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115年 - [無官方正解]第八章 綜合應用：8-1 字串#137684(35題)

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25. (4 points) 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 E Rixm be a matrix whose characteristic polynomial splits over R. 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. (4 points) Let Wi and Wa be subspaces of a finite-dimensional vector space V. Let 6 denote the direct sum. Which of the following statements are correct? (A) Win Wa is a subspace of V. (B) WiUW2 is a subspace of V. (C) W1+W2 is a subspace of V. (D) If V = Wi @ Wa, and Bi and Be are bases for Wi and Wa, respectively, then Bi O Bz = 0, and B1 U Bz is a basis for V. (E) If Wi e Wa = V, then the dimension dim(V) = dim(Wi)+dim(Wz).

23. (4 points) 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.

10. Let D be the region in zy-plane bounded by x2 = y, x2 = 3y, y2=x,y2=3x . Use the transformation to evaluate the double integral.

9. Use the method of Lagrange multiplier to find the shortest and longest distance from the origin to curve 9x2 + 16xy +21y2 = 125.

8. Let C be the curve of intersection of surfaces xy + yz +zx = -14 and x2+y2+z2=29.The tangent line of curve C at point (2,3, -4) is given by= z + 4. Findthe values of a, b.

test

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