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）︰103/11/11
考试时限（分钟）：60分钟
试题 :
台大资工双班线性代数第二次小考
2014年11月11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 V = (V, F, +,‧) be a vector space with 1 ≦ n = dim(V) < ∞. Let
α be an ordered basis of V. For each vector x∈V, let φ_α(x) = [x]_α be
the coordinate of x in V with respect to α. Prove that the function φ_α is
an invertible (i.e., bijective) linear transformation from V to F^n. Your
proof may directly use 座标定理, which states that each vector x∈V has a
unique way to be written as a linear combination of α.
第二题 Let V be a finite-dimensional vector space over scalar field F. You
are given the fact that (F(V,V), F, +_F,‧_F), where F(V,V) consists of the
functions from V to V, is a vector space. Prove that L(V,V), which consists of
the linear transformations T : V → V, is a subspace of F(V,V).
第三题 Let V = (V, F, +,‧) be a vector space with 1 ≦ n = dim(V) < ∞. Let
α =〈α_1, ... , α_n〉be an ordered basis of V. Prove that for any n vectors
x_1, ... , x_n ∈ V which are not necessarily distinct, there exists a unique
linear transformation T : V → V such that T(α_i) = x_i holds for each
i = 1, 2 , ... , n. (This is a specical case of our 裁缝定理.)
第四题 Let V = (V, F, +,‧) be a vector space with 1 ≦ n = dim(V) < ∞. Let
α =〈α_1, ... , α_n〉and β =〈β_1, ... , β_n〉be two ordered bases of V.
Let Φ_α^β be the function such that, for any linear transformation
T : V → V, Φ_α^β(T) is the representation matrix [T]_α^β for linear
transformationnT with respect to α and β. Prove that Φ_α^β is injective
(one-to-one) and surjective (onto). Your proof may directly use the special
case of 裁缝定理 in the previous problem and 座标定理.