5 ms·

Linear: having to do with lines, planes, etc. This definition makes me worry that the book is going to operate entirely in R^n. Is there anything here about su

• nostrebored 7y ago
  Abstract vector spaces is a follow up course to linear algebra at gatech
• impendia 7y ago
  Looking through the book, it seems to concentrate on R^n, and not pay much attention to abstract vector spaces. Probably a good choice for the intended audience (mostly engineers, likely). You might prefer Hefferon's Linear Algebra, also free and open-source, and which spends more time on this. http://joshua.smcvt.edu/linearalgebra/ http://joshua.smcvt.edu/linearalgebra/ Both books look truly excellent.
• deleted 7y ago
  [deleted]
• sitkack 7y ago
  Is it a real concern or are you humble bragging about the lack of advanced math in an introductory text on _linear_ algebra?
  • throwawaymath 7y ago
    No, that's a legitimate concern. It may or may not be relevant for what you specifically care about, but it's definitely a legitimate thing to ask about. It's not uncommon for even undergraduate linear algebra to cover abstract spaces and notions of linearity which generalize beyond R^n. For example, the function space P_n consisting of all polynomials with degree less than or equal to n. Hoffman-Kunze, Halmos, Axler and Friedberg-Insel-Spence are all examples of undergraduate textbooks which cover this material. This isn't just theoretical. Function spaces like P_n are useful in applied mathematics. And even if you don't use function spaces, it's very common for engineers, physicists and applied mathematicians to work in the complex space C^n rather than R^n.
    • vcxy 7y ago
      This is just the first semester course. In my experience teaching it at GT, it is mostly taken by freshman, and I'm not sure how many majors require it but my students have come from a huge variety of majors. I'm not sure that abstract vector spaces is that useful for, say, psychology majors. There is a second semester course that usually uses Axler.
  • chobytes 7y ago
    Various function spaces were definitely brought up in my linear algebra class. That kind of material comes up even in intro differential equations (eg systems of differential equations, Lp spaces).
• gowld 7y ago
  What are some examples of interesting interactions (that you'd like to see, not necessarily already exist) that illustrate linear algebra with non-R^n vector spaces? Thanks to representation theory, we can model vector spaces as R^n https://en.wikipedia.org/wiki/Representation_theory https://en.wikipedia.org/wiki/Representation_theory
  • impendia 7y ago
    I don't really think that representation theory is required here, it's easier than that. (Although representation theory is really cool!!) Every finite dimensional vector space over the real numbers is isomorphic to R^n, for some n. But there is not always a canonical or "unique" isomorphism. I think the real difference -- and it is a subtle one -- is that R^n always comes with an "obvious" choice of basis, and many vector spaces don't. I think the following may be the easiest interesting example. Consider the subspace of R^3, consisting of all (x, y, z) for which x + y + z = 0. This is a 2-dimensional real vector space. It is isomorphic to R^2, although in some sense it does not "present as such": you have to choose an isomorphism to R^2 if you want to "treat it as R^2", and there is no single choice that stands out. In practice, one would not necessarily construct an isomorphism to R^2, or (more or less equivalently) exhibit a basis, before working with this vector space directly.
  • kd0amg 7y ago
    I think you could do some interesting stuff with error-correcting codes or polynomial spaces (or more generally function spaces). I'm not too familiar with representation theory. Do you know where I could find a worked example of constructing, say, a representation of (F_p)^m on R^n? (for prime p and m>1)
    • gowld 7y ago
      Visually, (F_p)^m can be embedded in R^m trivially, but I suspect you are interested in animations of operations on F^p, which indeed would look different from R^m due to modular arithmetic, and so would be beyond the scope of the OP.
• ajkjk 7y ago
  I would hope that a linear algebra book only operated in R^n to introduce the concepts, and then include function spaces in order to introduce integral transforms near the end.
• vcxy 7y ago
  I've taught this course at GT a few times. For this first introduction to the material, we stay away from anything other than R^n for the most part.
• zelly 7y ago
  Axler's Linear Algebra book is best if you want a deep understanding in more fields than R^n
• Myrmornis 7y ago
  Yes I found this offputting. It's one thing to focus on R^n, but another to give students definitions that don't even hint at greater generalisation. The follow up definition also feels very dumbed-down. algebra: solving equations involving unknowns