61–70 of 74 posts
Does anybody know of an awesome list that has more of these kinds of "pdf textbooks" on math/phys/engineering?
In college I was taught Linear Algebra from the operator point of view, rather than with matrices. That way theorems are clearer and the student's understanding is deeper, but for applications it's better to study from the matrix point of view and with lots of examples. Kuttler's book was refreshing in that sense. His other books are excellent, too. If you have been studying pure math (or french-style applied math wh…
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I never understood why they do that - especially for something like this that's offered for free. What's the point of even including the exercises if there's no way to check the answers?
I wondered the same. I have no idea, other than to speculate. In this case, it might be out of habit (since this is usually how textbooks are written) or laziness. I can’t recommend any of these books for self-studies unless the person studying is fine with posting every exercise they do online for correctness checks or can hire a tutor.
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The best, by far, book on Linear Algebra that elegantly teaches it from Vector Spaces and Linear Operators point of view is Paul Halmos "Finite-Dimensional Vector Spaces" For instance, the way Halmos introduces the determinant of a matrix (or an operator) is the most consistent, elegant and simple way I ever encountered. OTOH, in Kenneth Kuttler's LinAlg books the determinant is pulled out of the thin air like in 100…
While Halmos' book is lovely, I still prefer the geometric definition of determinant to the algebraic one: The determinant of a matrix is the signed volume (or area) of the parallellepiped spanned by its columns. Equivalently, the determinant of a linear map is the volume of the image of a unit cube by that map (or any arbitrary shape of volume one, not necessarily a cube). All the algebraic properties of the determi…
And strictly speaking, determinant is not volume because the former is dimensionless. It is the scaling factor of the volume when a geometric entity is transformed by a linear map.
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Thanks, I'll look it up. The best textbook from which I studied (operators) was Elon Lima's Algebra Linear. Sadly the only physical copies are sold in Brazil.
If you want to see the matrix point of view done well, there's Linear Algebra Done Wrong : https://www.math.brown.edu/streil/papers/LADW/LADW.html . You can read a bit about the motivation for doing it that way on that website. The title is a reference to a somewhat well-known book, Linear Algebra Done Right , which avoids using determinants to develop the theory (resulting in a somewhat novel/cleaner presentation).…
Wow, these look great! Does anyone know of any similar resources for mechanics, specifically the Hamiltonian & Lagrangian formulations? I've had a bit of trouble finding good resources online to supplement my mechanics modules at uni.
Look for Dover books, particularly from Russian authors. Cheap and good!
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While Halmos' book is lovely, I still prefer the geometric definition of determinant to the algebraic one: The determinant of a matrix is the signed volume (or area) of the parallellepiped spanned by its columns. Equivalently, the determinant of a linear map is the volume of the image of a unit cube by that map (or any arbitrary shape of volume one, not necessarily a cube). All the algebraic properties of the determi…
If you define determinant as volume, how do you define volume? I agree that it's pedagogically sound to motivate the notion of determinant by the volume of a parallelepiped, but using volume as the definition of determinant just doesn't sound right. And strictly speaking, determinant is not volume because the former is dimensionless. It is the scaling factor of the volume when a geometric entity is transformed by a l…
How do you define "length" and "area"? I guess that if you don't have already a very firm grasp of these basic concepts, then there's no business for you (yet) in studying determinants. Much later, once you master thoroughly lengths, areas, volumes and hypervolumes; and also linear algebra and determinants (however they are defined), then you can embark in the elegant definitions using exterior algebra and the like. Notice that Halmos itself says that his treatment is appropriate for a *second* course in linear algebra, preparing the field for the later study of infinite-dimensional spaces.
> And strictly speaking, determinant is not volume because the former is dimensionless.
This really depends on the context. If you are working on euclidean space, you already have "units" and the determinant makes sense in itself, as the volume spanned by sets of vectors.