[試題] 101上金必耀化學數學一期中考

看板NTU-Exam作者noreg0108807 (周公的召喚)時間11年前 (2012/11/13 21:20)推噓16(17推 1噓 3→)

留言21則, 17人參與討論串1/1

課程名稱︰化學數學一課程性質︰化學系必修課程教師︰金必耀開課學院：理學院開課系所︰化學系考試日期（年月日）︰2012/11/13 考試時限（分鐘）：120 是否需發放獎勵金：是（如未明確表示，則不予發放）試題 : 3% for true-or-false problem a) b) c) 2% for true-or-false problem d) e) 1. True or false? a) If A is invertible and its rows are in reverse order in B, then B is invertible. b) If A and B are symmetric, then AB is symmetric. c) If A and B are invertible, then BA is invertible. 2. True or false? a) If L1U1 = L2U2 (upper triangular U's with nonzero diagonal, lower triangular L's with unit diagonal), then L1 = L2 and U1 = U2. The LU factorization is unique. 2 -1 b) If A + A = I, then A = A + 1 c) If all diagonal entries of A are zero, then A is singular. 3. True or false for M = all 3 by 3 matrices? T a) The skew-symmetric matrices in M (with A = -A) form a subspace. T b) The unsymmetric matrices in M (with A ≠ A) form a subspace. c) The matrices that have (1,1,1) in their nullspace form a subspace. 4. True or false: If the eigenvalues of A are 2,3,5, then the matrix is certainly a) invertible. b) diagonalizable. c) not diagonalizable. 5. True or false? a) A square matrix has no free variable. b) An invertible matrix has ni free variable. c) An m by n matrix has no more than n pivot variables. d) An m by n matrix has no more than m pivot variables. 6. True or false? a) If the columns of a matrix are dependent, so are the rows. b) The columns space of a 2 by 2 matrix is the same as its row space. c) The columns space of a 2 by 2 matrix has the same dimension as its row space. d) Thw columns of a matrix are a basis for the column space. 7. True or false? a) If the columns of A are linearly independent, then Ax = b has exactly one solution for every b. b) A 5 by 7 matrix never has linearly independent columns. 8. True or false? T a) A and A have the same number of pivots. T b) A and A have the left nullspace. T c) If the row space equals the column space than A = A. T d) If A = -A, then the row space of A equals the column space. 9. True of false? a) The intersection of two subspaces of a vector space cannot be empty. b) If Ax = Ay, then x = y. c) The row space of A has a unique basis that can be computed by reducing A to echelon form. 2 d) If a square matrix A has independent columns, so does A . 10. True or false? a) If the vector x and y are orthogonal, and P is a projection, then Px and Py are orthogonal. -1 b) The determinant of S AS equals the determinant of A. c) A matrix whose entries are 0s and 1s has determinant 1,0, or -1. 11. True or false? a) For every matrix A, there is a solution to du/dt = Au starting from u(0) = (1,...,1). b) Every invertible matrix can be diagonalized. c) Every diagonalizable matrix can be inverted. d) Exchanging the rows of a 2 by 2 matrix reverses the signs of its eigenvalues. H e) If eigenvectors x and y correspond to distinct eigenvalues, then x y = 0. 12. Suppose A is symmetric positive definite and Q is an orthogonal matrix. True od false: T a) Q AQ is a diagonal matrix. T b) Q AQ is symmetric positive definite. T c) Q AQ has the same eigenvalues as A. -A d) e is symmetric positive definite. 13. (5%) Find the eigenvalues and eigenvectors of ┌ ┐ │ 0 -i 0 │ A = │ i 1 i │ │ 0 -i 0 │ └ ┘ What property do you expect for the eigenvectors, and is it true? 14. (5%) Construct the projection matrix P onto the space spanned by (1,1,1) and (0,1,1) End 不負責統計：滿分129分　是非題119分＋非選題10分 --

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kmchao33

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ceiba

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seanyallow

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