I'm not sure that this is the appropriate forum for this question but the snooping around that I've don of the site makes me feel as if I'll get solid advice. I'm currently finishing an introductory course in Linear Algebra, which, from what I understand is one of the last "computational" courses in mathematics that most students take. I've finished the Calculus sequence through vector calculus and differential equations and I am transferring to my local university (University of Arizona) in the spring to being 300+ level math courses.

I guess I'm trying to solicit advice on how to make the math transition as smooth as possible. My first semester at the University consists of a course title "Formal Mathematical Reasoning and Writing" which is essentially an introduction to proof. The text is Analysis with an Introduction to Proof by Lay and from what I understand, is a fairly difficult transitional math course. It is the pre-req for just about every course thereafter (Real Analysis, Linear Algebra, Abstract Algebra, Topology, etc.). I want to be as prepared as possible going into the course and I've done a few basic proofs in my Linear Algebra course and they've been challenging to say the least. Are there any good self-study resources that anyone can recommend? Anyone willing to share experiences of making the transition from more computational mathematics to theoretical mathematics?

Any thoughts are very much appreciated, and if this is not meant for this venue, I apologize and will take this down.


  • $\begingroup$ You may find some interesting reading here: terrytao.wordpress.com/career-advice. $\endgroup$ – Austin Mohr Nov 9 '12 at 20:14
  • $\begingroup$ Can you be more specific when you say your linear algebra proofs were "challenging"? $\endgroup$ – wj32 Nov 9 '12 at 21:07
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    $\begingroup$ I suppose what I mean is that the thought process was new. All of the proofs were basic (ie prove that if two vectors in R^n are orthogonal, then they are linearly independent) I almost always knew where I was going but finding the correct steps/using the correct definitions often took more time then I expected. I found my self at office hours more than once trying to dig through where I was trying to land with the proofs assigned. Unfamiliar may be a better description. What I'm really trying to get at is I struggled getting from point A to point B. $\endgroup$ – Ben Anderson Nov 9 '12 at 21:45

This free set of notes (very complete) by a great teacher (Vaughan Jones), is actually for the real analysis course a Berkeley. While this, in its totality, may be a step beyond what you will be doing, it builds from the very basics - so you can go as far as you like.

But the exposition, especially in the early proofs, give a tremendous amount of insight into how to approach thinking about proofs. The treatment of math here shows how one of the math greats (Fields Medal winner) takes you into the beautiful world of rigorous math. Plus anything you glean here will always serve you.


ADDITIONALLY: This link to a question here gives a good discussion of a very important proof method. It's well worth making your own:

What's the difference between a negation and a contrapositive?

  • $\begingroup$ Thank you, this looks like it is going to be a great resource. $\endgroup$ – Ben Anderson Nov 10 '12 at 2:02

Ask this question of the people who taught the courses you have taken. They will know your strengths and weaknesses, if there are any, better than any of us on this site. You may wish to get a copy of the text and work as many of the problems in the first section or chapter as you can. If you can work most of them you will be that much ahead. If you have difficulties you will know what you need to concentrate on.


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