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06月19日(五)
考試時限：10:00-12:20，共140分鐘
試題 :
1. [15%] Let Y be a subspace of X. Recall that elements of the quotient X/Y are
those subsets [x] = x + Y of X where x∈X. Let z1,...,zm∈X.
Show the following.
(a) {[z1],...,[zm]} generates X/Y if and only if Y + <z1,...,zm> = X.
(b) [z1],...,[zm] are independent in X/Y if and only if z1,...,zm and
Y are independent. (i.e., z1,...,zm are independent and
<z1,...,zm>∩Y=0).
(c) Let y1,...,yk∈Y. If {y1,...,yk} is a basis of Y and {[z1],...,[zm]} is
a basis of X/Y, then {y1,...,yk,z1,...,zm} is a basis of X.
2. [15%] Let A∈Mn(R) be a symmetric matrix with eigenvalues λ1≧…≧λn.
Let N∈M (R) satsifying N^t．N = Im and let μ1≧…≧μm be the
n×m
eigenvalues of the matrix N^tAN. Show that λi≧μi≧λ for all i=1,..,m.
n-m+i
3. [20%]
(a) Solve the differential equation
d^2y dy