Elementary Linear Algebra, 11th Edition

APPENDIX AWORKING WITH PROOFS

Linear algebra is different from other mathematics courses that you may encounter in that it is more than a collection of problem-solving techniques. Even if you learn to solve all of the computational problems in this text, you will have fallen short in your mastery of the subject. This is because innovative uses of linear algebra typically require new techniques based on an understanding of its theorems, their interrelationships, and their proofs. While it is impossible to teach you everything you will need to do proofs, this appendix will provide some guidelines that may help.

What Is a Proof?

In essence, a proof is a “convincing argument” that justifies the truth of a mathematical statement. Although what may be convincing to one person may not be convincing to another, experience has led mathematicians to establish clear standards on what is to be considered an acceptable proof and what is not. We will try to explain here some of the logical steps required of an acceptable proof.

Formality

In high-school geometry you may have been asked to prove theorems by formally listing statements on the left and justifications on the right. That level of formality is not required in linear algebra. Rather, a proof need only be an argument, written in complete sentences, that leads step by step to a logical conclusion, and in which each step is justified by referencing some statement whose validity is either self-evident or has been previously proved. ...