110年 - 110 國立臺灣大學_碩士班招生考試_資訊工程學研究所:數學(含線性代數、離散數學)#102151

科目：台大◆資工◆數學(含線性代數、離散數學) | 年份：110年 | 選擇題數：2 | 申論題數：9

試卷資訊

所屬科目：台大◆資工◆數學(含線性代數、離散數學)

選擇題 (2)

9. (10%) Which of the following statements are true?(A) Let A and B be n✖ n matrices. Assume that A is invertible and B³ = 0. If AB=BA, then A+B is also invertible. (B) Let S be the set of all sequences in R^∞ which have exactly N non-zero elements where N is a known constant. S is a subspace of R^∞.
(C) Any square matrix A can be represented as a sum of a symmetric matrix and a skew-symmetric matrix.
(D) If A and B are similar, they have the same eigenvectors.
(E) If A, B

rank (A- B) ≤ rank( A )-rank( B )

複選題
10.(10%) Which of the following statements are true?
(A) Let D:V -V be defined by D

, where V is the space of functions with basis {sint,cost}. The eigenvalues of D are ±1. (B) F is linear if F:R³ → R² is defined by F(x,y,2) = (x + 1,y + 2). (C) Assume e that

has two distinct eigenvalues, λ₁ andλ₂. If dim(

)=n-1, A is diagonalizable. (D)

(E) Let Tbe a diagonalizable linear operator on a finite-dimensional vector space, and let m be any positive integer. Tand T^m are simultancously diagonalizable.

申論題 (9)

1. (10%) A graph is bipartite if and only if it has no cycle of______length. A bipartite graph can be colored with ________colors.

2. (10%) If✖___________log₂n.

3.(10%) There are___________ ways a binomial random walk, starting at the origin, returns to the origin the first time after exactly 2n steps.

4.(10%) With 0≤m,n, A =__________ and B =_____________in

5. (10%) Let (G,o) be a finite group with subgroup H. Prove that |H| divides IG].

(a) (5%) Suppose U and V are distinct six-dimensional subspaces of a vector space W and dimW = 8. Find dim(

(b) (5%) Let V be a vector space of all symmetric 2 x 2 matrices.
Define a linear transformation T:V → P₂(R) by

Find rank(T).

7. (10%) Suppose is the minimal polynomial of a nonsingular n X n matrix A. Find .

8. (10%) Solve for t if det=0.