[Solved] 18.06 Unit 2 Exercise 1-Orthogonal vectors and subspaces

Problem 16.1: (4.1 #7. Introduction to Linear Algebra: Strang) For every system of m equations with no solution, there are numbers y₁, …, y_m that multiply the equations so they add up to 0 = 1. This is called Fredholm’s Alternative:

Exactly one of these problems has a solution: Ax = b OR A^Ty = 0 with y^Tb =1.

If b is not in the column space of A it is not orthogonal to the nullspace of A^T . Multiply the equations x₁ − x₂ = 1, x₂ − x₃ = 1 and x₁ − x₃ = 1 by numbers y₁, y₂ and y₃ chosen so that the equations add up to 0 =1.

Problem 16.2: (4.1#32.) Suppose I give you four nonzero vectors r, n, c and l in R².

1. What are the conditions for those to be bases for the four fundamental subspaces C(A^T), N(A), C(A), and N(A^T)ofa2by2 matrix?
2. What is one possible matrix A?