課程名稱︰線性代數一
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01月03日(五)，10:00-12:10
考試時限：130分鐘
試題 :
There are 9 problems in 2 pages.
In the following, F denotes a field.
1. [10%] Let A∈M_n(F). If r is any positive integer with 1≦r≦n, an r × r
submatrix of A is any r × r matrix obtained by deleting (n-r) rows and
(n-r) columns of A. Suppose A≠0. Prove that the rank of A equals the
largest positive integer r such that some r × r submatrix of A i
invertible.
2. [10%] Let A∈M_n(F). Suppose A is diagonalizable and has characteristic
polynomial
d_1 d_k
(x-c_1) … (x-c_k),　　　c_i ≠ c_j.
Let V be the space of B∈M_n(F) such that AB = BA. Prove that
dim(V) = d_1^2 + ... + d_k^2.
3. [20%] Let T∈L(V) where V is finite-dimensional.
(a) Suppose p(x), q(x)∈F[x] are coprime (i.e., gcd(p,q) = 1) with the
characteristic polynomial ch(x) = p(x)q(x). Let W be a T-invariant
subspace. Show that W = (W∩Ker p(T)) ⊕ (W∩Ker q(T)).
(b) Let W be a T-invariant subspace. Suppose T is diagonalizable. Show that
the restriction operator T_W∈L(W) is diagonalizable.
4. Let V be an n-dimensional vector space and T∈L(V). Suppose T is
diagonalizable.
(a) [5%] If T has a cyclic vector, show that T has n distinct eigenvalues.
(b) [10%] If T has n distinct eigenvalues and if {v_1,...,v_n} is a basis of
eigenvectors of T, show that v = v_1 + ... + v_n is a cyclic vector for
T.
( 1 2 1 0 )
5. [10%] Let A = ( -2 1 0 1 )∈M_4(R) whose characteristic polynomial is
( 0 0 1 2 )
( 0 0 -2 1 )
x^4 - 4x^3 + 14x^2 - 20x +25 = (x^2 - 2x + 5)^2.
Find an invertible matrix P and the rational canonical form Q of A such that
P^(-1) AP = Q.
6. [10%] Let A∈M_n(R) satisfying A^2 + I_n = 0. Prove that n is even, and if
n = 2k, then A is similar to
( 0 -I_k )
( I_k 0 ).
7. (a)[10%] Prove that every complex n × n matrix is similar to its transpose.
(b) [5%] Ture or false: for any field F, every A∈M_n(F) is similar to A^t.
Justify your answer.
8. Let A, B∈M_n(F).
(a) [5%] Prove that if (I-AB) is invertible, then (I-BA) is invertible with
(I-BA)^(-1) = I + B(I-AB)^(-1) A.
(b) [10%] Prove that AB and BA have precisely the same eigenvalues.
(c) [10%] Do AB and BA have the same characteristic polynomial? Do they have
the same minimal polynomial?
9. [15%] Let V be an n-dimensional vector space and T∈L(V) with T^(k+1) = 0
for some integer k≧0. Let S_r = T^r for r≧0. Show that there exists
uniquely a chain of subspaces
V = W_k ⊃ W_(k-1) ⊃ ...
satisfying (i) T(W_i) ⊂ W_(i-2), (ii) S_r(W_r) = W_(-r), and
-1
(iii) W_r ∩ S (W ) = W .
r -r-1 r-1