[試題] 104上 謝銘倫 線性代數一 期中考

[xavier13540 (柊 四千)]

課程名稱︰線性代數一
課程性質︰數學系大一必修
課程教師︰謝銘倫
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2015/11/11
考試時限（分鐘）：130

試題 :


Notation: F denotes the set of complex numbers, n is a positive integer and I
                                                                             n

is the identity matrix in M (F).
                           n


                                     Part A


                            2     3
Problem 1 (10 pts). Let T: F  → F  be the linear transformation such that


                 T((1, 2)) = (1, 1, 1); T((2, 1)) = (1, -1, 3).


Let A = {(-5, 2), (-1, 1)} and B = {(2, 3, 1), (1, 2, 1), (1, 1, 1)} be bases of

 2      3
F  and F . Find the matrix representation [T]     of T with respect to A and B.
                                             A, B

                            4     3
Problem 2 (20 pts). Let T: F  → F  be the linear transformation defined by T(v)


= A・v, where


                           ╭ 3 -1  1  1 ╮
                           │            │
                       A = │ 1  1 -1  3 │ ∈ M     (F).
                           │            │     3 ×4
                           ╰ 2  0  0  2 ╯


(1) Find the rank and the nullity of T.


(2) Find bases of Ker T and Im T.


                                               3
Problem 3 (10 pts). Let V  be the subspace of F  spanned by
                         a

                        (1, 2, a), (1, a, 2), (a, 1, 2).


Determine all possible values a ∈ F such that dim  V  = 2.
                                                  F  a


Problem 4 (15 pts). Let


                                  ╭ -1  2 -2 ╮
                                  │          │
                              A = │  7 -3  1 │.
                                  │          │
                                  ╰ 11 -8  6 ╯

                                                -1
Find an invertible matrix P ∈ M (F) such that P  AP is a diagonal matrix.
                                3


Problem 5 (10 pts). Let

                         ╭  1  0  1 ╮      ╭ 1  1  1 ╮
                         │          │      │         │
                     A = │ -1 -3  1 │; B = │ 1  2  1 │.
                         │          │      │         │
                         ╰  1  4  1 ╯      ╰ 1  4  3 ╯


Find all roots of the characteristic polynomial of AB.



                                     Part B



Problem 6 (10 pts). Let V be a finite dimensional vector space over F with


dim  V = m. Let v , v , ..., v  be n vectors in V. Show that if n ≧ m + 2, then
   F             1   2        n
                                                                      n
there exists α , α , ..., α  in F not all equal to zero such that Σ α v  =
               1    2         n                                      i=1  i i
       n
0 and Σ α  = 0.
      i=1  i


Problem 7 (10 pts). Let A ∈ M (F). If rank A + rank(I  + A) = n, show that
                              n                       n

Tr(A) + rank A = 0.



Problem 8 (15 pts). Let A, B ∈ M (F). Let C = AB - BA. If CA = AC, prove that C
                                 n

is NOT invertible.

--
移居二次元(|R^2)的注意事項：            3. 如果你在從事random walk，往上下左右的
1. connectedness不保證pathwise connec-     的機率都是1/4，則你能回家的機率是1。
   tedness。可能你跟你的幼馴染住很近，  4. 下面這個PDE是二次元上的波方程式
   卻永遠沒辦法到她家。                    http://i.imgur.com/2H9HllP.png
2. ODE的C^1 autonomous system不會出現      它的解不滿足Huygens' principle，因此
   chaos，在預測事情上比較方便。           講話時會聽到自己的回音，很不方便。

--
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※ 編輯: xavier13540 (140.112.212.7), 11/13/2015 16:50:03