[試題] 108-2 余正道 線性代數二 第三次小考
課程名稱︰線性代數二
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︰2020年05月22日(五)
考試時限：11:20-11:50，共30分鐘
試題 :
[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
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