Lecture Notes Linear Algebra

[Jackie Bullinger]

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

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.

Writing lecture notes gives you valuable practice at academic writing and presenting your thoughts in a coherent manner. When you first try to write papers, it is tempting to try to intimidate the reader. Writing lecture notes helps to break this habit.

If you happen to teach the course again, going through the pain of writing lecture notes the first time you teach it will make it much easier to teach it in subsequent years, because you will be teaching from your own notes and you know exactly what you are doing.

Students often prefer lecture notes, for several reasons. They don't have to pay the cost of a textbook, and they don't have to worry about keeping a textbook in good condition for resale. This means that they are much more likely to be willing to make notes, highlights, underlinings etc on the notes themselves during the lectures. Providing printed lecture notes and letting them make notes on them creates a good balance: students are not frantically copying down everything without having time to think; nor are they likely to slip into a passive listening mode.

The vast majority, if not all, textbooks originated from lecture notes, usually from professors who held that course repeatedly and refined their notes over the years (often this is stated in the introduction of the textbook). Now, you might argue that these textbooks already exist and there is no point in repeating this process as somebody already has done the work.

When I was a student, I hated professors not choosing a textbook and only giving (very often badly written) lecture notes. I think following a single textbook, suggesting optionally one or two more, plus some extra material here and there, is very convenient for the students since a good book is coherent, tested, well written and edited, with useful pictures, and often evolved through several years of lectures. You might lose some diversity, but you gain simplicity and coherence (and a more pleasant formatting). The best students will anyway look for additional texts on their own, either during the course or afterwards. Of course additional or very specific or innovative material can require lecture notes. In my experience, lecture notes are often good for the teacher; not always for the student.

I think many people write their own lecture notes because they want to present the subject as it is living in their own mind, not as someone else presents it. You can really only ever try to convey your own perspective, and even in mathematics, this can be significantly different from anyone else's.

Matrix multiplication is defined the way it is so it corresponds to composition of linear functions. The proof of this is a computation which may involve a few too many summation signs for beginning students to follow fully.

d. So to find the (AB)_ij we just need to compute e_j^\top AB e_i = (e_j^\top A)(B e_i), which is the j^th row of A dotted with the i^th column of B. This is the standard formula, but it has been "chunked" in such a way that it makes it understandable (at least to me!).

This sequence a - d really represents thinking about a matrix as representing a bilinear form, and it is through this lens that the formula for matrix multiplication makes the most sense to me. You do not have to mention this to the students at this stage to make the sequence a-d understandable and memorable.

I find that this kind of thing occurs constantly. When I read a textbook, I usually find that I have no idea what is going on, and I have to develop some sort of narrative structure which makes sense of it. This becomes my understanding of the material. If I am teaching something, I must teach my perspective. So I often end up writing lecture notes.

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