课程名称︰线性代数二
课程性质︰必修
课程教师︰翁秉仁
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2018/04/25
考试时限（分钟）：110 分钟
试题 :
1. [20%] Compute the determinant of the following A_n∈S_n over C.
┌ 1 i 0 ... 0 0 ┐
｜ i 1 i ... 0 0 ｜
｜ 0 i 1 ... 0 0 ｜
A = ｜ . . . . . . ｜
｜ . . . . . . ｜
｜ . . . . . . ｜
｜ 0 0 0 ... 1 i ｜
└ 0 0 0 ... i 1 ┘
2. [20%] A∈S_3. It is know that there are 3 linear independent vector
u, v, w such that
Au = u - v - w
Av = 6v + 8w
Aw = -4v - 6w
Is A diagonalizable? Find the corresponding eigenspaces in term
of u, v, w.
3. [20%] Consider k vectors v_1, v_2, ... ,v_k∈R^n and
A = [v_1 v_2 ... v_k]∈M_nxk. Show the k-volume of parallelpiped (平行
超多面体) span by v_1, v_2, ... , v_k∈R^n is equal to √(detA^tA).
4. [20%] A∈S_n, λ_1, λ_2, ... , λ_k are all distinct eigenvalues of A.
Show that
1 ≦ g(λ_i) ≦ a(λ_i), ∀i = 1, ... ,k
5. [30%] Prove the following statements:
b. [10%] If W is A^t-invariant subspace of R^n, show that W^⊥ is
A-invariant
c. [10%] If f_A(t) split (i.e. all eigenvalues are real), show that there
is an (orthogonal) matrix S, such that S^(-1)AS = Γ, where Γ
is an upper triangular matrix.
d. [10%] Using c. to give another proof of Cayley-Hamilton theorem in
this case. (Or assume c. in C, and prove general case).
______________________________________________________________________________
常用记号或名词
a. M_nxm is the space of nxm matrices; S_n = M_nxn
b. i = √(-1)
c. E^A_λthe λ-eigenspace of A. If there is no confusement, E_λis uesd.
d. g(λ) = dimE_λ