Textbook/resource request for first year linear algebra

[rutwig]

Lie Algebras in Particle Physics (Frontiers in Physics), by H. Georgi. Perseus Publishing; 2nd edition (October 1999)



Introduction

Intended as an introduction to Lie theoretic methods in Particle Physics, the book starts with the operator treatment of angular momentum, the analysis of SU(2) and SU(3) [the eighfold way of Gell-Mann and Ne'eman (1964)] in order to introduce the semisimple Lie algebras, via the direct deduction of the Cartan form. Having in mind these important physical examples, and the d-harmonic oscillators, the authors develops the standard topics on Dynkin diagrams and representation theory of (classical) Lie groups. This book does not pretend to develop an exhaustive study of Lie algebras and its representation theory, but a practical and manipulable introduction for physicists to appreciate how useful group theory becomes to describe a physical system. The stars of the text are the groups SU(5) [Georgi and Glashow introduced the SU(5) theory] and SU(6), which are fundamental for unified theories and the quark model and which form excellent representatives to analyze the different pathologies appearing in Lie theory. More advanced topics like spontaneous symmetry breaking [specially important for the group SU(6) and the branching rules (Judd 1971)] and lepton number as fourth color are also introduced. The main objective is to explain and describe the techniques which are necessary to understand the modern particle physics, without leading the reader to a mountain of formal facts whose direct application are beyond their scope.

Audience
Physicists and mathematicians at the beginning graduate level. Physical knowledge is presupposed, and some manipulation capability in representations of symmetric groups is recommended [this refers to Young tableaux]

Scope
To provide a practical introduction to representation theory methods applied to physics, and to provide a solid background to attack more formal and difficult texts.

Pros
Any step is carried out having in mind the immediate physical significance, mainly the unitary groups [for obvious physical reasons]. It comments important facts like the unified theories and the quark model directly, avoiding very lenghtly treatments which usually lead to confusion and insecurity of the studied material.

Cons
As a consequence of the previous points, the lack of rigour is the most remarkable objection, mainly referring to some questions of wave functions and the weight theory underlying semisimple Lie algebras. This text will not provide a profound understanding of Lie algebras/Groups and its representation theory, since it avoids the development of the structure theory and some specialized topics indispensable for a full comprehension of the topic. Maybe its reading should be simultaneous to the probably best short book on general Lie algebras [J. P. Serre, Algèbres de Lie semi-simples complexes, Benjamin Inc 1966, english translation by Springer, 1987].
The price is also excessive for an introductory text.

Conclusion
Each of the 27 lectures in which the book is divided (~1 hour per lecture) concludes with some nice problems in the spirit of the book. The author tells the necessary material in a highly attractive way. In short, it is a direct experience of group theory written with much practical sense.
The book can be found at:

Rating: 4/5

 

[malawi_glenn]

Hi

I want to study Lie Groups which are applicable to modern physics, such as U(N), SU(N) etc.

But my problem might be that I lack much of required theory of Matrices, for example these statements are "magic" to me:

In N dimensions there are N^2 independent hermitian matricies

In N dimensions there are N^2 - 1 independent traceless hermitian matricies

+ more

I have been taught Linear Algebra from textbook:
"Elementary Linear Algebra" - Anton

(https://www.amazon.com/dp/0471170526/?tag=pfamazon01-20 )

Which ends with 5pages on complex matrices, which is not that much ;-)

So I am looking for textbook (or other resources) which covers more advanced material on complex valued matrices, suitable for background to study Lie Groups.

Thanx!

 

[qspeechc]

Hello everyone.

I have the first edition of Herstein's Topics in Algebra, which I found for next to nothing*, and I was wondering if it's worth getting the latest edition (2nd edition). Comparing the table of contents at amazon.com, it doesn't seem like too much has changed between the editions, is this right? Well, the amazon blurb says "New edition includes extensive revisions of the material on finite groups and Galois Theory. New problems added throughout".

Thanks.

*$1.50 at a second-hand bookstore. Someone obviously had a very traumatic experience with this book... In my experience 2nd-hand bookstores are usually a lot cheaper than online bookstores, but stock is highly variable of course.

 

[YaleU]

Now I am reading the book Radiation and Scattering of Waves by Leopold B. Felsen and Nathan Marcuvitz. But the Operator operation is so sofisticated, So I cann't do my work ahead .
This is the first time that I heard about the concept of decomposition of gradient.
What's more, the concept of dyadic Green's Function is also a big problem for me.
I am looking forwad to someone who had read through the book recommond some applied books for me .Thanks !

 

[Obis]

I think Beardon's Algebra and geometry is an ideally written book - there's motivation for every single definition given, the book is intuitive at the same time being quite formal, a lot of effort is done not only to present the facts, but to interconnect them.

Hence I'd be really happy to know names of books that are written in a similar fashion. It'd be also very good if the book's emphasis would be on depth, not width.

Of the most interest are books about :

Probability and statistics,
Differential equations,
Complex analysis,
Functional analysis,
Mathematical statistics (graduate level?),
Foundations of mathematics (logic, set theory, category theory?).

 

[Miranden]

Hi,

I want to buy Dr. Strang's Introduction to Linear Algebra to go with the online MIT lectures, and I am debating whether to buy the second or third edition of the book. The third edition has particularly good reviews on Amazon, but it is slightly more expensive. I would just go ahead and buy the second edition (especially since I think this is the edition that was out during the filming of the lectures and hence has corresponding problems), but I wondered if there was anything important that was clarified with the third edition. Specifically, a few Amazon reviewers of the second edition mentioned that he focused too much on his "discovery" style in some cases and left things like Determinants, Eigenvalues, and Eigenvectors inadequately explained. I also read some reviews of this edition that complained of too many complex tangents creating confusion in the text.

So basically I was just hoping to know whether there was a lot of text revision between the second and third editions, or if there was only the standard changing of a few problems here and there. If the former, it is possible that Strang could have rethought his style a bit and come up with a friendlier book (in which case I should buy the third edition). If the latter, then I can just chalk the negative comments up to personal opinion and buy the cheaper, second edition.

Input would be appreciated. Thank you. :)

P.S. I also have Poole's book on the way, so hopefully the two will complement each other.

 

[micromass]

• Author: Kenneth Hoffman, Ray Kunze
• Title: Linear Algebra
• Amazon link https://www.amazon.com/dp/0135367972/?tag=pfamazon01-20
• Prerequisities: A first linear algebra course, proofs
• Level: Undergrad

Table of Contents:

[LIST]
[*] Linear Equations
[LIST]
[*] Fields
[*] Systems of Linear Equations
[*] Matrices and Elementary Row Operations
[*] Row-Reduced Echelon Matrices
[*] Matrix Multiplication
[*] Invertible Matrices
[/LIST]
[*] Vector Spaces
[LIST]
[*] Vector Spaces
[*] Subspaces
[*] Bases and Dimension
[*] Coordinates
[*] Summary of Row-Equivalence
[*] Computations Concerning Subspaces
[/LIST]
[*] Linear Transformations
[LIST]
[*] Linear Transformations
[*] The Algebra of Linear Transformations
[*] Isomorphism
[*] Representation of Transformations by Matrices
[*] Linear Functionals
[*] The Double Dual
[*] The Transpose of a Linear Transformation
[/LIST]
[*] Polynomials
[LIST]
[*] Algebras
[*] The Algebra of Polynomials
[*] Lagrange Interpolation
[*] Polynomial Ideals
[*] The Prime Factorization of a Polynomial
[/LIST]
[*] Determinants
[LIST]
[*] Commutative Rings
[*] Determinant Functions
[*] Permutation and the Uniqueness of Determinants
[*] Additional Properties of Determinants
[*] Modules
[*] Multilinear Functions
[*] The Grassman Ring
[/LIST]
[*] Elementary Canonical Forms
[LIST]
[*] Introduction
[*] Characteristic Values
[*] Annihilating Polynomials
[*] Invariant Subspaces
[*] Simultaneous Triangulation; Simultaneaous Diagonalization
[*] Direct-Sum Decompositions
[*] Invariant Direct sums
[*] The Primary Decomposition Theorem
[/LIST]
[*] The Rational and Jordan Forms
[LIST]
[*] Cyclic Subspaces and Annihilators
[*] Cyclic Decompositions and the Rational Form
[*] The Jordan Form
[*] Computation of Invariant Factors
[*] Summary; Semi-Simple Operators
[/LIST]
[*] Inner Product Spaces
[LIST]
[*] Inner Products
[*] Inner Product Spaces
[*] Linear Functionals and Adjoints
[*] Unitary Operators
[*] Normal Operators
[/LIST]
[*] Operators on Inner Product Spaces
[LIST]
[*] Introduction
[*] Forms on Inner Product Spaces
[*] Positive Forms
[*] More on Forms
[*] Spectral Theory
[*] Further Properties of Normal Operators
[/LIST]
[*] Bilinear Forms
[LIST]
[*] Bilinear Forms
[*] Symmetric Bilinear Forms
[*] Skew-Symmetric Bilinear Forms
[*] Groups Preserving Bilinear Forms
[/LIST]
[*] Appendix
[LIST]
[*] Sets
[*] Functions
[*] Equivalence Relations
[*] Quotient Spaces
[*] Equivalence Relations in Linear Algebra
[*] The Axiom of Choice
[/LIST]
[*] Bibliography
[*] Index
[/LIST]

 

[micromass]

• Author: Serge Lang
• Title: Undergraduate Algebra
• Amazon Link: https://www.amazon.com/dp/1441919597/?tag=pfamazon01-20
• Prerequisities: Proofs
• Level: Undergrad

Table of Contents:

[LIST]
[*] Foreword
[*] The Integers
[LIST]
[*] Terminology of Sets
[*] Basic Properties
[*] Greatest Common Divisor
[*] Unique Factorization
[*] Equivalence Relations and Congruences
[/LIST]
[*] Groups
[LIST]
[*] Groups and Examples
[*] Mappings
[*] Homomorphisms
[*] Cosets and Normal Subgroups
[*] Application to Cyclic Groups
[*] Permutation Groups
[*] Finite Abelian Groups
[*] Operation of a Group on a Set
[*] Sylow Subgroups
[/LIST]
[*] Rings
[LIST]
[*] Rings
[*] Ideals
[*] Homomorphisms
[*] Quotient Fields
[/LIST]
[*] Polynomials
[LIST]
[*] Polynomials and Polynomial Functions
[*] Greatest Common Divisor
[*] Unique Factorization
[*] Partial Fractions
[*] Polynomials Over Rings and Over the Integers
[*] Principal Rings and Factorial Rings
[*] Polynomials in Several Variables
[*] Symmetric Polynomials
[*] The Mason-Stothers Theorem
[*] The abc Conjecture
[/LIST]
[*] Vector Spaces and Modules
[LIST]
[*] Vector Spaces and Bases
[*] Dimension of a Vector Space
[*] Matrices and Linear Maps
[*] Modules
[*] Factor Modules
[*] Free Abelian Groups
[*] Modules over Principal Rings
[*] Eigenvectors and Eigenvalues
[*] Polynomials of Matrices and Linear Maps
[/LIST]
[*] Some Linear Groups
[LIST]
[*] The General Linear Group
[*] Structure of GL_2(F)
[*] SL_2(F)
[*] SL_n(R) and SL_n(C) Iwasawa Decompositions
[*] Other Decompositions
[*] The Conjugation Action
[/LIST]
[*] Field Theory
[LIST]
[*] Algebraic Extensions
[*] Embeddings
[*] Splitting Fields
[*] Galois Theory
[*] Quadratic and Cubic Extensions
[*] Solvability by Radicals
[*] Infinite Extensions
[/LIST]
[*] Finite Fields
[LIST]
[*] General Structure
[*] The Frobenius Automorphism
[*] The Primitive Elements
[*] Splitting Field and Algebraic Closure
[*] Irreducibility of the Cyclotomic Polynomials Over Q
[*] Where Does It All Go? Or Rather, Where Does Some of It Go?
[/LIST]
[*] The Real and Complex Numbers
[LIST]
[*] Ordering of Rings
[*] Preliminaries
[*] Construction of the Real Numbers
[*] Decimal Expansions
[*] The Complex Numbers
[/LIST]
[*] Sets
[LIST]
[*] More Terminology
[*] Zorn's Lemma
[*] Cardinal Numbers
[*] Well-ordering
[/LIST]
[*] Appendix
[LIST]
[*] The Natural Numbers
[*] The Integers
[*] Infinite Sets
[/LIST]
[*] Index 
[/LIST]

 

[trulyfalse]

Hello PF,

I am currently taking linear algebra (MATH 110) at Queen's University and I am finding it very difficult to grasp simple concepts presented in the course. Although the semester has just begun, I'm concerned that I may be lacking certain prerequisites needed to succeed in linear algebra.

To note: I have no knowledge of formal mathematical proofs, little/no knowledge of vectors, and did not encounter matrices and complex numbers in previous math courses. The high school I went to placed little emphasis on the aforementioned topics, and thus I fear that I am at a disadvantage compared to other students in the class. The textbook we were assigned is called Linear Algebra: A Modern Introduction by Poole, and I am looking for more resources, textbooks, and lectures pertaining to linear algebra in order to aid my understanding of the material.

Also, if you have any general advice for a first year undergraduate, I'd be more than willing to hear it :^)

 

[Avatrin]

Hi

I really love the way http://probability.net/ teaches measure theory, and also how Allan Clark teaches Abstract Algebra in Elements of Abstract Algebra.

What they do is that instead of throwing a lot of definitions and theorems at you at once, and then giving you exercises, they give you an exercise after almost every definition. It makes things easier (at least for me).

So, I want to know if there are books that teach linear algebra and complex analysis this way.

Edit: Also, if there are books that can teach classical mechanics and electromagnetism this way, I would love suggestions for that as well.

 

[Tee2612]

I often see people in search of a "rigorous precalculus book". I found these books online and I thought I would post them here.

Prof. David Anthony Santos was a professor at the Mathematics Department of the Community College of Philadelphia and a prolific author of very high quality mathematics textbooks, infused with a deep understanding of mathematics and delightful eccentricities, that were made freely available. Unfortunately, he passed away in 2011, and, so far as I know, it seems that his institution did not provide a permanent archive to his life's work. To honor Prof. Santos memory, I will host his textbooks on my personal faculty page. I urge colleagues to mirror this website, so that Prof. Santos work can outlive the dumbed down and overpriced mainstream gobledygook that is being served to our students, and to write their own free textbooks or online lecture notes.

https://faculty.utrgv.edu/eleftherios.gkioulekas/OGS/Santos/santos-a-course-in-arithmetic.pdf

^Rigorous arithmetic/algebra book

https://faculty.utrgv.edu/eleftherios.gkioulekas/OGS/Santos/santos-precalculus.pdf

^Precalclus where EVERYTHING is proven rigorously.

Arithmetic

Theorem (First Law of Exponents)
Theorem (Second Law of Exponents)
Theorem (Cancellation Law)
Theorem (Sum of Fractions)
Theorem (Multiplication of Fractions)
Theorem (Division of Fractions)
Theorem (Third Law of Exponents)
Theorem (Fourth Law of Exponents)
Theorem (Square of a Sum)
Theorem (Difference of Squares)
Theorem (Sum and Difference of Cubes)

And many more!

Precalulus

Theorem [Hipassos of Metapontum]p2 is irrational.
Theorem (Triangle Inequality)
Theorem (Distance Between Two Points on the Plane)
Theorem (Midpoint of a Line Segment)
Theorem (Joachimstal’s Formula)
Theorem The equation of any non-vertical line on the plane can be written in the form y = mx+k, where m and k are real
number constants. Conversely, any equation of the form y = ax+b, where a,b are fixed real numbers has as a line as a graph.
Theorem Two lines are parallel if and only if they have the same slope.
100 Theorem Let y = mx+k be a line non-parallel to the axes. If the line y = m1x+k1 is perpendicular to y = mx+k then
m1 = −1/m. Conversely, if mm1 = −1, then the lines with equations y = mx+k and y = m1x+k1 are perpendicular.
Theorem (Distance from a Point to a Line)
Theorem The point (b,a) is symmetric to the point (a,b) with respect to the line y = x
Theorem Let f be a function and let v and h be real numbers. If (x0,y0) is on the graph of f , then (x0,y0 +v) is on
the graph of g, where g(x) = f (x)+v, and if (x1,y1) is on the graph of f , then (x1 −h,y1) is on the graph of j, where
j(x) = f (x+h).

And many, many more!