课程名称︰线性代数二
课程性质︰数学系大一必修
课程教师︰余正道
开课学院：理学院
开课系所︰数学系
考试日期︰2020年03月27日(五)
考试时限：11:20-11:45，共25分钟
试题 :
Linear algebra (II)　　　　　　　　Quiz 1　　　　　　　　　　　　　2020/3/27
Department:＿＿＿＿＿＿＿ ID Number:＿＿＿＿＿＿＿＿＿ Name:＿＿＿＿＿＿＿＿＿
1. (10%) Let W be a subspace of an inner product space V and let β be a vector
in V. Prove that α∈W is a best approximation to β by vectors in W if and
only if β-α is orthogonal to every vector in W.
2. (a) (6%) Let V be an inner product space, T be a self-adjoint liner operator
on V. Prove that every eigenvalue of T is real, and eigenvectors
associated with distinct eigenvalues are orthogonal.
( 2 1 1 )
(b) (6%) Let A = ( 1 2 1 ), find invertible P and diagonal D such that
( 1 1 2 )
t
D = P AP.
3. (8%) Prove that if A∈M (C) is normal and nilpotent, then A is the zero
n
matrix.