複選題

13. (5%) Which of the following are correct?
(A) Below vector set S is dependent.
S =
(B) Invertible matrix must be square matrix.
(C) Square matrix must be invertible.
(D) The inverse of matrix


(E) A n x n matrix is invertible if its columns span Rⁿ.

9. (5%) The differential equation x" + 3x' - 5x = sin t is equivalent to the system:(A)x' = 3x - 5y； y' = sin t(B) x' = 5x - 3y； y' = sin t(C) x' = y； y' = 5x - 3y + sin t(D)x' = y； y' = 3x - 5y + sin t(E) None of above

複選題10. (5%) The differential equation is y' = in the standard form, and 2xydx + x²dy = 0 in a differential form. This differential equation is(A) linear(B) homogenous(C) exact(D) separable(E) Bernoulli

複選題11. (10%) Which of the following are correct?(A) The system defined by F(x,y)=(x², x) is linear.(B) The system defined by F(x,y)=(dx/dt, x) is linear.(C) A and B are m x n matrices. If Aw = Bw for all w in Rn, then A = B.(D) The columns of matrix A contains zero vector. If Ax=b have solution, it will have infinite solutions.(E) The zero vector of Rn is within the span of any finite subset of Rn.

複選題12. (10%) For the matrix A, which of the following are correct?A =(A) The rank of A is 4.(B) The rank of A is 3.(C) The dimension of null space of A is 0.(D) The dimension of null space of A is 1.(E) The dimension of null space of A is 2.

複選題14. (10%) For the matrix below, which of the following are correct?(A) -1 is its eigenvalue.(B) 1 is its eigenvalue.(C) -3 is its eigenvalue.(D) Zero vector belongs to its eigenvector.(E) [1 0 2]T is a basis of its eigenspace.

複選題15. (5%) Which of the following are correct?(A) An eigenvalue of a matrix has infinite number of eigenvectors.(B) If det(A – λIn) = 0, λ is the eigenvalue of n x n matrix A.(C) If λ is the eigenvalue of matrix A, the columns of (A -λIn) are independent.(D) If λ is the eigenvalue of matrix A, there exist a non-zero vector v such that Av = λv.(E) The number of the eigenvalue of n x n matrix A may larger than n.

複選題16. (10%) Which of the following are correct?(A) Any orthogonal set of vectors is linearly independent.(B) Any orthonormal set of vectors is linearly independent.(C) If n x n matrix A is orthogonal, A is invertible.(D) If n x n matrix A is orthogonal, AAT=In.(E) If n x n matrix A is orthogonal, det( A ) = 1

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