課程名稱︰線性代數
課程性質︰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）︰2017/9/26
考試時限（分鐘）：60
試題 :
第一題
Give definitions of the following algebraic structures (5 points for each):
(1) Abelian group, (2) field, (3) vector space, and (4) vector subspace.
第二題
Let (F,＋,．) be a field.
1. (7 points) Prove a．0 ∈ F for any a ∈ F.
F
2. (7 points) Prove a．0 = 0 for any a ∈ F.
F F
-1
3. (6 points) Prove or disprove a ≠ a for any a ∈ F\{0 , 1 }.
F F
(You may directly use any properties proved or mentioned in class
for this subproblem.)
第三題
Let (F,＋,．) be a field. In class we proved a．0 ∈ F for any a ∈ F
F
in two parts. The first part assumes a≠0 . Consider the following
F
alternative proof for this part, i.e., a．0 ∈ F for any a ∈ F\{0}:
F F
By a．a + 0 = a．a + a．0 obtained in the following three steps
F F
a．a + 0 = a．a by definition of 0
F F
= a．(a + 0 ) by definition of 0
F F
= a．a + a．0 by the distributivity axiom of field F
F
and the cancellation law of Abelian group (F,＋), we have a．0 = 0 ∈ F.
F F
Is it OK to use the above proof to replace the proof given in class?
Justify your answer.
第四題
Let (G,＋) be a group. Let each G with i ∈ {1,2} be a subset of G.
i
Let each (G , ＋ ) with i ∈ {1,2} be a group with ＋ inheriting ＋ of G,
i i i
def
i.e., let x + y = x + y for any x, y ∈ G .
i i
Prove or disprove that (G , ＋ ) with ＋ inheriting ＋ of G and
3 3 3
def
G = { x + (-x ) : x ∈ G , x ∈ G }
3 1 2 1 1 2 2
is a group, where -x denotes the inverse of x in G . For this problem,
2 2 2
you may directly use any properties proved or mentioned in class.
第五題
Let Q be the field of rational numbers equipped with the standard addition
2
＋ and multiplication ． of rational numbers. Let V = F(Q , Q) consist of
2
the functions that map Q to Q. Define ＋ and ． by
V V
def
(f + g)(q , q ) = f(q , q ) + g(q , q )
V 1 2 1 2 1 2
def
(a ． f)(q , q ) = a．(f(q , q ))
V 1 2 1 2
for any f, g ∈ V and any a, q , q ∈ Q.
1 2
Prove or disprove that V obeys the distributivity axiom of vector space
(V, Q, ． ).
V