FINN 6203 Homework 1 Solutions 1. Consider the matrix 1 2 3 4 A = 2 3 4 5 3 4 5 6 As a linear transformation, A maps R4 to R3 . Find a basis for N ull(A), the null space of A, and find a basis for Col(A), the column space of A. Describe these spaces geometrically. Solution. Solving bf Ax = 0, 1 2 3 4 0 1 0 −1 −2 0 2 3 4 5 0 ∼ 0 1 2 3 0 3 4 5 6 0 0 0 0 0 0 Two linearly independent solutions are [1, −2, 1, 0]0 and [2, −3, 0, 1]0 , so these vectors form a basis for the null space of A. Since Dim(N ull(A)) = 2, the null space is a plane in R4 . Now, just finding the reduced row echelon for for A, 1 2 3 4 1 0 −1 −2 2 3 4 5 ∼ 0 1 2 3 3 4 5 6 0 0 0 0 Since there are pivots in columns 1 and 2 or the reduced row echelon form, columns 1 and 2 of A form a basis for the column space of A. In this case Col(A) is a plane in R3 . 2. For A in problem 1, what is Rank(A)? Solution. Since there are pivots the reduced row echelon form for A, Rank(A) = 2. 3. Consider a one-period economy with two times, 0 and T . There are S = 3 states and N = 3 securities with payoff matrix 23 12 10 D = 30 21 10 46 27 10 Describe the space of all attainable payoffs. Solution. Since there are 3 outcomes in the sample space and Rank(D) = 3, the space of attainable payoffs is all of R3 . 1