Exercises on 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, ..., ym 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 ATy = 0 with yTb = 1.
If b is not in the column space of A it is not orthogonal to the nullspace of AT . 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.
a) What are the conditions for those to be bases for the four fundamental subspaces C(AT), N(A), C(A), and N(AT) of a 2 by 2 matrix?
b) What is one possible matrix A?
