[試題] 99下 電機系 線性代數 期中考

[FAlin (FA（ハガレン）)]

課程名稱︰工程數學-線性代數
課程性質︰系定必修
課程教師︰馮蟻剛 馮世邁 蘇柏青 鄭振牟 林茂昭
開課學院：電機學院
開課系所︰電機系
考試日期（年月日）︰2011/04/20
考試時限（分鐘）：100
是否需發放獎勵金：是
（如未明確表示，則不予發放）

試題 :
USE OF ALLAUTOMATIC COMPUTING MACHINES IS PROHIBITED

1.If the 4x4 matrix M satisfies the following equation, then what is the value
  of the det(M^3)?　　　　　　　　　　 　　　　　　　　　　　　　　　　　 (10%)
                 ┌          ┐   ┌          ┐
                 │1 -2 -1  2│   │4  1 -2 -1│
                 │0  2  1  3│   │0  8  0  0│
                 │0  0 -1  1│M =│7  0  2  3│
                 │0  0  0  4│   │8 -7  1  4│
                 └          ┘   └          ┘
2.Let F and G be two similar nxn matrices. Find two bases X and Y for R^n such
  that [Tf]  = F and [Tf]  = G                                            (10%)
           X             Y

3.Given two subspaces V and U of R^n, let Z = V交集U, the intersection of V
  and U.
  (a) Prove that Z is a subspace of R^n with dimension no greater than those
      of   V and U.
  (b) When n=4, V = Null(A) and U = Null(B), where A and B are listed below,
      find a basis for Z.                                                 (10%)
      HINT: Z = { Z屬於R^4 | Az=0 and Bz=0 }
  (c) When n=4, V = Col(C) and U = Col(D), where C and D are listed below,find
      a basis for Z.                                                      (10%)
      HINT: Z = { Z屬於R^4 | z=Cx=Dy for some x屬於R^3 and y屬於R^2}.


       ┌           ┐      ┌          ┐     ┌        ┐     ┌     ┐
   A = │-1  0  3  1│  B = │1  1  0  1│ C = │ 0  3  6│ D = │ 7 -4│
       │ 1  2 -1  5│      │1  3  2  7│     │-1 -1 -3│     │-2  1│
       └           ┘      └          ┘     │ 1 -3 -5│     │-4  1│
                                               │ 3  2  7│     │ 1  1│
                                               └        ┘     └     ┘
4.(30%)Consider the two functions f and g defined as follows.
                 ┌ x ┐
   f:R^3→R^2, f(│ y │) = (┌ x+y+1 ┐)
                 └ z ┘     └   z   ┘
                             ┌   s   ┐
   g:R^2→R^3, g(┌ s ┐) = (│  s-1  │)
                 └ t ┘     └  s+t  ┘

  (a) Find function composition h1 = g(f(.)) and h2 = f(g(.)).Your answers
      should include specifing their domains and codomains.               ( 4%)
  (b) Determine whether each of f,g,h1 and h2 is an onto function and whether
      it is a one-to-one function
      (True or false only: no explanation needed.)                        ( 8%)
  (c) Determine whether each f,g,h1 and g2 is a linear transformation.
      Explain your answers.                                               (12%)
  (d) For any of f,g,h1 and h2 that is a linear transformationm find its
      standerd matrix, and determine whether the matrix is invertible.

5.(20%)Conside the matrix A =┌1   1   1  ┐屬於R^4x3
                             │1   a   b  │
                             │1   a^2 b^2│
                             └1   a^3 b^3┘
  (a) Show that the column vectors of A form a linearly independent set if
      and only if a不等於1,b不等於1 and a不等於b.                         (10%)
  (b) Assume a=b不等於1. Find rank A and nullity A.                       (5%)
  (c) Assume a不等於1,b不等於1,and a不等於b, Let ak be the kth column of A and
      B ={a1,a2,a3} is an ordered basis for Col A.Let v=[0 1 a+b a^2+ab+b^2]^T
      屬於R^4.Show that V 屬於 Col A, and find [v]B, the coordinate of v with
      respect to the subspace Col A.


打到一半斷線的悲劇...
--
沒有伴隨著痛苦的教訓是沒意義的。
人如果不犧牲一些東西，就無法得到任何東西。
但是當超越了障礙，並且把得到的東西變成屬於自己的東西時...
人應該就能夠得到無法取代的鋼之心靈吧。

<Fullmetal Alchemist>

--
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◆ From: 140.112.240.113
※ 編輯: FAlin           來自: 140.112.240.113      (04/20 20:32)
→ littleshen21:考卷不是交回去了?
推 ly2314      :好快！
→ kuoly1      :所以又是傳說中的偷領兩份考卷??
→ FAlin       :工數不用交吧?
→ littleshen21:電二105的有收~其他就不知道了
→ FAlin       :106沒收