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05月22日(五)
考试时限:11:20-11:50,共30分钟
试题 :
[Quiz 3] Name: ID:
1. (8%) Solve the differential equation y'''+5y''+3y'-9y=0.
2. Let V be a vector space over a field (not necessarily finite-dimensional),
and let T:V→V be a linear transformation. Let W⊆V be a T-invariant
subspace. We have knwon that T induces the linear transformations
T| : W→W and T':V/W → V/W.
W
(a) (6%) Show that if T| and T' are isomorphisms, then T is an isomorphism.
W
(b) (8%) Conversely, suppose W is finite-dimensional, show that if T is an
isomorphism, then T| and T' are isomorphisms.
W
3. (8%) Let A∈M (R) be a symmetric matrix with the eigenvalues λ1≧…≧λn.
n
T
Let N∈M (R) with N N = I and μ1≧…≧μm be the eigenvalues of the
n×m m
T
matrix N AN. Show that μi≧λi≧μ for all i=1,...,n.
m-n+i