Explain the following que.stions as attached below.

Pages 2-5 Q1 a) Let V be a finite dimensional vector space over a field F. If S and S' are two different bases of V, then prove that the number of elements in them must be the same. (3) b) Professor X asks two of his students P and Q to calculate the number of multiplications and additions involved in evaluating A" for a given square matrix A. P, being obedient, does the given job in the naive way and comes out with s multiplications and / additions. @ observes that r = j x k, where j > 1 and k > 1 and evaluates A" as (A')" and observes that he had used s' multiplications and ?' additions. Using the above compute (s - s'| and It - t'|. (2) Pages 6-8 Q2 a) Let T : V - W be a linear transformation with V and W being finite dimensional vector spaces over the same field F. Using the following, prove the rank nullity theorem, answering the questions that are asked with justifications. ij Is Ker(7) a subspace of V7 If so, start with (1, "2,.. .. .} as the basis of Ker(7) (0.5] ii) What is the relationship between the basis of Ker(?') and the basis of V? This would give the hint to come out with a basis of V. (1) ifij i) and if) would give a hint on the basis for Range(T) based on the statement of the rank nullity theorem. Use that to complete the proof of the theorem. (1.5) b) Construct, if possible, a linear transformation 7 : R' - R3 with Ker(?) = (0}, where the field is R. If not possible, give justification. (2) Pages 9-11 Q3 a) It is known that the matrix that rotates a vector through an angle of in [cos(#) - sin(#) two dimensions is given by: A(?) = sin(a) cos(#) Show that the no power of rotation matrix through an angle of is same as the matrix that performs rotation through an angle no (n times #) where n is an integer. In other words, prove that: (A(8)]" = And). (3)Q2. (a) Show that N(u, (2), is not a member of the Regular Exponential Class. [Marks 4] (b) Consider a random sample of size n from a Normal distribution, N(1, 02) where u is known. (i) Find a sufficient statistic for o using the Factorization Criterion. [Marks 4] (ii) Show that the statistic found in (i) is also sufficient using the conditional joint density method. [Marks 4] (iii) Show that the statistic found in (i) is also complete from definition and without using membership in Regular Exponential Class. [Marks 6] (iv) Construct a Uniformly minimum variance unbiased estimator for a. [Marks 8] (v) Derive the Cramer Rao Lower Bound for Unbiased Estimators of o. [Marks 4] (vi) Derive the asymptotic distribution of the Maximum Likelihood Estimator for o. [Marks 4]Problem 4: presidential predictions (6 Pts.) An Individual has preferences over "contingent" consumption in two possible altua- tions: consumption -in dollars- if Biden wins ro and consumption -In dollars- if Trump wins TT, given by U(TB, IT) = PBIB + (1- PB)IT where PB is the probability, according to the agent, of a Biden win, her belief so to speak. Thus we can interpret this agent as someone that evaluates consumption in the two situations using as weights the probabilities that each take place (since the probability of a Trump win is 1 - PB)- Her endowment is (10, 10), namely she has 10 dollars in her pocket, which remain 10 dollars whether Biden or Trump win. a) our agent has acces to a prediction market where she can buy or sell bets on a Biden win. Let a ( [0. I] be the price of a 1 dollar bet on a Biden win, namely a bet that repays $ 1 if Biden wins, and zero otherwise. To understand how this works, immagine the agent buys a 2 dollar bet on Biden winning. Then she doles out a2 dollars, and, if Biden wins, she earns $ 2. This allows her to move, in our contingent consumption space, from (10, 10) to (12-02, 10-a2). Draw a graph of her budget set for a generic a e (0, 1). How would the budget set look like if the agent could only buy, and not sell, bets on Biden? b) assume the agent believes the probability of a Bidon win is 0.6. Find the agent's net demand and net supply for bets on Biden. At which price a* is the agent Indifferent between buying and selling bets on Biden? c) Suppose now that the agents utility is described by u(FB, IT) = 0.6u(FB) + 0.4u(IT) where u is increasing, continuous and differentiable, and such that the agent's pref- crences are strictly convex (and as in b) her belief of a Biden win is 0.6). Will theb. Draw a graph of a market where government quota controls have limited the quantity supplied to less than the market equilibrium price and quantity. Suppose these production quotas are eliminated. Explain, verbally and graphically, what happens to the market price and quantity supplied after the controls are eliminated.| 3. Price Elasticities of Demand and Supply (3) a. Define price elasticity of demand. b. Draw a demand diagram for a good that has an inelastic demand and use it to explain what It means to say that the demand is price "Inelastic." c. Draw a supply diagram for a good that has an elastic supply and use it to explain what it means to say that the supply is price "elastic." 4. Total Revenue Test (2) a. Suppose demand is elastic. Explain what happens to total revenue when there is a decrease "in the price of a product, all else held constant. b. If price increases and total revenue increases, is demand elastic, unitary elastic, or inelastic. Explain Income Elasticity (1] a. Define "Income Elasticity of Demand" and use it to explain verbally and with a diagram what happens to demand when there is an increase in income for an "Inferior good."