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04月10日(五)
考试时限：11:20-11:45，共25分钟
试题 :
1. Let V be a finite-dimensional inner product space. If T and U are linear
operators on V, we wrtie T < U if U - T is a positive operator. Prove or
disprove
(a) (7%) If T < U and U < S, then T < S.
(b) (7%) If T < 0 and U < 0, then TU > 0.
2. (8%) Show the following fourth order Hilbert matrix is positive definite.
[ 1 1/2 1/3 1/4 ]
H4 = [ 1/2 1/3 1/4 1/5 ].
[ 1/3 1/4 1/5 1/6 ]
[ 1/4 1/5 1/6 1/7 ]
3. (8%) Let V be the polynomial function space of degree less than 2020
together with the 0-polynomial over |R equipped the inner product
1
(p|q) = ∫p(x)q(x)dx.
0
1
Define a linear functional L by L(p) = ∫p(x)dx for p∈V. For any
0
non-singular linear operator T on V, prove that there exists q∈V such that
L(p) = (Tp|q).
(Hint: You can use the result of Assignment 4 without proof but you need to
write the statement which you use.)