Q1.(10%) Consider two events A and B, prove that if A and B are independent, then so are .

A municipal bond service has three rating categories (A, B, and C). Suppose that in the past year, of the municipal bonds issued throughout the United States, 70% were rated A, 20% were rated B, and 10% were rated C. Of the municipal bonds rated A, 50% were issued by cities, 40% by suburbs, and 10% by rural areas. Of the municipal bonds rated B, 60% were issued by cities, 20% by suburbs, and 20% by rural arcas. Of the municipal bonds rated C, 90% were issued by cities, 5% by suburbs, and 5% by rural areas. If a new municipal bond is to be issued by a city, what is the probability that it will receive an A rating? 

Three events: A1, A2 and B are given. We know that A1 and A2 are disjointed events. P(A1)>0 and P(A2)>0. Let P(B|A1)= P(B|A2)=0.2. Please find P(B|A1UA2).

(12%) Let V be a finite-dimensional vector space and T :V →V be linear. Suppose rank(T) = rank(T2). Prove that R(T) ∩N(T) = {0}.

(13%) Given a vector space V over F. Define the dual space of V* of V as the set of all functions (also known as linear functionals) from V to F, i.e, V* {f|f : V →F}. It is obvious that V* is itself also a vector space with the addition + :V*'x V*→ V* and scalar multiplication * : F x V*→  V* defined as pointwise addition as well as pointwise scalar multiplication. Given any linear transformation T : V→  W. The transpose Tt is a linear transformation from W* to V* defined by Tt(f) = fT for any f W*. For every subset S of V, we define the annihilator Suppose V, W are both finite-dimensional vector spaces and T : V → W is linear. Prove that .N(Tt) = (R(T))0.

(10%) The nullities of the matrices BBT - λ/ for  λ= 0, 1,2,3, 4  are______,________,_________,_______ respectively. 

(d) (2%) Compute dim(N(Bt A)).

(c) (3%) Compute rank(AtAAAt).

(b) (2%) Compute rank(AB).

(a) (3%) Compute rank(A).

(b) Find the nullity of matrix BTA (10%)

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