複選題
5. In a vector space V, we have the two sets S = {u1, u2,..., un3 and Q= {V₁, V₂,..., V_m}. Which of these statements can be true? (A) S is linearly independent, Q spans V, and n ＞ m. (B) V has dimension n, Q spans Y, and m ≥ n. (C) S and Q are bases for V, and n = m. (D) S is linearly independent, Q is a subset of S and Q is linearly dependent. (E) S is linearly dependent, Q is a subset of S and Q is linearly independent.

複選題

6. ConsiderEach is obtained from the other by rearranging columns. Which statement is correct? (A) Ker（A）=Ker（B）. (B) Range（A）= Range（B）. (C) A and B have the same reduced row echelon form. (D) Column space of A = Column space of B. (E) Row space of A= Row space of B.

7. Let A be an m x n matrix whose null space has dimension k. Which conclusion is correct? (A) The dimension of Null(A^T) is k. (B) The dimension of row space of A is m-k. (C) The dimension of column space of A is m-k. (D) The dimension of row space of A is n-k. (E) The dimension of column space of A is n-k.

8. Let matrices A, B, and C be square matrices. Choose the incorrect arguments (A) The determinant det(AB) = det（A）det（B） (B) Let matrix A can be decomposed into A=QR, where Q is the orthogonal matrix and R is the upper triangular matrix. The determinant det（A） = det(R). (C) Let matrix A be diagonalizable； that is, A=X^-1DX. The determinant det（A）=det（D）. (D) Let matrices A and B are similar. The determinant det（A）=det（B）. (E) Let matrix B be the Hermitian transpose of matrix A； that is B=A^H. The determinant det（A）=det（B）.

9. Choose the incorrect arguments. (A) Let A be the Hermitian matrix. Then, matrix A is diagonalizable； that is, A=X^-1DX. (B) For a square matrix A, the eigenvectors correspond to different eigenvalues are linearly independent. (C) Two similar matrices have the same characteristic polynomial. (D) Let A be an mxn real matrix. Then A^TA is diagonalizable. (E) Let matrix A be diagonalizable. Then, matrix A is not singular.

10. Let A be an nxn matrix and b be an arbitrary nx1 vector. Select the incorrect arguments. (A) If determinant det（A）≠ 0, then the system equation Ax=b has exactly one single solution. (B) If matrix A is diagonalizable, then the system equation Ax=b has exactly one single solution. (C) Let matrix A can be decomposed into A=QR, where Q is the orthogonal matrix and R is the upper triangular matrix: The system equation Ax=b has exactly one single solution. (D) Let matrix A have n independent eigenvectors. The matrix A should have n distinct eigenvalues. (E) The system equation Ax=b can be solved by Cramer's rule either the system is consistent or inconsistent.

申論題 (7)

11 . A stack of cards consists of seven red and five blue cards. A second stack of cards consists of nine red cards. A stack is selected at random and three of its cards are drawn.If all of them are red, what is the probability that the first stack was selected? (Show your calculation steps.) (10 points)

12.(a) If two fair dice are rolled 4 times, what is the probability of at least one 6 (on either die) in exactly two of these 4 rolls? (5 points) (b) Suppose that, on average, in every two pages of a book there is one typographical error.What is the probability of at least one error on a specific page of the book? (5 points)

13. Let the density function of X be

Find the density functions of and Z=e^-2X. (10 points)

(a) Determine the value of c. (4 points)

(b) Find the marginal probability density functions of Y. (6 points)

(a) If X is a normal random variable with parameters μ = 3 and σ² =16,find P(1＜X＜11).(4 points)

(b) If 48 random numbers are selected independently from the interval (0, 1),what is the approximate probability that the sum of these numbers is at least 20? (6 points)