Notes On Linear Algebra
Colleen Bramham
Now, the reputation of this class precedes itself and it is very likely that you've heard about it from upperclassmen complaining about it. And you may be thinking to yourself: "Wait! I already took MATH 33A, which is Linear Algebra. Why do I have to take it again? In fact, MATH 33A is a pre-requisite for MATH 115A!"
In introductory courses like Math 33A, linear algebra often revolves around matrix studies. However, at the 115A level, the focus shifts to exploring vector spaces and their transformations. If you're unfamiliar with the concept of a vector space, don't worry - we'll delve into it soon. For starters, consider $\mathbbR^n$ - the set of all $n$-tuples of real numbers - as your introductory vector space. Typically, this is the sole vector space explored in elementary linear algebra courses. In 115A, however, we'll expand our horizons, exploring linear algebra in various other vector spaces, which proves to be incredibly beneficial.
Our approach involves starting from the very basics. It is perhaps helpful to momentarily set aside all your previous mathematical knowledge and treat 115A as a foundational course designed to systematically build a specific mathematical field from the ground up. This is our initial aim in 115A.
A noteworthy point regarding this goal is the following: You might be anticipating that exploring linear algebra in vector spaces beyond $R^n$ will be a radically different and exciting experience. However, I must clarify that abstract linear algebra in general vector spaces largely mirrors the linear algebra you've encountered in $\mathbbR^n$. The concepts of linear independence, transformations, kernels, images, eigenvectors, and diagonalization - all familiar topics within the realm of $\mathbbR^n$ - function similarly in 115A.
(2) Construct and Follow Abstract Mathematical Arguments and Statements
This goal extends beyond mere proof-writing. Upper-division mathematics, in contrast to lower-division studies, prioritizes the discovery and articulation of truths over computation. In 115A, every solution you formulate should be viewed as a mini technical essay, marking a departure from mere scratch work to determine problem solutions. Mastering the art of clear, logical, and effective communication of mathematical truths is a challenging yet essential skill to develop.
Here is a list of strategies and advice that I found useful in navigating this challenging yet rewarding course. Some of these advice are in retrospect (i.e. things I would do if I were to re-take the course).
3 years ago, I experimented with the 'weekly newsletter' approach and it was not only unsustainable, but also slightly spammy. My new model is as follows: I promise to only send an email to your inbox when I'm absolutely positive it is something you'll find interesting, and perhaps more importantly, actionable. Topics will include personal finance, productivity or general life insights.
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
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.
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.
This can be used for an intro to proofs course, or a reference in a proof-based course Designing any guide or text on mathematical proof leads to a discussion of sets first or propositions first. We introduce a little of each first, and then constantly bring the discussion back to categorizing what each kind of thing is, with emphasis on mathematical language.
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