課程名稱︰線性代數一
課程性質︰數學系大一必修
課程教師︰余正道
開課學院：理學院
開課系所︰數學系
考試日期︰2019年12月20日(五)，11:20-11:40
考試時限：20分鐘
試題 :
Linear Algebra Quiz
Name:　　　　　　　　　　 Student ID:　　　　　　Department:
1. (a) Let A = [1 1], find invertible matrix P and diagonal matrix D such that
[1 0]
A = PDP^(-1).
(b) Let x_0 = 1, x_1 = 1, x_n = x_(n-1) + x_(n-2) when n > 1, prove that
x_n = 1/sqrt(5) (c_1^(n+1) - c_2^(n+1)) for some constant c_1, c_2.
2. Let V be a finite dimensional vector space over C and T be a linear operator
on V such that T^2 = T, i.e. T is an abstract projection.
Show that Tr(T) = rank(T).
3. Find all A∈M_3(C) satisfying A^3 - 2A^2 + A = 0. You only need to write
down the answer up to similar classes of matrix.