Linear Algebra 2 Pdf Notes
Nayme Cutforth
So, now, I want someone can recommend books or lecture notes which have complete linear algebra contents and natrually combined it with abstract algebra. I find that knapp's $\itBasic$ $\itAlgebra$ is very suitable for my expectation, but I think it's too hard for me at current stage. Hope that someone can recommend books like knapp's but simpler and more complete than it.
For an abstract algebra textbook that covers also typical (non-numerical) linear algebra topics, you may try Cohn's Classic Algebra (Mathematical Gazette review). I haven't read Knapp or Cohn carefully, but I think Cohn goes deeper in linear algebra than Knapp does. However, Cohn doesn't discuss any matrix decomposition, if I remember correctly.
For a linear algebra textbook whose treatment is more algebraic, you may try Berberian's Linear Algebra (MAA review). Someone recommended this book to me on this site before and I have read it once from cover to cover. I remember that I quite liked it, but I don't remember why I liked it. Most introductory texts discuss matrices over some fields, but Berberian also discusses matrices over principal ideal rings. It has a brief discussion of multilinear algebra, but the coverage is not strong. Also, although it includes some abstract algebra topics (such as factorisations over integral domains), this is a linear algebra text. So, don't expect yourself to learn abstract algebra from it.
Just a thought, I noticed that in the Dover series of paperbacks, there's one entitled Linear Algebra and Group Theory. So this may be the sort of thing you are looking for. But again all I have is the title (it caught my eye): I haven't looked at it.
Later in this page, links to each of the lecture notes, quizzes, and review sheets are available. If, however, you want to download all the lecture notes or all the quizzes, saving each link can be a pain. The links below can be useful if you want to download in bulk. Note that page numbers as shown on the pages are for individual files, not for the combined file.
To determine the chronological order of quizzes, please use the dates in the quiz titles. Quizzes related to a given topic may not all have been administered at the time the topic was taught. Some quizzes were deliberately delayed in order to facilitate spaced repetition.
For solutions to any quiz, add -solns to the part of the URL just before the .pdf at the end of the URL. For instance, the solutions file for the quiz with URL -196/10-18-linear-systems-rank-dimension-considerations.pdf has URL -196/10-18-linear-systems-rank-dimension-considerations-solns.pdf
This set of notes has been compiled over a period of more than 30 years. Some chapters were used in various forms and on many occasions between 1981 and 1990 by the author at Imperial College, University of London. The remaining chapters were written in Sydney.
The material has been organized in such a way to create a single volume suitable to take the reader to a reasonable level of linear algebra. Chapters 1 - 4 cover very basic material. The concept of vector spaces is then introduced in Chapters 5 - 7. More advanced topics, including the concept of linear transformations from one vector space to another and the concept of inner products, are covered in Chapters 8 - 12.
An Introduction to Algebraic Combinatorics: Power series and generating functions, partitions, permutations, alternating sums and Schur polynomials. Work in progress, but mostly complete.
Enumerative Combinatorics: Rigorous and detailed introduction to enumerative combinatorics. Chapters 1 and 2 done, covering various types of subset counting, inclusion-exclusion, binomial identities and more. Further topics are covered in the Fall 2022 lecture notes.
Darij Grinberg, Notes on the combinatorial fundamentals of algebra (PRIMES 2015 reading project: problems and solutions).
Sourcecode.
A version without solutions,for spoilerless searching.
A set of notes on binomial coefficients, permutations anddeterminants. Currently covers some binomial coefficientidentities (the Vandermonde convolution and some of its variations),lengths and signs of permutations, and various elementary propertiesof determinants (defined by the Leibniz formula).
The sourcecode of the project is also trackedon github.
Darij Grinberg, 18.781 (Spring 2016): Floor andarithmetic functions.
Sourcecode.
These are the notes for a substitute lecture I gave in the18.781 (Introduction to Number Theory) courseat MIT in 2016. (Though they contain morematerial that fits into a single lecture; I omitted some resultsand only sketched some of the proofs in the actual lecture.)
In Section 1, I define the floor function and show some of its basicproperties; I then prove de Polignac's formula for the exponent of aprime in n! and use it to show that binomial coefficients are integers(there are better proofs of this, but it illustrates the power ofthe formula).
In Section 2, I introduce the standard arithmetic functions (φ, Mbius, sumof divisors, etc.), define multiplicativity and Dirichlet convolution,and prove the standard results: Mbius and φ are multiplicative;Dirichlet convolution is associative; the sum of φ(d) over alldivisors d of n is n; the sum of μ(d) over all divisors d of n is0 unless n = 1; the Mbius inversion formula; the Dirichletconvolution of two multiplicative functions is multiplicative.A variant of the Dirichlet convolution (called the "lcm-convolution")is also studied and its associativity proved.
Darij Grinberg, The Lucas and Babbagecongruences.
Sourcecode.
In this expository note, we prove the Lucas and Babbage congruences forbinomial coefficients. The proof is elementary (by induction) and worksfor arbitrary integer parameters (as opposed to merely for nonnegativeintegers). Afterwards, we also prove that0k + 1k + ... + (p-1)kis divisible by p for any prime p and any nonnegative integer kthat is not a positive multiple of p-1.
Darij Grinberg, Analgebraic approach to Hall's matching theorem(version 6 October 2007).
Sourcecode.
There is also anabridged version, which is probably easier to readas it omits some straightforward details.
Hall's matching theorem (also called marriage theorem) has receiveda number of different proofs in combinatorial literature. Here is aproof which appears to be new. However, due to its length, it isfar from being of any particular interest, except for one ideaapplied in it, namely the construction of the matrix S. See thecorresponding MathLinks topic for details.
It turned out that the idea is not new, having been discovered byTutte long ago, rendering the above note completely useless.
1. (rev.lin.alg.pdf): Linear algebra notes, including spectral theorem for symmetric operators, jordan form, rational canonical form, minimal and characteristic polynomials, and Cayley Hamilton, all in 15 pages! (Expanded version of same: laprimexp.pdf.)
2. (RRT.pdf): A discussion of the easy aspects of the Riemann Roch theorem for curves, surfaces, and n dimensional smooth manifolds. We give Riemann's classical proof for curves, assuming his results on the existence of meromorphic differentials of first and second kinds. Then we reprove from scratch the Hirzebruch version for curves, using (and recalling) sheaf cohomology, but only sketching Serre's proof of the duality theorem. Then we prove similarly Hirzebruch's version for smooth surfaces embedded in projective three space. Finally we sketch the formalism of chern roots and their use in defining Todd classes and in stating the general HRR. This is aimed at a grad student who has had complex analysis of one variable, and a little topology.
f. 845-2.pdf, Existence of "Jordan" form of a linear operator if min'l polynomial has all factors linear; existence of "diagonal" form if min'l polynomial has all factors linear and distinct, or if the matrix is "symmetric" /R or "normal"/C.
These condensed notes include basic theorems about pid: uniqueness of factorization and decomposition of finitely generated modules, application to Jordan and rational canonical forms of matrices; also Gausstheory of content and unique factorization in Z[X], Dumas Eisenstein irreducibility, Noetherian rings, Sylow theorems, Jordan - Holder, the fundamental theorem of Galois theory, Zorn lemma, existence of algebraic closures of fields, normality, separability, cyclotomic polynomials, insolvability of general polynomials of degree > 4, duality and spectral theorems. These notes are most useful for someone reviewing the material a second time. Consult the 843-4-5 course notes for more details, examples and omitted topics (tensors).
Review of basic definitions, dimension theory, statements of basic facts about row reduction, and detailed proofs of existence and uniqueness of jordan forms for split minimal polynomials, as well as generalized jordan forms (rational canonical forms) for all minimal polynomials.
Every decomposition theorem is proved three times, in increasing degree of complexity: i) reduced minimal polynomial, i.e. maximal number of cyclic factors, or diagonal case, ii) minimal polynomial equals characteristic polynomial, i.e. minimal number of factors, or cyclic case, iii) general case.
There is also a sketch of existence of cyclic product decomposition for finite abelian groups, a complete treatment of determinants and the cayley- hamilton theorem. a complete treatment of "spectral" i.e. structure theorems for normal operators in finite dimensional real and complex spaces, some discussion of duality, solving homogeneous ode's with constant coefficients to motivate jordan form, of inverting linear constant coefficient operators "locally" on spaces of polynomials, and a definition of the derivative for any locally integrable function as an adjoint operator on distributions.
Some people have expressed a wish to have a copy of the proceedings of the 1964 conference in algebraic geometry at Woods Hole, Massachusetts.
Here is an apparently complete set, courtesy of my friend Doug Clark, who attended the meeting.
woods hole 1. Theory of singularities: Abhyankar, Hironaka, Zariski;
Classification of surfaces and moduli: Kodaira, Matsusaka, Mumford, Nagata, Rosenlicht, Igusa.