※ [本文轉錄自 NTU-Exam 看板 #1M8tj5lE ]
作者: NTUForest (NTUForest) 看板: NTU-Exam
標題: [試題] 104上 呂學一 線性代數 第一次期中考
時間: Sun Oct 18 18:58:10 2015
課程名稱︰線性代數
課程性質︰資訊系必修
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）︰2015/10/18
考試時限（分鐘）：180
是否需發放獎勵金：是
試題 :
總共十二題，每題十分，可按任何順序答題。只能參考個人事先準備的A4單頁單面大抄。
每一題都是一個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。(第十二題除外)課堂上證過的定理，或是提過的練習題，皆可引
用。
第一題
Let W be a vector space. If V is a non-empty subset of W, then V is a subspace
of W if and only if ax + by ∈ V holds for any scalars a,b ∈ F and any vectors
x,y ∈ V.
第二題
If(U,F,‧_U) nad (V,F,‧_V) are two vectors spaces, then so is (W,F,‧_W) with
(def)
W = F(U,V)
(def)
(f +_W g)(x) = f(x) +_V g(x)
(def)
(a ‧_W f)(x) = a‧_V(f(x))
for any f,g ∈ W, x ∈ U, and a ∈ F.
第三題
If U and V are subspaces of vector space W, then U∪V is a subspace of W if
and only if U∩V = U or U∩V = V.
第四題
If U and V are subspaces of vector space W with U + V = W and U∩V = {0_W},
then for each vector z ∈ W there exists a unique pair (x,y) with x ∈ U,
y ∈ V, and z = x + y.
第五題
Let V be a subspace of vector space W. If x is a vector of W, then the subset
{x} + V
of W is a subspace of W if and only if x ∈ V.
第六題
A subset V of vector space W is a subspace of W if and only if V + V =V.
第七題
If R and S aresubsets of vector spce V with R ⊆ S and span(R) = V, then
span(s) = V.
第八題
Let S be the subset {(1_F,0_F,0_F),(1_F,1_F,0_F),(1_F,1_F,1_F)} of vector
space F^3.
‧If F = Q, then S is linearly independent.
‧If 1_F + 1_F = 0_F, then S is linearly dependent.
第九題
If R and S are subsets of vector space V, then span(R)∩span(S)⊆span(R∩S)
if and only if R⊆S or S⊆R.
第十題
If S = {x,y,z} is a linearly independent subset of a vector space over
scalar field F, then for each s ∈ span(S) there is a unique 3-tuple
(a,b,c) ∈ F such that s = ax + by + cz.
第十一題
If x,y,z are three distinct vectors of vecror space V, then {x,y,z} is linearly
in dependent if and only if {x+y,y+z,z+x} is linearly independent.
第十二題
Prove the replacement theorem:
For any finite subset S of a vector space and any linearly independent subset Q
of span(S), there exists a subset R of S\Q with |Q| + |R| = |S| and span(Q∪R)
= span(S).