5. Let V be the vector space of all A ∈ , with the operations of matrix addition and
multiplication of a matrix by a real scalar. Let T be a linear transformation from V to
Y, and (B₁,B₂,... ,] be a basis for V. Suppose U is a linear transformation from D to
given by U(c₁B₁ + c₂B₂ +.... +) = Which of the following
statements is/are true?
(A) Let(A₁, A₂,... ,] be linearly independent over R. The set ｛U(A₁),U(A₂),... ,｝
can be linearly dependent over .
(B) Suppose the dimension of the range space of T is k: over IR. Then ｛T(B₁),T(B₂),... ,｝
are linearly independent over .
(C) n=m.
(D) Let  c = U(B₁) and  a = U(T(B₁), then

(E) None of the above is true.

1. Letbe the reduced row echelon form of a matrix A = . Which of thefollowing statements is/are true?(A) is the 4 x 1 zero vector.(B) The three column vectors , are linearly independent.(C) The column space of R is the same as the column space of A.(D) Let B = be a submatrix of A. The reduced row echelon form of B isnot always a submatrix of R.(E) None of the above is true.R=

2. Continue from Question -. Which of the following statements is/are true?(A) The nonzero rows of R form a basis for col.(B) Let C be the reduced row echelon form of . Then is the identity matrix.(C) Let E be a 3 x 5 matrix and[10110]D = 01110100001be the reduced row echelon of E. We can obtain the intersection of the right nullspace of A and the right null space of E from R and D without knowing A and E.(D) Let Q be a 5 x 5 matrix and rank(Q) = 3. Then rank(AQ) = 3.(E) None of the above is true.

3.Let U and V be subspaces of and T be a linear transformation from to . Whichof the following sets is/are subspace(s) of ?(A) (B)(C)(D) (E) None of the above is true.

複選題4.Let A, B, and C be n x n matrices for some positive integer n. Which of the followingstatements is/are true?(A) det( A ) = 0 implies det(R) == 0, where R is the reduced row echelon form of A.(B) If AB = CA - , then B = C.(C) If det(AB) 7 0, then A is invertible.(D) rank(AB)=rank(BA).(E) None of the above is true.

6. For the matrixwhich of the following statements is/are true?(A) rank( A ) = 3.(B) The sum of eigenvalues of A is 6.(C) A is similar to B =(D) The system of linear equations A x = has a solution.(E) None of the above is true.

複選題7. For a non-zero matrix A ∈ , deine(1)the maximum Frobenius norm ||A  x llover all unit-norm vectors  x . Which of the followingstatements is/are true?(A) σ2 is the maximal eigenvalue of .(B) If x0 is an optimal solution to equation (1), i.e, llAx0ll = σ, then there exists a vectorx1 such that Ax1 = 0  and ≠0.(C) Assune m = n and A is nonsingular. Then = .(D) Assume m = n. Then t A is singular only if σ ≥ 1.(E) None of the above is true.

複選題8. Let V together with the inner product (‧,‧)y be an inner product space over . For acollection of N linearly independent vectors [w1 ,...,] in V, let M = bethe corresponding Gram matrix, i.e, the (i, j)-th entry of M is given by Define the following set of real-valued functions f : V →which is a vector space over when combined with standard addition and scalar multiplica-M= 1.8m (2m,3)v tion of real-valued functions. For any f( x )  = in the vector space F, defineWhich of the following statements is/are true?(A) M is positive definite.(B) If m1,1 = 2, m2,2 = 1, and m1,2 = -, then is the vector norm induced by the inner product .(C) (f,g)s is an inner product for elements f,g ∈ F.(D) Given g(y) = , we have = f(ω1) for all f ∈ F.(E) None of the above is true.＜f(z) =

9. Which of the following staternents is/are true?(A) Let  x : and  y  be two non-zero vectors in . The matrix is diagonalizable if≠0.(B) If A is symmetric and positive definite, then , is positive semi-definite.(C) For u,n ∈ such that =, where ⊕ is theoperation of direct sum of two vector spaces.(D) For the real matrix Q = , where u ∈ , we have det(Q) + rank(Q) = n.(E) None of the above is true.

10. Let P ∈ be the orthogonal projection matrix, with respect to the Euclidean innerproduct, of vectors in onto col( A ) for some nonzero matrix A ∈ . Which of thefollowing statements is/are true?(A) ≥rank(P).(B) P can have an eigenvalue larger than 2.(C) Px≠ x for all x e .(D) If k ＜ n and ! e , the system of linear equations Ax = o has a unique least squaressolution.(E) None of the above is true.

11.Consider the first order differential equation xy'(x) = y(x) - with initialcondition y(2) = 0. Which of the following statements is/are true?(A) It is a linear differential equation for the dependent variable y.(B) y(x) is a parabolic function of x.(C) y'(x) =-x.(D) y"(x) =.(E) None of the above is true.