[試題] 101上 金必耀 化學數學一 期中考

[noreg0108807 (周公的召喚)]

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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唉......
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※ 編輯: noreg0108807    來自: 140.112.4.202        (11/14 10:44)
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