[試題] 97下 電機系統一教學 線性代數 期中考

[cokewolf (可樂狼)]

課程名稱︰線性代數
課程性質︰必修
課程教師︰(統一教學)
開課學院：電資學院
開課系所︰電機系
考試日期（年月日）︰2009.04.22
考試時限（分鐘）：100 分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）

試題 :

1.(28%) Label the following statements as being true or false.
(Explain your answer. Answers with no explanation get 0%):

(a) Let A be an m*n matrix with reduced row echelon form R.
    Then there is a unique matrix P such that PA = R.
(b) If T is a linear transformation, then the dimension of the range
    T plus the dimension of the null space of T equals the dimension
    of the domain of T.
(c) If A is an n*n matrix and the system Ax = b is consistent for
    every b, then A is invertible.
(d) Let S be a linearly independent subset of R^n and V be a
    k-dimensional subspace of R^n. If S has k vectors, then S is a
    basis of V.
(e) The vectors in the vector form of the general solution to Ax = 0
    form a basis for the null space of A.
(f) Let V and W be two subspaces of R^2. V∪W is a subspace of R^2.
(g) The pivot columns of the reduced row echelon form of A form a basis
    for the column space of A.

2.(15%) Find the determinant of the n*n matrix
          ┌                                          ┐
          │  a+b  ab    0    0   …   0    0    0    │
          │   1   a+b  ab    0   …   0    0    0    │
          │   0    1   a+b  ab   …   0    0    0    │
   A =    │   …   …   …  …   …   …   …   …   │
          │   0    0    0    0   …   1   a+b   ab   │
          │   0    0    0    0   …   0    1    a+b  │
          └                                          ┘
3.(15%) Find an explicit description of the reflection T of R^2 about
        the line with equation y = mx.

4. Let T and U be linear transformations with standard matrices A and
   (A^T)A, respectively. Let T be onto.
(a)(7%) Show whether U is onto, on-to-one, or invertible.
(b)(7%) What is the dimension of the null space of U? Explain your answer.

5.(8%) Prove that for any m*n matrix A and any n*p matrix B,
       rank AB ≦ rank A.

6. Let T be the linear opeartor on R^3 such that
     ┌ ┐    ┌ ┐    ┌  ┐   ┌  ┐    ┌  ┐   ┌  ┐
     │1│    │1│    │ 0│   │ 2│    │-1│   │-1│
   T(│0│) = │1│, T(│-1│) =│-1│, T(│ 1│) =│ 0│
     │1│    │3│    │ 1│   │ 0│    │-1│   │-1│
     └ ┘    └ ┘    └  ┘   └  ┘    └  ┘   └  ┘
         ┌ ┐ ┌ ┐ ┌ ┐
         │1│ │1│ │3│
   B = { │1│,│1│,│2│} is a basis for R^3.
         │0│ │1│ │1│
         └ ┘ └ ┘ └ ┘
(a)(10%) Find the B-matrix representation of T.
(b)(10%) Find a basis for the null space of T and represent it in
         B-coordinate vectors.





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