課程名稱︰線性代數一
課程性質︰數學系必修/經濟系選修
課程教師︰楊一帆
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰ 2020/1/15
考試時限（分鐘）：130
試題 :
1.(15) Prove or disprove by giving a counterexample the statement that if
U_1, U_2, U_3 are three subspaces of a finite-dimensional vector space V,
then
dim(U1+U2+U3) = dim(U1)+dim(U2)+(dimU3)
-(dimU1∩U2)-(dimU2∩U3)-(dimU3∩U1)+(dimU1∩U2∩U3)
2.(15) Let An be the nxn tridiagonal matrix of the form:
2 1
1 2 1
1 2 1
An = ( . . . )
1 2 1
1 2
(The main diagonal has entries 2, the first diagonal above the main
diagonal and the first diagonal below the main diagonal are 1,
and all others are 0.)Compute det An.
3.(15)
Let V be a finite-dimensional vector space with a basis B. Let B1...Bk be a
partition(i.e., B1∪...∪Bk == B and Bi∩Bj = Φ(empty set) if i ≠ j).Prove
that
V = span(B1)⊕...⊕span(Bk)
4.(15)
Let T be a linear operator on a finite-dimensional vector space, and W be a
T-invariant subspace of V. Suppose that v1,...,vk are eigenvectors of T
corresponding to distinct eigenvalues λ1,...,λk Prove that
If v1+...+vk is in W, then vi ∈ W for all i.
(Hint: One possible way is to use mathematival induction on k.
5.(45)Let A ∈ M_{mxn}(F) and B ∈ M_{nxm}(F) with m≧n.
a)(5) Prove that ch_{AB}(t) = t^{m-n} ch_{BA}(t). (Hint: Consider the
two products.
Im -A tIm A Im 0 tIm A
( )( ) and ( ) ( )
0 tIn B In -B tIn B In
Where Im and In denote the identity matrices
and 0 is the zero matrix of suitable size)
b)(15) Let m_{AB}(t) be the minimal polynomials of AB and BA,
respectively.Prove that either m_{AB}(t) = m_{BA}(t),
m_{AB}(t) = tm_{BA}(t), or m_{BA}(t) = tm_{AB}(t)
c) (15) Assume that BA is invertible. Prove t hat AB is diagonlizable iff
BA is diagonalizable.
d) (10) What happnes if BA is not invertible ? Prove or disprove(by
giving a counter example) the statement in (c) that AB is diagonalizable
iff BA is diagonalizable, under the assumption that BA is not invertible.
6.(15) Let T be linear operator on a finite-dimensional vector space V
ove a field F with dim V ≧ 2. Prove that V has a proper nontrivial
T-invariant subspace W ("proper" means W != V;"nontrivial" means W !={0})
iff the characteristic polynomial of T is reducible over F ("reducible"
means ch_T(t) = g(t)h(t) for some g(t),h(t) ∈ F[t] with deg g, deg h >= 1).
Rmk: In problem 6, you may freely use the following general properties
of F[t]:
i) every nonconstant polynomial can be written as a product of
irreducible polynomials( and the expression is unique up to order and
scalars).
ii) If two polynomials f and g have no common (irreducible) factors, then
(f) + (g) = (1), where for a polynomial h; we let(h) denote the ideal
generated by h.