Undergraduate Level Math Books

Question

What are some good undergraduate level books, particularly good introductions to (Real and Complex) Analysis, Linear Algebra, Algebra or Differential/Integral Equations (but books in any undergraduate level topic would also be much appreciated)?

EDIT: More topics (Affine, Euclidian, Hyperbolic, Descriptive & Diferential Geometry, Probability and Statistics, Numerical Mathematics, Distributions and Partial Equations, Topology, Algebraic Topology, Mathematical Logic etc)

Please post only one book per answer so that people can easily vote the books up/down and we get a nice sorted list. If possible post a link to the book itself (if it is freely available online) or to its amazon or google books page.

Answer 1

Algebraic Topology by Hatcher (available online here).

Answer 2

Generatingfunctionology by Wilf is fun, free, requires very little in the way of prerequisites, and is as good an introduction to the methods of analytic combinatorics as could be asked for. It's long been one of my favorite textbooks.

Answer 3

Spivak, Calculus

Answer 4

Concrete Mathematics, Graham, Knuth and Patashnik. Extremely useful, very good exercises, and a sense of humor that appeals to me.

Answer 5

A basic undergrad algebra book which I feel is not as well known as it should be is Michael Artin's Algebra. I have it in soft cover so I hope it's actually the one in this Amazon link. Anyway it's beautifully written, provides context and motivation and is just a pleasure to read or browse. How often do you find a basic text written by a world-class expert?

"Always study from the masters".

Answer 6

For undergraduate level topology (mostly point set topology) I recommend "Topology" by Munkres. I learned topology from this book as an undergrad and I remember this being one of my favorite books at the time.

Answer 7

Milnor, Topology from the differentiable viewpoint.

Answer 8

Serre, A Course in Arithmetic.

Answer 9

Needham, Visual Complex Analysis. I read this while in high school, and it's simply beautiful. I recommend this book as a supplement to any first course in complex analysis (a different book should probably be used for the main textbook since Needham's is very pretty, very engaging, but not very rigorous).

Answer 10

The Chicago undergraduate mathematics bibliography is a nice annotated list of books.

Answer 11

Ordinary Differential Equations by Vladimir I. Arnold

Answer 12

Miles Reid, Undergraduate Algebraic Geometry.

ps - anyone who thinks one can teach an undergraduate class out of Hatcher's Algebraic Topology (which is a great book) at more than 10 universities in the US is sadly deluded. Ditto for a few more things I've seen here.

pps - somewhere between a third and half of the math majors here could handle abstract algebra out of Artin. It would be great for those that could, but we're not going to ditch half our students.

Answer 13

Not exactly one of the topics in the question, but I particularly liked Silverman and Tate's Rational Points on Elliptic Curves.

Answer 14

Differential Geometry of Curves and Surfaces, by Manfredo Do Carmo is an excellent introductory book.

https://www.amazon.com/Differential-Geometry-Curves-Surfaces-Manfredo/dp/0132125897

Answer 15

Complex Analysis, by Lars Ahlfors

https://en.wikipedia.org/wiki/Lars_Ahlfors

Answer 16

The Princeton Lectures in Analysis by Stein and Shakarchi are great introductions to Fourier, complex, and real analysis (in that order!).

Answer 17

Real and Complex Analysis by W. Rudin is a beautiful and extremely well written book which presents the fundamentals of real and complex analysis highlighting the interactions between different results and ideas.

Answer 18

Spivak, A Comprehensive Introduction to Differential Geometry. There is a nice geometrical philosophy and plenty of motivation.

Answer 19

As an undergraduate, I loved Shafarevich's book Basic notions of algebra. This is not a textbook, but gives small beautiful tastes of a broad choice of topics in algebra, emphasizing connections with other fields.

I found it very stimulating, in the sense that every example or overview of some topic in this book made me want to learn more details about it. In fact I became interested in algebraic geometry because of this book.

Answer 20

Guillemin and Pollack, "Differential Topology"

Answer 21

Topics in Algebra by I. N. Herstein. A new edition will be coming out this year.

Answer 22

Dummit and Foote's Abstract Algebra is an excellent book for learning group theory, ring theory, and module theory. There's also a section on basic algebraic geometry and homological algebra.

Answer 23

I didn't see any suggested books from the great Russian school of mathematics, here is a brief list of superb, well written, example oriented books:

• Elements of the Theory of Functions and Functional Analysis by A. N. Kolmogorov and S. V. Fomin
• Theory of Functions of a Real Variable by I. P. Natanson (this I think, it's hard to find)
• Theory of Functions of a Complex Variable, Second Edition (3 vol. set) by A. I. Markushevich
• Elements of Functional Analysis by L.A. Lusternik and V.J. Sobolev
• Problems in Mathematical Analysis by B. Demidovich
• Calcul intégral et differentiel (2 vol. set) by N. Piskounov. In English should be something like Differential and Integral Calculus
• A Course of Mathematical Analysis (2 vol. set ) S. Nikols'skii
• Differentialrechnung und Integralrechnung (3 vol. set), by Gregor M. Fichtenholz. Unfortunately, there is no English translation of this book, only the German translation that it's mentioned. I think this was THE calculus book on Russia. THIS BOOK SHOULD BE TRANSLATED INTO ENGLISH, and I suspect that there is no copyright, it appears around 1959 I think.
• Mathematical Analysis (2 vol. set) by Vladimir A. Zorich. This is a very recent book from a great mathematician which is in Moscow university. The book is based upon is lecture.

I have also found a Spanish translation of a book written by S. Banach about "Differential and integral calculus". It is very good as an undergraduate book. The spanish translation (for those who want to search) is Calculo diferencial e integral

Books that undergraduate should not touch (in my humble opinion) are books written in bourbaki's style.

Answer 24

Ireland and Rosen, A Classical Introduction to Modern Number Theory is a great second course in number theory. In spite of being part of "Graduate Texts in Mathematics" series and unlike Rudin's Real and Complex Analysis (see a comment above), this is a book at the undergraduate level. It only presupposes undergraduate algebra as in Herstein Topics in Algebra or M. Artin's Algebra, undergraduate analysis like in Rudin's Principles of Mathematical Analysis and basic number theory. In fact it recalls or proves many of the necessary results in each of those fields. A Classical Introduction to Modern Number Theory bridges the gap between basic number theory (that covers modular arithmetic, Fermat's little theorem and QR) and books like Lang's Algebraic Number Theory or Cassels and Fröhlich.

Answer 25

I am surprised no one has mentioned Halmos' Naive Set Theory or Finite-Dimensional Vector Spaces or Rudin's Principles of Mathematcal Analysis. There's also Sheldon Axler's Linear Algebra Done Right and Royden's Real Analysis.

Answer 26

For discrete mathematics, I would recommend Van Lint-Wilson's "A Course in Combinatorics" as a good introductory text. It consists of 38 (in my edition) chapters that give (often largely self-contained) introductions to various areas of the field. Although it doesn't go nearly as in depth as, say, Stanley's "Enumerative Combinatorics" or a text focused solely on graph theory, I found it excellent for giving a broad overview and indicating to me where I wanted to explore deeper.

My one caveat would be that some chapters require background in either linear algebra or basic group theory, though those are easily skippable due to the structure of the book.

Answer 27

Basic Algebra by Jacobson.

Answer 28

General geometry: Coxeter, Introduction to Geometry.

Not so much a textbook as a collection of essays (in particular, it doesn't have exercises), but all of the essays are instructive and enlightening.

Answer 29

I am surprised this has not been mentioned before (is it too advanced?):

Bott and Tu, Differential forms in algebraic topology.

The best introduction to de Rham cohomology, spectral sequences, characteristic classes from the algebraic point of view, and countless other topics.

Answer 30

Algebra: Chapter 0, Paolo Aluffi

Best book on algebra I've had my hands on yet, and I love how it uses category theory. I wouldn't mind having a course taught from this one. Topics from group theory all the way through field theory, linear algebra, and homology. This book deserves more attention!

https://www.amazon.com/Algebra-Chapter-Graduate-Studies-Mathematics/dp/0821847813/ref=sr_1_1?ie=UTF8&s=books&qid=1278799249&sr=8-1

Answer 31

Ok, this is not a single book, but I have often found books from the Springer Undergraduate Mathematics Series (SUMS) to be excellent. Here is a list of titles.

Answer 32

Another one I like is "An introduction to Lie algebras." by Erdmann and Wildon.

Answer 33

Galois Theory by Ian Stewart is excellent. The third edition is quite different from the second and includes many more problems.

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Answer 34

Jaenich: "Topology"

Introduces the concepts of point set topology ("paracompact" and all this stuff) motivating each via examples which are rigorously defined but also drawn. Other advantage: It is short!

Answer 35

Apostol "Calculus"

Answer 36

Bartle "The Elements of Integration and Lebesgue Measure"

Answer 37

Searcóid: Elements of Abstract Analysis. I loved this book as an undergraduate, for many reasons, but mainly because it gave me an idea of the unity of mathematics. It starts from the axioms of set theory and takes you all the way to C*-algebras and the Gelfand-Naimark theorem. Here's the Google Books page.

Answer 38

I'm a big fan of John Hubbard's "Vector Calculus, Linear Algebra and Differential Forms" text. I was a TA for the course twice at Cornell and was amazed at how well it went. The text has an extremely pleasant "zest" to it. When Hubbard asked me to take a look at it my first response was the text is "overflowing with the spirit of calculus". I still believe that. I have a hard time containing my praise.

The main problem with the text is that it's so engrossing. It places more demands on the student than a traditional service course text would ever consider. But it's also far more rewarding. At Cornell it was taught as a branch of their traditional calculus sequence -- it was a course that was earmarked for keener students, mostly from other departments.

In short, if you want to have physics, engineering and economics students appreciating the derivative as a linear approximation, thinking Lipschitz bounds for functions are cool, being interested in the computation of norms of linear operators, etc, this is a great resource.

Answer 39

Karen Smith et al., An Invitation to Algebraic Geometry

Answer 40

Fraleigh's "A First Course in Abstract Algebra"

https://www.amazon.com/First-Course-Abstract-Algebra-7th/dp/0201763907

Answer 41

Visual Complex Analysis by Tristan Needham is awesome!

Answer 42

Linear Algebra / Hoffman & Kunze - A book that truly develops linear algebra in a gradual manner. It starts with a basic discussion of systems of linear equations, matrices, Gaussian elimination, etc. and gradually progresses to the more abstract theory. Eventually it even touches upon subjects such as tensor products, the exterior algebra and the Grassmann ring. In short, it manages to cover a lot of linear algebra in a very leisurely and clear manner. I think that this is the quintessential example of a how an undergraduate level math book should be written. The only thing I don't like about it is the fact that quotient spaces aren't mentioned throughout the book (they're mentioned in the appendix, though).

Answer 43

For a thorough introduction on Partial Differential Equations, read L.C. Evans, "Partial Differential Equations". Features both linear and nonlinear equations.

Answer 44

Real Mathematical Analysis by Charles Pugh

Answer 45

Lectures on Linear Algebra by I. M. Gel'fand

Answer 46

Also, I just started this book and absolutely love it

Geometry: Euclid and Beyond, Hartshorne

Answer 47

For a long time, Kolmogorov-Fomin's Introductory Real Analysis was my standard for a great mahtematics textbook. I can't imagine a better introduction to serious analysis.

The translation I'm linking to is very good, and includes excercises (the original has many fewer), but it is incomplete (it's missing the chapter on Fourier Series). So if you can read Russian, I recommend you get the original.

Answer 48

From the quick look I've had, not much representation theory has been mentioned so here goes for undergrad level rep theory (perhaps suitable for 3rd/4th year in a standard sequence of undergraduate study), roughly in the order of difficulty (from easiest to hardest):

• James & Liebeck - "Representations and Characters of Groups" (a very good introduction)

• Sagan - "The symmetric group: representations, combinatorial algorithms, and symmetric functions"; (the first two chapters here at least are representation theory) OR James & Kerber - "Representation Theory of the Symmetric Group" (this one includes some modular representations of $S_n$)

• Alperin - "Local Representation Theory" (basically, modular representation theory)

• Hall - "Lie groups, Lie Algebras and Representation Theory" (a solid introduction to Lie theory); for a more advanced perspective Harris & Fulton - "Representation Theory: A first course" (but it could be slightly terse at points, but not necessarily)

For algebraic geometry, the one book I'd suggest is "Algebraic Geometry: A first course" by Joe Harris, very nice and full of examples. For algebraic number theory, a very good introduction is Janusz - "Algebraic Number Fields" (followed perhaps by Childress - "Class Field Theory", or Silverman - "The Arithmetic of Elliptic Curves" to go in a slightly different direction).

Answer 49

Here is an undergraduate level math book recommendation from an early undergrad's position:

I like "Linear Algebra Done Right". I've looked at a bunch of books on linear algebra, and the usual matrix approach is to me a big turn-off when what you're really interested in is the abstract machinery of transformations between vector spaces. I'm not a research mathematician. In fact, I don't even study linear algebra yet, but as a student of mathematics that like algebra, spaces, maps and all that good stuff, I find this to be a very readable account of linear algebra.

There are more abstract books on the subject, and my impression is that LADR prepares you for the next level way before you're usually "allowed to" by other accounts like Lax etc. The trade-off is that LADR is not a book for engineers, but this would be a sad world for a mathematician if that was something he had to worry about (in his spare time). Great for self-study. Reads like a novel. I'd probably prefer it if Axler used sets for span and bases instead of lists, but that's something you'll probably be able to shake off with the next book you read on the subject.

Answer 50

An Introduction to Manifolds by Loring W. Tu

https://www.amazon.com/Introduction-Manifolds-Universitext-Loring-W/dp/0387480986/ref=sr_1_1?ie=UTF8&s=books&qid=1256082981&sr=1-1

Answer 51

Riemannian Geometry: A Beginner's Guide by Frank Morgan

https://www.amazon.com/Riemannian-Geometry-Beginners-Frank-Morgan/dp/1568810733/ref=sr_1_1?ie=UTF8&s=books&qid=1256083041&sr=1-1

I love this book!

Answer 52

Lebesgue Integration on Euclidean Space / Frank Jones - an extremely readable book on Lebesgue theory on $\mathbb{R}^n$ (lots of figures and geometric intuition). He constructs Lebesgue measure in a very down-to-earth manner, much more explicitly than other more abstract constructions (via Caratheodory's extension theorem or Riesz's representation theorem). In my experience, it's best to first study Lebesgue measure on $\mathbb{R}^n$ and only then point out that it's merely one instance of the general theory of measures, which is the way this book is written. It can't compare with the "tougher" books on measure theory (e.g. Big Rudin) since it doesn't discuss the Radon-Nikodym theorem and many other important theorems in measure theory, but then again the book is clearly intended for an undergraduate audience, and as for Lebesgue theory on Euclidean spaces, it provides a pretty complete picture.

Answer 53

Subject: FUNCTIONAL ANALYSIS

Erwin Kreyszig

Introductory Functional Analysis with Applications

Answer 54

E. Hairer, G. Wanner: Analysis by its history for an introduction to real and numerical analysis from a historical point of view.

Answer 55

Kock, Vainsencher: An invitation to Quantum Cohomology.

Written in the most friendly and motivating style I have ever seen in a book. Almost has no prerequisites: You should that there exists something like algebraic varieties - without having to know any technical details - and that P^1 is such a thing. Everything else is provided in easy exercises or the text. It gives an excellent intuition about the subject with lots of outlooks on a field of current research, and at the same time manages to be easily undergraduate readable.

Answer 56

Alexandre Stefanov keeps an extensive list (Wayback Machine, another link) of free math books / lecture notes. The list is divided according to subject and updated frequently. I have found some very nice books there.

Answer 57

Linear Algebra and Its Applications by Gilbert Strang. You can also watch his video lectures at MIT OpenCourseWare

Answer 58

"Introduction to Mathematical Logic" by Ebbinghaus, Flum and Thomas

Careful introduction, addresses many doubts that one might have about why one does logic in this way and not some other, e.g. whether one is doing something circular when formulating set theory in 1st order logic, or e.g. it proves Lindstroem's Theorem, that says that classical 1st order logic has the highest power of expressability among the logics with completeness and Loewenheim-Skolem.

Answer 59

Metric Spaces by Mícheál Ó Searcóid

https://www.amazon.com/Metric-Spaces-Springer-Undergraduate-Mathematics/dp/1846283698/ref=sr_1_1?ie=UTF8&s=books&qid=1256082496&sr=1-1

It's an exhaustive introduction analysis at the level of the metric space that's well worth reading.

Answer 60

Introduction to Analysis by William R. Wade

https://www.amazon.com/Introduction-Analysis-4th-William-Wade/dp/0132296381/ref=sr_1_1?ie=UTF8&s=books&qid=1256082707&sr=1-1

This is a good transition from undergraduate calculus to analysis.

Answer 61

Godement "Analysis" (I,II,III,IV) https://www.amazon.com/Analysis-Convergence-Elementary-functions-Universitext/dp/3540059237

"... The content is quite classical ... [...] The treatment is less classical: precise although unpedantic (rather far from the definition-theorem-corollary-style), it contains many interesting commentaries of epistemological, pedagogical, historical and even political nature. [...] The author gives frequent interesting hints on recent developments of mathematics connected to the concepts which are introduced. The Introduction also contains comments that are very unusual in a book on mathematical analysis, going from pedagogy to critique of the French scientific-military-industrial complex, but the sequence of ideas is introduced in such a way that readers are less surprised than they might be.

Answer 62

How about an anti-recommendation? Someone in another answer mentioned Steven Axler's Linear Algebra Done Right. My comment, not as someone who has used this book in a class, but as someone who has taught the students from this class during the following term: It doesn't prepare the students to use linear algebra in engineering, in physics, in chemistry, or even in branches of mathematics other than abstract algebra.

Answer 63

For (applied) ODEs: Nonlinear dynamics and chaos by Steven Strogatz.

A very inspiring book! The explanations are crystal clear with lots of pictures. And it's funny too – the "Romeo and Juliet" illustration of 2-dimensional linear systems (Section 5.3) is a classic.

Answer 64

General Topology, by Stephen Willard

https://books.google.com/books?id=-o8xJQ7Ag2cC&lpg=PP1&dq=stephen%20willard&pg=PP1#v=onepage&q=&f=false

Answer 65

Atiyah and MacDonald, Introduction to Commutative Algebra

Answer 66

I would recommend Walter Rudin's classic text entitled Real and Complex Analysis for the mathematically mature undergraduate student. (Of course, this should really only be read after one has familiarity with most of Rudin's earlier textbook: Principles of Mathematical Analysis.)

The beautiful thing about this book is that nearly every example or result stated by Rudin is used in "the big picture". As you progress through the book, you really start to appreciate the magic Rudin has used to weave together analysis in a manner that is rarely done in other textbooks.

While Rudin states in his preface that the textbook is intended as a first year graduate course on the subject of analysis, I believe that it is quite possible for a mathematically mature undergraduate to follow it. There is no assumption in the book that the reader has any familiarity with linear algebra, abstract algebra, general topology etc. beyond that which was covered in the first seven chapters of his earlier book, but realistically one would like to have at least mastered the basics of these areas before attempting to delve deeper into the analysis covered in this book. (There is a chapter on Banach algebras, but all the necessary abstract algebra here is developed from scratch.)

The textbook is indeed challenging with plenty of exercises, but it is not something with which a student with the correct prerequisites should have tremendous difficulty. Furthermore, if a student can successfully read most of the book, he is well-equipped to go deeper into most branches of modern analysis, and should find the other classic texts in the subject, for instance Royden's Real Analysis, Bartle's The Elements of Integration and Lebesgue Measure etc., very easy to follow.

The Amazon page for this book can be found here.

Answer 67

Algebraic Theory of Numbers, by Pierre Samuel

Assumes only elementary knowledge of group and ring theory (even less than a complete undergraduate course which covers Galois theory) and develops algebraic number theory, a beautiful subject which puts much elementary number theory into an interesting perspective.

Answer 68

I would also recommend the book entitled Analysis on Manifolds by James Munkres. I think that this is a good undergraduate textbook in mathematics for any student wishing to pursue multivariable calculus in greater depth. My only complaint is that Munkres often chooses to include details which can be seen easily after a little bit of thought. Perhaps this can be viewed as an effort to show the student how to "properly do analysis": doing analysis, just like doing any other branch of mathematics, requires you to carefully apply definitions and theorems, and it is important for the student to appreciate this early in his/her mathematical learning.

That said, the book is an excellent text overall for "advanced calculus". The student will need to be familiar with single variable analysis and perhaps some linear algebra. (Even a rudimentary knowledge of linear algebra will do since Munkres develops most of the necessary theory from scratch.)

Roughly speaking, the book splits into two parts. The first part covers most of the results students see when doing multivariable calculus that are stated "without proof" in their texts. For example, "the equality of the mixed partials", "double integrals can be done in any order", "a bounded function is Riemann integral if and only if it is continuous almost everywhere", "the change of variables theorem" etc., are (very) imprecise forms of some of the results Munkres establishes.

In the second part of the book, manifolds and their theory are introduced. Thus, for example, a rudimentary introduction to tensors is given, and this is supplemented by the basic theory of differential forms, the De Rham groups (of the punctured plane), Stokes' theorem etc.

I think that the exposition could be tightened: if you actually pick up the book and really make an effort to read it, it is quite possible to finish the first half of the book in the space of a week (that is, approximately 200 pages in a week) simply because certain topics are explained in more detail (at least in my opinion) than necessary. (One example is Munkres' proof of the linearity, monotonicity, additivity etc. of the Riemann integral. This is proved in three contexts separately: the case of the integral over a rectangle, that over a bounded set, and that of improper integrals, when essentially the proofs can be left as relatively easy exercises in some cases.)

As the above comments suggest, I think that this is an excellent book for undergraduate students, but perhaps less so for graduate students. (Spivak's Calculus on Manifolds is good for both undergraduate and graduate students, in my opinion, but some people may suggest that it is too hard for undergraduates.) And after reading this book, you should have more than enough preparation to read more advanced texts such as William Boothby's An Introduction to Differentiable Manifolds and Riemannian Geometry.

Answer 69

A Field Guide to Algebra by Antoine Chambert Loir. Covers a surprising amount of material considering the short length and minimal prerequisites

Answer 70

"TOPOLOGY WITHOUT TEARS" by SIDNEY A. MORRIS is a very nice introduction in topological spaces theory, I think.

Answer 71

Geroch, Mathematical Physics

Don't be scared by the title: it teaches algebra, topology and measure theory, using category-theoretic language.

Answer 72

There is a good list here (Wayback Machine), divided by subject, that also contains many links to freely available textbooks and lecture notes.

Answer 73

Kelley, General Topology

Answer 74

Introduction to Topology: Pure and Applied, by Adams and Franzosa. The figures in the book are beautiful, the problems are good, and the applications are good (and unusual) to see in an undergraduate text.

Answer 75

"Differential Calculus on Normed Linear Banach Spaces" by Kalyan Mukherjee.

The author is a prof at the Indian Statistical Institute (Kolkata)

This is an amazing book which introduces differential calculus in arbitrary finite dimensional spaces (thinking of the derivative as the Jacobian) as the next leap from the Apostol and Rudin level and also build in topological ideas. It then goes into manifold theory and shows how to compute tangents to curves inside lie groups. It has nice sections of things like differentiating the determinant function and the matrix multiplication function and inverse function theorem and idea of equivalence of norms.

I would strongly recommend this book to an undergrad after he/she has done the Apostol/Rudin level of calculus.

"Calculus on Manifolds" by Spivak and "Differential geometry and Lie groups" by Kumaresan are two other good books which can be read alongside it.

Answer 76

I liked Elements of Abstract Algebra by Allan Clark, which is mainly a problem book with a moderate amount of exposition, but the problems are so well-chosen that a diligent undergraduate student working through all of them will come out with a solid knowledge of group theory, classical ring theory, and Galois theory.

Answer 77

I think that Linear Algebra by Friedberg, Insel, and Spence is a spectacular linear algebra book. It gets straight to the point, it provides a worked out example or two exactly when they're needed, and it has lots of interesting exercises.

There are way too many gigantic linear algebra books with colour pictures and contrived examples which often seem to obfuscate the concepts being introduced. In my mind this book is 'the mathematician's linear algebra book'; clean and concise.

Answer 78

Hugo Steinhaus https://en.wikipedia.org/wiki/Hugo_Steinhaus book "Mathematical Kaleidoscope". It is kind of mathematical trivia sometimes very deep;-) It is not for learning math but for learning how to learn math in fun way.

Answer 79

Complex Variables: Harmonic and Analytic Functions by Francis J. Flanigan

A nice little Dover paperback which turns the standard course on complex variables on its head. It begins by doing some multivariable calculus in the plane and harmonic functions, then proceeds to talk about complex numbers and to build analytic functions.

Answer 80

Siegfried Bosch, Lineare Algebra

It's a very elegant, concise but beautifully written approach to Linear Algebra, and I love it.

Unfortunately for people who don't speak German, it has never been translated.

Answer 81

I plan to post a complete reading list for undergraduates and graduate students at my blog this summer with my commentaries,but here's one I think that's available online and doesn't get nearly enough credit despite the fame of it's author: Gilbert Strang's Calculus. I wouldn't use anything else for a regular,non-honors calculus course. Carefully written,beautifully motivated with TONS of creative and SIGNIFICANT applications. I hope one day Strang finds the time to write a second edition-I have a list of improvements to suggest.

Answer 82

Linear Algebra with Differential Equations: Bentley and Cooke 1973

Answer 83

A Concrete Approach to Abstract Algebra by W. W. Sawyer ($6 on Amazon!)

Though it goes a bit slow at times, it is by far the simplest, most intuitive book on Abstract Algebra in existence. Written for the non-mathematician, it does a great job of teaching the subject in simple, easy-to-understand prose. I couldn't put it down!

There are also two chapters on linear algebra, leading up to the final chapters, "vectors over fields" and "fields regarded as vector-spaces".

Answer 84

Introduction to Analytic Number Theory - Tom M. Apostol.

When I bought this I really didn't want to put it down. It's a great book for exciting one's interest in the subject.

Answer 85

Linear Algebra: With Applications Otto Bretscher

Used at Carleton College – nice explanations, and quite a few proofs. Presents information primarily by providing examples, definitions/axioms and then proofs.

Answer 86

I had many trials and these are in my opinion the best for an introductory, undergraduate level:

ODEs: Holzner: Differential Equations for Dummies
PDEs: Farlow: Partial Differential Equations for Scientists and Engineers

Answer 87

Applied Linear Algebra, by B. Noble and J.W. Daniel, https://www.amazon.co.uk/Applied-Linear-Algebra-Ben-Noble/dp/0130412600/ref=sr_1_14?ie=UTF8&s=books&qid=1256927879&sr=1-14 .

Answer 88

G. J. O. Jameson: Topology and Normed Spaces for an introduction to functional analysis from a topological point of view.

Answer 89

lindsey childs a concrete introduction to abstract algebra

here maybe.

Answer 90

Strichartz, The Way of Analysis

Herstein, Abstract Algebra

Answer 91

Milnor's book "Dynamics in One Complex Variable: Introductory Lectures". An early version is available from his website. Suitable for advanced undergraduates, graduate students, and mathematicians.

http://www.math.sunysb.edu/cgi-bin/preprint.pl?ims90-5 (Wayback Machine)

Answer 92

Most readers here will not be able to appreciate them for a simple reason, but my favorite beginners' Analysis text is that by Bröcker (Wayback Machine). No-nonsense, concise, with a slight orientation towards topology.

Answer 93

Since Numerical Mathematics has not been covered, I would recommend the following

Introduction to Numerical Analysis by Stoer et. al. https://www.amazon.com/Introduction-Numerical-Analysis-J-Stoer/dp/038795452X/ref=sr_1_14?ie=UTF8&s=books&qid=1255807973&sr=8-14

Answer 94

Vinogradov's Elements of Number Theory - the problems more so than the text itself.

Answer 95

Real Analysis, by Frank Morgan. The chapters are short and very directed. The proofs are written well. The exercises are well-selected. The book is written at a level accessible for most students.