課程名稱︰代數一
課程性質︰數學系大二必修
課程教師︰于靖
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2015/09/21
考試時限（分鐘）：70
試題 :
1. Prove that the order of an element in S equals the least common multiple of
n
the lengths of the cycles in its cycle decomposition.
2. Show that if n ≧ m then the number of m-cycles in S is given by
n
n(n-1)(n-2)...(n-m+1)
───────────.
m
[Count the number of ways of forming an m-cycle and divide by the number of
representations of a particular m-cycle.]
3. Show that GL (F) is non-abelian for any n ≧ 2 and any F.
n
╭ a b ╮
4. Let G = {│ │| a, b, c ∈ |R, a ≠ 0, c ≠ 0}.
╰ 0 c ╯
╭ a b ╮ ╭ a b ╮
│ 1 1 │ │ 2 2 │
(a) Compute the product of │ │ and │ │ to show that G is
│ 0 c │ │ 0 c │
╰ 1 ╯ ╰ 2 ╯
closed under matrix multiplication.
╭ a b ╮
(b) Find the matrix inverse of │ │ and deduce that G is closed under
╰ 0 c ╯
inverses.
(c) Deduce that G is a subgroup of GL (|R).
2
(d) Prove that the set of elements of G whose two diagonal entries are equal
(i.e., a = c) is also a subgroup of GL (|R).
2
--
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。
--
※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 140.112.212.7
※ 文章網址: https://www.ptt.cc/bbs/NTU-Exam/M.1444344975.A.842.html