15.  .
(A)
(B)
(C)
(D)

16. Let T : be a function. Which of the following defines a linear transformation? (A) T(a1, a2) = (1, a1) (B) T(a1, a2) = (a1, a2) (C) T(a1,a2) = (a1 + 1, a2) (D) T(a1,a2) = (a1 cosθ-a2sinθ, a1 sinθ+a2cos θ),   for a fixed angle θ

17.  Which of the following statements is FALSE? (A) S is a basis for a vector space V if and only if S is a minimal spanning set.  (B) V is a finite-dimensional vector space (dim(V) ＜∞), and W is a subspace of V. If dim(V) = dim(W), then W = V. (C) Let V and V'be vector spaces having the same finite dimension, and let T:V→ V be a linear transformation. Then T is one-to-one if and only if range(T) = V'  (D) Let V and V' be vector spaces of dimensions n and m, respectively. A linear transformation T : V -+ is invertible if and only if m = n.  

18.  Which of the following matrices are unitary? will 10:13 2100 (A)(B)(C)(D)

19. Considering the following matrix: The eigenvalues of the matrix﹕  The eigenvalues of the matrix A are (A)-1(B)1(C)6(D)7(E)2

20.  Which of the following statements are correct? (A) A linear system with fewer equations than unknowns may have no solution. (B) Every linear system with the same number of equations as unknowns has a unique solution. (C) A linear system with coefficient matrix A has an infinite number of solutions if and only if A can be row-reduced to an echelon matrix that includes some column containing no pivot. (D) If [A|b) and [B|c] are row-equivalent partitioned matrices, the linear systems Ax = b and Bx = c have the same solution set. (E) A linear system with a square coefficient matrix A has a unique solution if and only if A is row equivalent to the identity matrix.

21.  Which of the following statements are NOT correct? (A) Let A be a real n x n matrix. If A2 is invertible, then and A3 are invertible. (B) If A and B are invertible, then so is A+B, and . (C) Let A be a real m x n matrix and B a real n x m matrix. Then trace(AB) = trace(BA).  (D) Let A be a real m x matrix and B a real n x m matrix. Then det(AB) = det(BA). (E) Let A and B be real n x n matrices. Then AB = BA.

22.  Which of the following statements are NOT correct? (A) Any linear operator on an n-dimensional vector space that has fewer than n distinct eigenvalues is not diagonalizable.  (B) Two distinct eigenvectors corresponding to the same eigenvalue are always linearly independent. (C) If A is an eigenvalue of a linear operator T, then each vector in the eigenspace is an eigenvector of T.  (D) A linear operator 7 on a finite-dimensional vector space is diagonalizable if and only if the multiplicity of each eigenvalue A equals the dimension of the corresponding eigenspace .  (E) If A is diagonalizable, then   is also diagonalizable.

23.  Which of the following statements are NOT correct? (A) If S is linearly independent and generates V, each vector in V can be expressed uniquely as a linear combination of vectors in S. (B) Every vector space has at least two distinct subspaces. (C) No vector is its own additive inverse. (D) All vector spaces having a basis are fnitely generated. (E) Any two bases in a finite-dimensional vector space V have the same number of elements.

24.  Let W1 and W2 be subspaces of a finite-dimensional vector space V. Let ⊕ denote the direct sum. Which of the following statements are correct? (A) W1 ∩W2 is a subspace of V. (B) W1 ∩W2 is a subspace of V. (C) W1+W2 is a subspace of V. (D) If V = W1 ⊕W2, and β1 and B2 are bases for W1 and W2, respectively, then β1 and B2 = 0, and β1 ∪ β2 is a basis for V.  (E) If  W1 ⊕ W2 = V, then the dimension dim(V) = dim(W1)+dim(W2).

25.  Which of the following statements are correct? (A) If Q is orthogonal, then det(Q) = ±1.  (B) Let A be a real n x n matrix. Then A is symmetric if and only if A is orthogonally equivalent to a real diagonal matrix. (C) Let A ∈ be a matrix whose characteristic polynomial splits over . Then A is orthogonally equivalent to a real upper triangular matrix.  (D) Let T be a self-adjoint (Hermitian) operator on a finite-dimensional inner product space V. Then every eigenvalue of T is positive. (E) Let T be a self-adjoint (Hermitian) operator on a finite-dimensional inner product space V. Then every eigenvalue of T is negative.

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