# Introduction to Proof via Linear Algebra

Many universities offer a transition course from computational courses like Calculus to proof-oriented courses like Abstract Algebra. Such courses often go by a name like "Introduction to Proof" or "Transition to Higher Mathematics". They typically contain an introduction to first-order logic (conditionals, conjunctions, negations, quantifiers, etc.) as well as various methods of proof (contradiction, induction, etc.).

I'm hoping to find a text for a first course in linear algebra that fills the role of a "transition course" by deliberately incorporating first-order logic and proof techniques as part of the instruction.

The text should be accessible to students with two semesters of Calculus (roughly the basics of single-variable differentiation, integration, and infinite series). In particular, the overwhelming majority of students will have never written a formal proof and will have extremely limited exposure to logic and set theory.

Ideally, the author would discuss these topics just as they are needed in the treatment of linear algebra (as opposed to supposing the reader is familiar with them already). For example, the author might have a digression on proof by contradiction just prior to using it in some proof about linear independence.

Less ideal (but still acceptable) would be a text that at the very least makes use of all the relevant ideas from first-order logic and proof techniques that one expects from a transition course. Hopefully, the progression of such a text would be such that the instructor could use a supplemental text to discuss, say, proof by contradiction just as it is about to make its first appearance in the text.

• I worry my question isn't terribly clear, so please do not hesitate to ask for clarification. – Austin Mohr Mar 21 '17 at 3:57
• Two texts that at least go into this direction are Jim Hefferon, Linear Algebra and Isaiah Lankham, Bruno Nachtergaele and Anne Schilling, Linear Algebra - As an Introduction to Abstract Mathematics. Hefferon also has a dedicated introduction to proofs. I doubt that there is an American text that goes all the way to your goal, viz. (if I ... – darij grinberg Mar 23 '17 at 19:13
• ... understand it correctly) a text teaching linear algebra and basics of proofs at the same time. This might be related to the US-typical antipattern of trying to stuff all linear algebra (including some applications) into a single semester. European universities tend to offer a 2- or maybe sometimes 3-semester sequence on linear algebra, which allows for a less hasty treatment both of the basics and of some advanced topics like tensors; thus, I'd suggest looking for a text in German or French... but no idea how much your students will profit from that. – darij grinberg Mar 23 '17 at 19:14
• Actually, I've just realized that Klaus Jaenich's Linear Algebra exists in English. That might be worth a look! It begins by introducing sets, injectivity and surjectivity, etc. In Germany the text is occasionally considered too introductory (its first edition had the subtitle "notes for a first semester"), but this can be a good thing for the given question. – darij grinberg Mar 23 '17 at 19:16
• @darijgrinberg From a quick survey, it seems like Hefferon's text is closest to what I'm looking for. He has an appendix on sets, functions, and techniques of proof. He seems to indicate the first time these concepts arise in the text, which would be a clue to the instructor to spend some time fleshing out the details. You might consider posting Hefferon as an answer. Thanks. – Austin Mohr Mar 23 '17 at 22:11

Cherry-pick what you need from multiple books, if you need to. But here are some books that satisfy your requirements.

If you want a thorough, from the roots, introduction to higher mathematics, read

It covers logic, naive set theory, the real numbers and linear algebra. It is a great book, but unfortunately it is a bit less well-known. From the author:

This book helps the student complete the transition from purely manipulative to rigorous mathematics.

For a more standard course in linear algebra, consider

I would complement this book with Basic Concepts of Mathematics for the logic and set theory basics.

Not about linear algebra, but you should take a look at the

It is sort of a cookbook. For instance, one section is entitled 'How to prove $A \subseteq B$?'

• I think "Linear Algebra as an Introduction to Abstract Mathematics" supplemented with "Book of Proof" as needed is the recommendation I will be making. Thanks for your answer. – Austin Mohr Mar 30 '17 at 18:20

Linear Algebra - As an Introduction to Abstract Mathematics (direct link to pdf here) seems like a good shout. I haven't looked through it thoroughly, but the following bold claim is made in the intro:

In the setting of Linear Algebra, you will be introduced to abstraction. As the theory of Linear Algebra is developed, you will learn how to make and use definitions and how to write proofs.

Indeed, at the end of each chapter are special "Proof-Writing Exercises".

• I had initially missed the proof-writing exercises when I first looked over the text. These exercises make the text much more useful to me, though it is unfortunate that specific proof techniques (e.g., contraposition) are not developed as part of the reading. – Austin Mohr Mar 27 '17 at 20:04
• Unfortunately, I could only give the bounty to one answer, so I chose the one that also referenced "Book of Proof". Thanks for your answer. – Austin Mohr Mar 30 '17 at 18:21

From a colleague:

I've used Andrilli/Hecker for Linear Algebra + transition courses. It's my fave. It has a separate section on proof techniques in the middle of the first chapter, and then uses/develops them throughout the text. Nothing on first-order logic directly, but that hasn't seemed to present a problem because of the way notation is introduced gently.

My favourite book in linear algebra, as the first course, would be Schaum's Outline of Linear Algebra by Seymor Lipschutz.

Pros: Starts with an intense use of algorithms: vector, matrices, elimination, linear equations, etc. This comprehends the first three chapters. The rest of the book is the basic material for a first time course in linear algebra. It starts with axiomatic definition of vector spaces, then linear transformations, then connections between linear transformations and matrices, then inner products, diagonalization, Jordan's canonical form, etc. It has a plethora of examples.

Cons: It is highly devoted to algorithm, so mathematicians don't like that a much (This methods, though, are essential: imagine a mathematician incapable of solving a $3 \times 3$ ODE or being unable to diagonalize a small size matrix). Proofs are delayed to the end of each chapter, so the theorems are stated, then there are plenty of examples and then, at the end, the proofs.

• From a quick glance, I don't see that Schaum's does anything to help students transition to higher mathematical reasoning. Indeed, my general experience with Schaum's text is that they take the most direct route to computation. – Austin Mohr Mar 30 '17 at 18:24
• I learnt linear algebra from Schaums and I hated the book, I'll admit that. After a while, however, I noticed that many of other students who took linear algebra from more "reasonable" books like FIS, Hoffman, Lang, etc... The problem with those books is that they lack an intensive use of algorithms making people a bit of "cripple" (they lack a strong arm of solving problems). The book is great because, as I said, in chapter four it starts with axiomatic definitions and start making connection with the previous parts. – Will M. Mar 30 '17 at 18:30
• Also, never judge a book without having tried it; I don't know about other Schaum's texts, but the linear algebra one is definitely a great option for those who are going to be first exposed to the topic. – Will M. Mar 30 '17 at 18:30