Lecture Notes On Linear Algebra Pdf

[Carlito Austin]

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

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.

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.

I see that many teachers devote time to writing lecture notes. To me, this looks odd. Whatever I can write will not be half as good as a well-written textbook. so my first choice would be looking for a good book, recommend it to the students, and stick with it as much as possible for my lectures. "Everyone writes their own notes" looks like a model in which there is a lot of needless duplication of work.

If you have a single-text class, it may not be necessary to provide students an additional set of notes, provided your lectures stick to the main text material. However, if you bring in alternative or additional topics into your lectures, you may want to include notes for those topics, and refer students to the textbook for places where you follow the "standard" outline.

As you are teaching linear algebra, I will use it as an example. Gilbert Strang teaches/taught linear algebra at MIT. He also wrote the textbook Introduction to Linear Algebra. You might expect the course to follow along perfectly with the textbook. If you look at the syllabus from when Strang was teaching the course or now what you will see is there are significant deviations. It goes section: 3.6, 8.2, 4.1-4.4, 8.5, 5.2-5.3, 6.1-6.2, 6.6, 8.3, 6.3.

If the author of the textbook cannot even happily follow his own ordering of the topic, it is not surprising that many teachers feel the need to create their own notes that go in the order and cover the material in the depth that they want.

For a course in which I have some flexibility as to the content, I may find that no single textbook includes all the material I want to teach. Asking students to buy three or four books is rather obnoxious, especially if for some of them I will only be referring to a few pages. When I write lecture notes, I can include exactly the material that I want to include in the course.

Even when I am generally following a textbook, I often find places where I want to go off on a tangent, or discuss additional related material, or maybe just cover the same material with a different treatment. I feel like me teaching a class should add value beyond the student reading the textbook, and one way to do that is to tell students how I personally think about and understand the material in question. It can be helpful for the student to have that in writing. In part this is probably arrogance, but I really do feel I have insights to offer that are not contained, or not as well expressed, in even the most "well-written" textbook.

If I am going to need fairly detailed and precise notes to lecture, I might as well type them - they'll be neater, easier to read, and I can refer to them next time I teach the course. If I'm going to that trouble, I might as well make them available to the students.

I find that writing lecture notes for an audience other than myself is a really effective way to teach myself material, and understand it at a deeper level. It very often leads me to new insights on something that I thought I understood.

Written lecture notes that are posted on a website can be helpful to anyone in the world, not just the students in my course. I've been able to answer questions on MathOverflow and Math.SE by pointing people at my lecture notes.

For high-level courses (especially graduate topics courses), there may not be any textbook on the relevant material - I am assembling it from the research literature. But in order to use material from a research paper in a course, I usually have to rewrite a lot of it - filling in background and omitted details, and so on. So it becomes lecture notes.

Writing lecture notes enables you to communicate to the students exactly what is examinable. Most textbooks are bloated with irrelevant material. See any first year textbook in economics or statistics, for example. That thing doesn't need to have 500 pages and colour pictures and a $150 price tag.

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