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 .
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