複選題
10. (5%) Please pick the statements below that are true.
(A) The rank of the coefficient mnatrix of a consistent system of linear equations is equal to the ariables in the genera number of basic varia ral solution of the system.
(B) The matrix-vector product of an m X n matrix A and a' vector in Rⁿ is a linear combination of the columns of A.
(C) A subset of Rⁿ containing fewer than n vectors must be linearly independent.
(D) If S₁ and S₂ are finite subsets of R" having equal spans, then S₁and S₂ contain the same number of vectors.
(E) If the columns of an n X n matrix A for a generating set of Rⁿ, then the reduced row echelon form of A is In.

6. (5%) A forced, undamped, and resonant motion of a mass on a spring can be described in the following equation: , where x(t) is the location of the mass. In the beginning, the mass is  located at x(0)=0 and the initial velocity of the mass is equal to I, ie.,， The Laplace transformX(s) of x(t) is: (A)(B)(C) (D)(E)

7. (5%) A semi-infinite plate coincides with the region defined by. The left end is held at temperature: exp(-y), and the right end is held at temperature zero for y 2 0. The bottom of the plate is insulated. (A)g(a)=1+α,(B)f(x)=αx,(C)f(x)=α(ε-x),(D) p(y) = sin(αy), (E) p(y)=cos(ay).

複選題8. (5%) A string is stretched and secured on the x-axis at x = 0 and x = 1 for t＞ 0, that is initially held at these points 0.01sin(3tx) and then simultaneously reicased at all points at time t = 0. The string is released from rest from the initial displacement. (Wave equation constant: a)u(x,t) = 0.01f(x)g(t) (A) g(t)=cos(πat),(B) f(x)=sin(3πx),(C) f(x)=sin(3παx), (D) g(t) = cos(3πat), (E) g(t) = sin(3πat).

9. (5%) An undamped string/mass system, in which the mass m = 1 slug and the spring constant k = 10 lb/t, is driven by the 2-periodic external force F(t) = 1-t,0 ＜t＜2； f(t+2) = f(t). Assume that when f(t) is extended to the negative t-axis in a periodic manner, the resulting function is odd. Find a particular solution Xp(t).(A) p(t)=nπt, (B)s=nπ, (C) k=nπ, (D) k=n2n2, (E) s=n2n2,

複選題11. (5%) Please pick the statements below that are true. (A) If A and B are m x n matrices and C is an n X p matrix, then (A + B)C = AC + BC. (B) For any matrices A and B, if A is the inverse of BT, then A is the transpose of B-I. (C) An n x n matrix is invertible if and only if its rows are linearly independent. (D) The codomain of any function is contained in its range. (E) for every standard vector e；, then f(v) mg(v) for every v in Rn.

複選題12. (5%) Please pick the statements below that are true. (A) A function is onto if its range equals its domain. (B) A linear transformation is one-to-one if and only if every vector in its range is the image of a unique vector in its domain. (C) For any m X n matrices A and B and any scalars c and d, (CA + dB)T = cAT + dBT. (D) No scaling operations are required in the forward pass of Gaussian elimination. (E) If A and B are invertible n X n matrices, then A + B is invertible.

13. (5%) Consider the matrix Which of the following values of c will make the matrix A invertible? (A) 3； (B) 4； (C) 5； (D) 6； (E) None of the above.

14. (5%) Which of the following subsets of Rnis a subspace? (A) (B) (C) (D) (E)

複選題15. (5%) Which of the following statements are correct? (A) A basis for a subspace is a linearly independent subset of the subspace that is as large as possible. (B) If V is a subspace of dimension k, then every generating set for V contains exactly k vectors. (C) is any basis for Rn, then there exists a vector v so that v can not be expressed as a linear combination of bi, ba...., bn. (D) Let T be a linear operator on Rn, and u and V be two bases forRn. Denote [T]Bas the representative matrix of T with respect to any basis B for Rn. Then, [T]u and [T]v are similar. (E) If B is a basis for Rn and T is the identity operator on R", then [T]B = In.

複選題16. (5%) Which of the following statements are correct? (A) If λ is an eigenvalue of an n x n matrix A, then for any (B) If there exists some such that Av =λv, then λ is an eigenvalue of A. (C) If two matrices have the same characteristic polynomials, then they have the same eigen- values. (D) A diagonal matrix is always diagonalizable. (E) Let A and P be n X n matrices. If the columins of P form a set of n linearly independent eigenvectors of A, then AP is a diagonal matrix.