A Competitive Approach To Linear Algebra Pdf _VERIFIED_ Download
Halima Leisch
My background:I'm a masters student in computer science working with theoretical stuff that involves lots of linear algebra. Before the masters, my education in mathematics was, to be honest, terrible. Fortunately, I have being able to keep up and have taught myself a fair amount of discrete mathematics. As a result, today I feel much more comfortable with proof writing (or the so called mathematical maturity). Now, the quest is to learn linear algebra.
My goal:Learn (or master, if possible) linear algebra until the end of this year. I need to learn well at least the basics (up to orthogonality, projection matrices, etc) until September. Although I'm not a beginner in the subject, I'm very far from proficient. I'm interested in learning linear algebra for further studies in theoretical computer science, e.g. algorithms, graph theory, combinatorial optimization, etc. Let me make some points clear:
It's like Axler and Strang are two extremes, one treating something the other does not. Strang starts with systems of linear equation with those endless and boring mechanical calculations, but seems to have an extensive treatment of matrix algebra. On the other hand, Axler jumps right into the interesting stuff (vector spaces), but has almost nothing about matrices.
Based on your remarks, I'd say make your focus Axler, or some other similarly theoretical book, and spend $\textitsome$ time studying the matrix stuff if you feel you need more (I don't know to what extent matrix computations are important for your work). Alternatively, you can first skim through Strang to remind yourself of the subject or acquaint yourself with the basics, and then begin the more serious treatment of Axler. But the bulk should undoubtedly (in my humble opinion) be from a book similar in character to Axler. Linear algebra isn't about matrices, it's about (finite-dimensional) vector spaces and linear transformations. The matrix computations you can, after a certain foundational understanding, just pick up real quick when you need them. You can't just pick up a solid understanding of vector spaces real quick.
In the modern economy, quantitative methods are highly valued skills. Students must satisfy the empirical methods component of the economics major in one of two ways, either as a three-quarter sequence or a two-quarter sequence. Students must complete the empirical methods sequence by the end of third year.
Option A: The three-quarter empirical methods sequence is comprised of a course in linear algebra, a course in statistics, and a course in econometrics. This sequence of courses covers a broad set of topics that will enhance the student's quantitative toolkit. The topics covered in this sequence will lay the foundation for further quantitative training in the major.
Option B: The two-quarter empirical methods sequence is comprised of a course which combines the basic material in linear algebra and statistics that is utilized in many economic applications, and a course in econometrics. Students who complete the empirical methods component of the major with just two courses (ECON 21010 Statistical Methods in Economics and ECON 21020 Econometrics) must complete an additional economics elective, as discussed in Electives.
Is there any good reference for difficult problems in linear algebra? Because I keep running into easily stated linear algebra problems that I feel I should be able to solve, but don't see any obvious approach to get started.
Halmos's Linear Algebra Problem Book. It contains problems, then hints, then solutions. There is a variety of difficulty levels, and some of the problems are very easy, but some are challenging. The book is designed to be a supplement for learning linear algebra by problem solving, so it may not have the focus you're looking for.
The Putnam Competition covers a range of material in undergraduate mathematics, including elementary concepts from linear algebra, modern algebra, analysis, and number theory. Below are some books available for purchase that may help students prepare for this exam:
The basic premise is the familiar one that linear algebra should be taught with geometry in mind. That linear equations correspond to linear spaces and their simultaneous solutions can be viewed most profitably via a geometric approach is nothing new to most readers of MAA Reviews. However sometimes in our rush we may forget to incorporate this basic idea into our courses. Depending on the choices we make, linear algebra can become either the most exciting mathematics course in the lower division or into a tedious mix of matrix calculations and definitions-theorems-proofs which are unmotivated, misunderstood and unappreciated. Hopefully this review will help you decide whether this book is going to be included in your choices for your next linear algebra course.
Shifrin and Adams start with vectors on the plane and dot products, and move on to n dimensions (including a discussion of hyperplanes in Rn) after which they introduce the basic ideas of linear systems. The geometric connection is there from the beginning, and the first parts of this chapter have about as many figures as pages. Matrix algebra is studied in the second chapter, with the basic matrix operations, matrix inverses, and the transpose each getting their own subsections. The third chapter introduces vector spaces. First the focus is on subspaces of Rn and the basic notions like linear independence, basis and dimension are all studied within this more concrete setting. Four basic subspaces associated with a matrix (the nullspace, the row and column spaces and the nullspace of the transpose) are studied in detail. An optional section on abstract vector spaces concludes this chapter.
The text makes a serious effort to embed the basic notions of mathematical proof into the main flow. The authors intend it to be used for a course introducing the basics of linear algebra while also preparing the students for more advanced mathematics courses where they will be reading and writing proofs of their own. This makes the text more appropriate for courses which are transitional in nature, where the audience includes students who are looking to become mathematics majors, rather than for courses where the sole purpose is to introduce the main tools of liner algebra to future physicists, engineers and economists. The informal language of the text is interrupted often with more precisely stated definitions and theorems, and the students are gradually guided into thinking more rigorously and provided with progressively sophisticated exercises to test their developing skills in writing proofs. The instruction on writing proofs is not found in one separate section or in an appendix. Instead many blue boxes are sprinkled throughout the text, where various methods of proof are introduced and hints are given about how to attack a particular kind of problem (eg. asserting set equality, or showing linear independence of a collection of vectors). The almost seamless way the informal and the formal are combined in the book make the book feel like a well-polished set of lecture notes, but in a good way.
The brief description I gave above probably makes it pretty obvious that the Shifrin-Adams book does not attempt to revolutionize the teaching of linear algebra. In fact the table of contents is pretty traditional. The emphasis on geometry is also not incredibly novel; there are many other texts which focus on visuals and concrete geometric analogies to motivate students (I reviewed one such book for MAA Reviews: Visual Linear Algebra). The notion that linear algebra can be used as a suitable context for introducing students to the rigors of upper level mathematics is also not really unorthodox, as many colleges and universities are already using this idea. It is mainly the successful combination of all these features that makes this book interesting and worthy of looking seriously into.
A final comment: I don't know if it might be considered cheating, but I tend to check out other reviews of a book before wrapping up my own. In this case I visited the page for this book in amazon.com and the slew of negative comments from students was, for me, a wake-up call. It is not uncommon that students have completely unexpected experiences with a text no matter how scrupulous the instructor may have been in her search for the best textbook to use. Of course instructors make choices with many concerns in mind, including pedagogy, pricing, examples and exercises, but sometimes we end up making what turn out to be unpopular choices. Being open to student feedback and processing it thoughtfully may lead to a change in course book adoptions, or alternatively may motivate us to make other modifications in our classroom presentations incorporating the text into the course in novel and interesting ways. One of my long-time favorite texts in linear algebra was slammed by my first class, but has become a treasured reference (if not a smashing hit with) for the following ones.
The field of linear algebra has had millions of man hours of effort poured into it, and so in many scenarios it really is clear exactly what combination of multiplications and additions leads to an efficient solution. One would arrive at that solution either way (as you demonstrate, it is easy to recognize the n-1 wedges as solving a matrix and using proven standard methods, without destroying the added geometric insight).
In my view, GA and LA are complementary, where LA gives you the opportunity to use a large body of work for optimal solutions (and ofcourse extra strength when dealing with problems in the general linear group), and GA gives you the geometric view and coordinate free approach (and extra strength when dealing with incidence relations and transformations from orthogonal groups like SO(n), SE(n) or the conformal group).
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