For each of your solutions x in Exercise 1: (a) determine the projection p = Ax. (b) calculate the residual r(x). (c) verify that r(x) N(AT ).

Week 1 Lecture 1 (1/12/2016) Introduction: Ring theory essentially generalizes the ideas familiar to us through our knowledge of the integers. We shall see some theorems and proofs. Well-Ordering axiom: This axiom states that every non-empty set of non-negative integers contain a smallest element. This axiom is helpful in many of the proofs that we will be learning next. Division Algorithm: Let a, b ∈ ℤ, b>0, then there exists unique integers q, r with the property that 1. a=bq+r 2. 0≤r