課程名稱︰線性代數一
課程性質︰數學系必修
課程教師︰余正道
開課學院：理學院
開課系所︰數學系
考試日期︰2019年10月25日，11:20-11:45
考試時限：25分鐘
試題：
Show your work details.
1. (10%) Let A and B be two n by n complex matrices. If A is similar to B,
show that trace(A) = trace(B).
2. Let P_2 be the set of all univariate polynomials of degree less then or
equal to 2 with real coefficients and B = {1,x,x^2} be the standard ordered
basis of P_2. Define the linear transformation
T : P_2 → P_2
by (Tf)(x) = f'(x) + 3f(x).
(a) (6%) Write down [T]_B.
(b) (4%) Show that T is a bijection, i.e. T is invertible.
3. Let V be a vector space over the field F and S be a subset of V.
(a) (2%) Write the definition of the annihilator of S, i.e. What is S^0?
(b) (2%) If S = {0}, what is S^0? (You do not need to prove your answer.)
(c) (6%) Suppose that S_1 and S_2 are subsets of V and S_1⊆S_2. Show that
S_2^0⊆S_1^0.