课程名称︰线性代数
课程性质︰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）︰103/11/13
考试时限（分钟）：60分钟
试题 :
(注意数字 1 与英文字母 l 的差别)
台大资工单班线性代数第二次小考
2014年11月13日下午四点起一个小时
总共四题，每题十分，可按任何顺序答题
第一题 Let
T : R → S,
where R and S are sets. Prove that T is invertible if and only if T is
bijective.
第二题 Let U and V be vector spaces over scalar field F with dim(U) < ∞.
Let
T : U → V
be linear. Prove that
dim(U) = nullity(T) + rank(T).
(This is our Dimension Theorem“维度定理”, where nullity(T) is the dimension
of N(T) and rank(T) is the dimension of T(U).)
第三题 Let U be a vector space over F with 1 ≦ l = dim(V) < ∞. Let V be a
vector space over F. Let α =〈α_1, ... , α_l〉be an ordered basis of U.
Prove that for any l vectors y_1, ... , y_l ∈ V which are not necessarily
distinct, there exists a unique linear transformation
T : U → V
such that T(α_i) = y_i holds for each i = 1, 2, ... , l. (This is our“裁缝
定理”.)
第四题 Let U be a vector space with 1 ≦ l = dim(U) < ∞. Let
β =〈β_1, ... , β_l〉and γ =〈γ_1, ... , γ_l〉be two ordered bases of U.
Let Φ_β^γ be the function such that, for any linear transformation
T : U → U,
Φ_β^γ(T) is the representation matrix [T]_β^γ for T with respect to β
and γ. Prove that
1. Φ_β^γis linear and
2. Φ_β^γis surjective (onto).
Your proof may directly use“裁缝定理”in the previous problem and“座标定
理”, which states that each vector y∈U admits a unique way to be written as
a linear combination of any ordered basis of U.