7.Let Which of the following statements are true?
(A)
(B)
(C)
(D)
(E) None of the above.

複選題8. Let A be an m x n matrix. Which of the following statements are true? (A) nullity( A ) + rank( A )=n. (B) rank( A )=rank(AT). (C) If m = 7 and n = 5, then rank( A ) is at most 5. (D) Suppose that m = n. Then, "A is singular" "A has rank n". (E) None of the above.

複選題9. Let A be an n x n matrix whose (i，j) component is aij. The trace of A is defined as trA = Let B and C be n x n matrices. Which of the following statements are true? (A) tr(AB)= tr(BA). (B) tr(ABC)=tr(CBA). (C) tr(ATB)=tr(ABT). (D) tr(A+B)=(trB)(trA). (E) None of the above. 

複選題10. Let A be an n x n matrix whose (i, j) component is aij. Let f be a real-valued function defined on A. Let  ∇Af( A ) be the gradient of f( A ) with respect to A； ∇Af( A )is delined as an n x n matrix whose (i, j) entry is . Let B and C be n x n matrices. Which of the following statements are true? (A) If f( A ) = tr(AB), then ∇Af( A )= B. (B) If f( A ) = tr(AB), then ∇Af( A ) = BT. (C) If f( A ) = tr(AATC), then ∇Af( A ) = CA + CTAT. (D) If f( A ) == tr(AATC), then ∇Af(  A  ) = CA + CAT. (E) None of the above.

11. Which of the following statements are true? (A) is singular if and only if Ax = b has infinitely many solutions for every n x 1 vector b. (B) Every matrix transformnation is a linear transformation. (C) If A is a nonsingular upper triangular matrix, then the adjoint matrix is lower triangular. (D) The dimension of the zero space {0} is 0. (E) None of the above.

12. Let A be a 2 x 2 matix defined s.Which of the tolowing statements are true? (A) The characteristic function is (B) The eigenvalues are (C) 'The eigenvectors are (1,i) and (1, -i). (D) The eigenvectors are (-1,i) and (-1, -i). (E) None of the above.

13. Which of the following are projection matrices? (A)(B) (C) (D) (E) None of the above.

14. Find the matrix for the refection of R2 through the a = y line. (A) (B) (C)(D) (E) None of the above.

複選題15. Let A be an n x n matrix. Which of the following statements are true? (A) A symmetric matix A is positive definite if and only if all the eigenvalues of A are positive. (B) A is nonsingular if and only if detA=0. (C) A is singular if and only if 0 is an eigenvalue of A. (D) detA = λ1+ λ2 +...+  λn , where λi's are eigenvalues of A. (E) None of the above.

1. The number of parents choosing to have only one child _____ increasing in many countries all over the world. (A) were (B) has (C) is (D) are

2. Most students choose their majors in college ____ their interests and scores of the college entrance exam. (A) owe to (B) made of (C) adjust to (D) based on

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