課程名稱︰線性代數
課程性質︰大二上必修
課程教師︰呂學一
開課學院：電機資訊學院
開課系所︰資訊工程學系
考試日期（年月日）︰2015/11/29
考試時限（分鐘）：180
試題 :
共十二題，前九題都是可能對也可能不對的敘述。如果你覺得對請證明它是對的，
如果你覺得不對請證明它是錯的。後面三題請按題目指示作答。
1.
Let T : V → V be linear for some finite-dimensional vector space V.
If N(T) + T(V) = V, then N(T) ∩ T(V) = { 0 }.
V
2.
If T : V → V is linear for some finite-dimensional vector space V,
β
then there exist ordered bases α and β of V such that [T] is diagonal.
α
3.
If T : V → V is linear for some finite-dimensional vector space V, then
TT = T if and only if T(V) ⊆ N(T).
0
4.
Let T : V → V be linear for some finite-dimensional vector space V.
If α is an ordered basis of V, then T is an isomorphism if and only if
T(α) is an ordered basis of V.
5. n ×n
If A and B are matrices in F , then A is similar to B (i.e.
-1 n ×n
B = Q AQ for some invertible matrix Q ∈ F ) if and only if
tr(A) = tr(B).
6. n ×n
There is a t ∈ {1, 2, 3} such that if X ∈ F , then
n ×n
X = X X ...X holds for elementary matrices X ,...,X ∈ F
1 2 k 1 k
with k ≦ 9 that are not of type t.
7. n ×n
If A and B are matrices in F , then rank(A + B) ≦ rank(A) + rank(B).
8.
Let V be an INFINITE-dimensional vector space. Let α be an ordered basis
of V. If f : α → V is a function, then there exist a UNIQUE linear
T : V → V with T(x) = f(x) for each x ∈ α.
9.
If c , c ,...,c are n distinct real numbers, then the function
0 1 n-1
1 ×n
T : |P (|R) → |R defined by
n-1
def
T(p) === ( p(c ), p(c ),...,p(c ) ) is an isomorphism.
0 1 n-1
10.
Let T : V → W be linear for finite-dimensional vector spaces V and W
over a common scalar field. Let α (respectively, β) be an ordered
β
basis of V (respectively, W). Prove [T(x)] = [T] [x] for any x ∈ V.
β α α
(Your proof may directly use our 魔杖定理 shown in class.)
11. ┌ ┐
│ 1 0 -1 2 1│
│-1 1 3 -1 0│
Let A = │-2 1 4 -1 3│.
│ 3 -1 -5 1 -6│
└ ┘
5 ×5
Find a B ∈ |R of rank 2 such that
4 ×5
AB is the all-zero matrix in |R .
12.
┌ ┐
│ 1 2 0 3 0│
│ 2 1 -3 1 3│
Find the inverse of │ 4 -1 0 5 1│.
│ 7 3 0 2 1│
│ 1 0 -1 1 1│
└ ┘