課程名稱︰線性代數
課程性質︰大二必修
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）︰2016/10/25
考試時限（分鐘）：180
試題 :
線性代數 第一次期中考
2016年10月25日下午兩點二十分起三小時
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共十一題每題十分，可按任何順序答題。每題難度不同，請判斷恰當的解題順序？可直接
使用課堂上證明過或出現過的任何定理或性質，除非題目禁止。使用時須標示定理或性質
的名稱或編號。
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第一題 請直接根據「直觀冗」與「實戰冗」的定義證明兩者等價。請勿使用課堂上出現
過的相關性質或證明方式，例如證明兩者都跟「威力冗」等價之類的方法。
第二題 You are given the fact that P_n(C) is a vector space over R. Prove or
disprove that its dimension is 2n+2.
第三題 Prove or disprove |A|<=|b| for any linear independent subset A of
vector space V and any finite spanning set B of V. Your proof or
disproof cannot directly use the replacement theorem.
第四題 Prove or disprove -1_F=/1_F for any field F with 1_F+1_F_=/0_F.
第五題 Prove or disprove that if S is a subset of vector space V, then S is a
subspace of V if and only if span(S)=S.
第六題 Let U and V be subspaces of vector space W with U∩V = {0_W}. Prove or
disprove that each vector x∈U+V can be uniquely written as
x=y+z
for some vectors y∈U and z∈V.
第七題 Let S be a linearly independent subset of a vector space. Prove or
disprove that each vector in span(S) admits a unique linear combination
of S.
第八題 Prove disprove that if {f_1, f_2, f_3} is a linearly independent subset
of the vector space F(R,R) of functions over Q, then {f_1+f_2, f_2+f_3,
f_3+f_1} si also linearly independent.
第九題 Prove or disprove that if r a real number and n is a positive integer,
then the set
{p∈P_n(R)|p(r)=0}
is an n-dimensional subspace of the vector space P(R).
第十題 Let F be a field. Let n be a positive integer. Prove or disprove that
F_nxn = {A∈F_nxn|A^t=-A} + {A∈F_nxn|A^t=A},
where A^t denotes the transpose(轉置方陣）of A and -A denotes the
additive inverse of A.
第十一題 Prove or disprove that if V is a subspace of a vector space W with
dim(V) = dim(W) < ∞,
then V=W.