Appendix P  Preliminaries

This appendix contains important ideas about complex numbers, sets, and the logic and techniques of forming proofs. It is not meant to be read straight through, but you should head here when you need to review these ideas.

We choose to expand the set of scalars from the real numbers, , to the set of complex numbers, . So basic operations with complex numbers (like addition and division) will be necessary. This can be safely postponed until your arrival in Section O, and a refresher before Chapter E would be a good idea as well.

Sets are extremely important in all of mathematics, but maybe you have not had much exposure to the basic operations. Check out Section SET. The text will send you here frequently as well. Visit often.

This book is as much about doing mathematics as it is about linear algebra. The “Proof Techniques” are vignettes about logic, types of theorems, structure of proofs, or just plain old-fashioned advice about how to do advanced mathematics. The text will frequently point to one of these techniques in advance of their first use, and for specific instructions there will be additional references. If you find constructing proofs difficult (we all did once), then head back here and browse through the advice for second or third readings.

 Section CNO Complex Number Operations
  Subsection CNA: Arithmetic with complex numbers
  Subsection CCN: Conjugates of Complex Numbers
  Subsection MCN: Modulus of a Complex Number
 Section SET Sets
  Subsection SC: Set Cardinality
  Subsection SO: Set Operations
 Section PT Proof Techniques
  Proof Technique D: Definitions
  Proof Technique T: Theorems
  Proof Technique L: Language
  Proof Technique GS: Getting Started
  Proof Technique C: Constructive Proofs
  Proof Technique E: Equivalences
  Proof Technique N: Negation
  Proof Technique CP: Contrapositives
  Proof Technique CV: Converses
  Proof Technique CD: Contradiction
  Proof Technique U: Uniqueness
  Proof Technique ME: Multiple Equivalences
  Proof Technique PI: Proving Identities
  Proof Technique DC: Decompositions
  Proof Technique I: Induction
  Proof Technique P: Practice
  Proof Technique LC: Lemmas and Corollaries