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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.

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).

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.

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