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）︰103/10/9
考試時限（分鐘）：60分鐘
試題 :
台大資工單班線性代數第一次小考
2014年10月9日下午四點起一個小時
總共四題，每題十分，可按任何順序答題
第一題 Based upon our definitions of Abelian group and field shown in class,
prove that if (F, +,‧) is a field, then
1. a‧b ∈ F
2. a‧0_F = 0_F‧a = 0_F
hold for any elements a and b of F.
第二題 Let C denote the set of complex numbers. Let R denote the set of real
numbers. Let Q denote the set of rational numbers. Let + denote the component-
wise addition. Let‧denote the component-wise scalar mulplication.
1. Prove or disprove that (M_5x5(Q), R, +,‧) is a vector space.
2. Prove or disprove that (M_5x5(C), Q, +,‧) is a vector space.
第三題 Let V be a subset of vector space W = (W, F, +,‧) such that
ax + y ∈ V
holds for any scalar a∈F and any vectors x,y∈V . Prove that the
following two statements P1 and P2 imply each other:
P1: (a) There is a vector e 2 V such that x + e = x holds for each vector
x∈V and
(b) for each vector x∈V, there is a vector y∈V with x + y = e.
P2: 0_W∈V.
第四題 Let R and S be two subsets of vector space W = (W, F, +,‧). Use「罩
咖定理」to prove
span(R∩S) ⊆ span(R)∪span(S).