课程名称︰线性代数
课程性质︰大二必修
课程教师︰吕学一
开课学院：电机资讯学院
开课系所︰资讯工程学系
考试日期（年月日）︰2016/10/25
考试时限（分钟）：180
试题 :
线性代数 第一次期中考
2016年10月25日下午两点二十分起三小时
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共十一题每题十分，可按任何顺序答题。每题难度不同，请判断恰当的解题顺序？可直接
使用课堂上证明过或出现过的任何定理或性质，除非题目禁止。使用时须标示定理或性质
的名称或编号。
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第一题 请直接根据“直观冗”与“实战冗”的定义证明两者等价。请勿使用课堂上出现
过的相关性质或证明方式，例如证明两者都跟“威力冗”等价之类的方法。
第二题 You are given the fact that P_n(C) is a vector space over R. Prove or
disprove that its dimension is 2n+2.
第三题 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.
第四题 Prove or disprove -1_F=/1_F for any field F with 1_F+1_F_=/0_F.
第五题 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.
第六题 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.
第七题 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.
第八题 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.
第九题 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).
第十题 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(转置方阵）of A and -A denotes the
additive inverse of A.
第十一题 Prove or disprove that if V is a subspace of a vector space W with
dim(V) = dim(W) < ∞,
then V=W.