課程名稱︰線性代數二
課程性質︰數學系大一必修
課程教師︰翁秉仁
開課學院：理學院
開課系所︰數學系
考試日期︰2010年03月12日(五)，11:20-12:10
考試時限：60分鐘
是否需發放獎勵金：是
試題 :
線性代數小考 2010/03/12
┌ -1 1 -1 ┐
1. [30%] A = │ -1 1 0 │. Find all eignevalues and the corresponding
└ 0 0 1 ┘
eigenspaces of A.
2. [30%] Suppose u and v are two non-zero vectors in |R^n, we define the n ×n
square matrix A by
A = u ． v^T
a. Find all eigenvalues and the corresponding eigenspaces of A.[20%]
b. Could we always find an eigenbasis for A?[10%]
3. [40%] Let P2 be the vetor space of polynomials whose degree are ≦2 and M2
be the vector space of 2 by 2 square matrices. A = ┌ 1 1 ┐. We define the map
└ 1 0 ┘
T : P2 → M2 by T(at^2 + bt +c) = aA^2 + bA + cI.
a. Prove that T is a linear transformation.[8%]
b. Let 1,t,t^2 be a basis of P2; and ┌ 1 0 ┐，┌ 0 1 ┐，┌ 0 0 ┐，
└ 0 0 ┘ └ 0 0 ┘ └ 1 0 ┘
┌ 0 0 ┐ be a basis of M2. Find the correspnding matrix [T] of T w.r.t
└ 0 1 ┘
these bases. [8%]
c. Describe the ker T. [8%]
d. Describe the im T. [8%]
e. Does A^5 ∈im T? [8%]
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.252.31