Some good book suggestions beyond linear algebra?

Question

High school student here...

Recently I was told I could go to the book store and pick out any math books I want. (2 or 3)

Does anyone have some good suggestions? I'm comfortable with anything involving calculus and I am currently studying linear algebra. It doesn't really even need to be a textbook just something that will help me learn and build mathematical maturity.

Here's my answer to a closely related question: https://math.stackexchange.com/questions/1714966/what-books-should-i-get-to-self-study-beyond-calculus-for-someone-about-to-start/1714986#1714986 — Ethan Bolker, Aug 05 '18 at 18:50

score 5 · Answer 1 · answered Aug 05 '18 at 18:03

5

Two suggestions:

Discourses on Algebra, by Igor Shafarevich
Geometry: Euclid and Beyond, by Robin Hartshorne

answered Aug 05 '18 at 18:03

José Carlos Santos

427,504

+1 for Hartshorne's book. A very good suggestion. – Aug 05 '18 at 18:06
@Brahadeesh Thank you. – José Carlos Santos Aug 05 '18 at 18:07
You like a lot Shafarevitch José... – user577215664 Aug 05 '18 at 18:09
1

@Isham I like the book Discourses on Algebra a lot. – José Carlos Santos Aug 05 '18 at 18:12
Everyone seems to really like these suggestions. I'll keep them in mind – CaptainAmerica16 Aug 05 '18 at 18:19

score 3 · Accepted Answer · answered Aug 05 '18 at 18:05

3

Have you seen What is Mathematics? by Courant and Robbins? It is by far the best introduction to mathematics that you can get your hands on, in my opinion. Here's a short form of the table of contents for your reference:

Chapter I: The Natural Numbers
Chapter II: The Number System of Mathematics
Chapter III: Geometrical Constructions. The Algebra of Number Fields.
Chapter IV: Projective Geometry. Axiomatics. Non-Euclidean Geometry.
Chapter V: Topology
Chapter VI: Functions and Limits
Chapter VII: Maxima and Minima
Chapter VIII: The Calculus
Chapter IX: Recent Developments

answered Aug 05 '18 at 18:05

Topology before calculus? This definitely seems worth checking out. – CaptainAmerica16 Aug 05 '18 at 18:08
@CaptainAmerica16 Here are the subsections in the Topology chapter: 1. Euler's Formula for Polyhedra; 2. Topological Properties of Figures; 2.1 Topological Properties; 2.2 Connectivity; 3. Other Examples of Topological Theorems; 3.1 The Jordan Curve Theorem; 3.2 The Four Color Problem; 3.3 The Concept of Dimension; 3.4 A Fixed Point Theorem; 3.5 Knots; 4. The Topological Classification of Surfaces; 4.1 The Genus of a Surface; 4.2 The Euler Characteristic of a Surface; 4.3 One-Sided Surfaces – Aug 05 '18 at 18:13
@CaptainAmerica16 The chapter ends with an appendix: 1. The Five Color Theorem; 2. The Jordan Curve Theorem for Polygons; 3. The Fundamental Theorem of Algebra. – Aug 05 '18 at 18:13
@CaptainAmerica16 So, definitely a very interesting set of topics :) – Aug 05 '18 at 18:14
It's official, this will be the first book I look for. Thanks! – CaptainAmerica16 Aug 05 '18 at 18:15
Glad to be of help! – Aug 05 '18 at 18:15

Anik Bhowmick · Answer 3 · 2018-08-05T18:31:16.343

2

If you wanna study linear algebra, then I'd like to suggest - I) Linear Algebra (Friedberg, Insel, Spence) II) Introduction to Linear Algebra (Gilbert Strang) III) Linear Algebra (Kwak and Hong) IV) Linear Algebra (Kenneth Hoffman, Ray Kunze)

For abstract algebra, go for - I) A First Course in Abstract Algebra (J.B. Fraleigh) II) Abstract Algebra (Dummit and Foote) III) Contemporary Abstract Algebra (Joseph A. Gallian)

For Analysis, study - I) Mathematical Analysis (Tom M. Apostol) II) Real and Complex Analysis (Walter Rudin) III) Real Analysis (Royden , Fitzpatrick) IV) Introduction to Real Analysis (Bartle, Sherbert)

edited Aug 05 '18 at 18:31

answered Aug 05 '18 at 18:03

Anik Bhowmick

1,051

Thank you! Abstract algebra is high on my list of further interests. – CaptainAmerica16 Aug 05 '18 at 18:11
I'd say that Rudin requires quite a bit of mathematical maturity in the reader. I wouldn't recommend it to a high school student... – Aug 05 '18 at 18:16
Then what should he study for analysis ?? @Brahadeesh – Anik Bhowmick Aug 05 '18 at 18:26
Perhaps Tao's Analysis I and II, or Sherbert and Bartle's Introduction to Real Analysis would be my recommendations. But Apostol is good too! – Aug 05 '18 at 18:29
Yeah, thanks @Brahadeesh, I forgot to mention Bartle-Sherbert. – Anik Bhowmick Aug 05 '18 at 18:30
@Brahadeesh & Anik Noted. – CaptainAmerica16 Aug 05 '18 at 19:12

score 2 · Answer 4 · answered Aug 05 '18 at 18:34

Vector Analysis by Murray Spiegel is one good book with lots of exercises. Also I googled "Cambridge University Maths Reading List" and found this. I am sure others exist if you look.

I read Fraleigh's "First Course in Abstract Algebra" when I was at school, and found it accessible and good.

Körner's "Calculus for the ambitious" is a quirky approach - but geared to building understanding in the bridge between "calculus" at school and "analysis" at university.

And Feynman's Lectures on Physics are worth reading, too.

+1 For the reading list. Thanks :) – CaptainAmerica16 Aug 05 '18 at 19:01 — CaptainAmerica16, Aug 05 '18 at 19:01

score 1 · Answer 5 · answered Aug 05 '18 at 18:02

1

V. Prasolov's Problems and Theorems in Linear Algebra will not only teach you everything you need to know about linear algebra, but also give you germs of ideas that will prove very useful later if you study more advanced mathematics where linear algebra occurs in a way or another.

answered Aug 05 '18 at 18:02

Arnaud Mortier

27,276

Wow, thank you so much for the link! This will be really helpful. – CaptainAmerica16 Aug 05 '18 at 18:23

MR_BD · Answer 6 · 2018-08-07T19:33:12.717

1

I have a favorite author. Peter J. Cameron. I love the way he tells the math-story for you. He has some lecture notes that you may find on his homepage for both linear algebra and abstract algebra.

Beside that I recently faced a cute book for Linear Algebra which I highly recommend. The approach of the book is based on Linear Maps (and not matrices) and try to not use Determinant concept directly. The name of the book is "linear algebra done right".

Also, for real analysis I recommend to read Real Mathematical Analysis by Charles Pugh. The book is compact and lovely with really nice exercises.

edited Aug 07 '18 at 19:33

answered Aug 06 '18 at 15:52

MR_BD

5,942

1

Thank you for the recommendations. The lecture notes are perfect :) – CaptainAmerica16 Aug 06 '18 at 16:05

Some good book suggestions beyond linear algebra?

6 Answers6