Does anybody know of an awesome list that has more of these kinds of "pdf textbooks" on math/phys/engineering?
openstax.org
Free Math Books
51–60 of 74 posts
Re: Free Math Books
#52Earlier quoted context omitted.
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…
> Really, I don't see what you like about Halmos definition of the determinant... Halmos shows (it is almost trivial) that the space of anti-symmetric n-forms Wn over L_n is 1-dimensional. Wn(Ae1,...,Aen) = const*Wn(e1,...,en). This scalar const is called determinant. It has all the properties you would ascribe to Volume like volume spanned by collinear column-vectors is zero. This is a nice bridge to geometry in Ln.…
The determinant of n vectors {vi} relative to a particular basis {ei} in an n-dimensional vector space is the scalar-valued ratio:
( v1 ∧ v2 ∧ ··· ∧ vn ) / ( e1 ∧ e2 ∧ ··· ∧ en )
The signed volume per se is just the n-vector: v1 ∧ v2 ∧ ··· ∧ vn
Generally working with the wedge product is more pleasant and conceptually clearer than working with determinants. Among other things we don't need to make an arbitrary choice of basis or unit n-vector. There's also no reason to limit ourselves to n terms. v1 ∧ v2 is also a reasonable quantity to use, etc.
Re: Free Math Books
#53Re: Free Math Books
#54Earlier quoted context omitted.
Probably not. That’s why most of these books are useless for people that self-study. There isn’t an external feedback mechanism to check your work. I’d recommend hiring a tutor. You can post online too, but the answers to your questions may vary.
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.
Re: Free Math Books
#55Earlier quoted context omitted.
I have been adding a number of these free books at https://learnawesome.org/ Do you really care about the format being PDF or is it about the books being FREE? I'd like to make common queries like yours easier. LearnAwesome is open-source, so of course you're free to contribute: https://github.com/learn-awesome/learn
Nerds love to bike shed crap like this and bite the hand that feeds. To paraphrase a typical rant: “Making knowledge free and accessible is useless unless a libre format is used like Markdown.” Lol The PDF beef is funnier in that people usually don’t know why they are morally opposed to PDF, it usually boils down to not liking Adobe Acrobat a decade ago, not accepting that page format preservation is a thing, or some…
Not just that, but if you actually look at the implementation its pretty understandable. 90% of use cases are covered by the easiest parts to parse too. PDF might not be perfect, but honestly not much comes close.
Re: Free Math Books
#56Earlier quoted context omitted.
> Really, I don't see what you like about Halmos definition of the determinant... Halmos shows (it is almost trivial) that the space of anti-symmetric n-forms Wn over L_n is 1-dimensional. Wn(Ae1,...,Aen) = const*Wn(e1,...,en). This scalar const is called determinant. It has all the properties you would ascribe to Volume like volume spanned by collinear column-vectors is zero. This is a nice bridge to geometry in Ln.…
The determinant of n vectors {vi} relative to a particular basis {ei} in an n-dimensional vector space is the scalar-valued ratio: ( v1 ∧ v2 ∧ ··· ∧ vn ) / ( e1 ∧ e2 ∧ ··· ∧ en ) The signed volume per se is just the n-vector: v1 ∧ v2 ∧ ··· ∧ vn Generally working with the wedge product is more pleasant and conceptually clearer than working with determinants. Among other things we don't need to make an arbitrary choice…
The beauty of Halmos' derivation, which is similar but not identical to exterior algebra (wedge product), is that his approach is basis independent. A determinant by his definition is scalar invariant over all bases. It is very geometrical in nature.
Re: Free Math Books
#57Does anybody know of an awesome list that has more of these kinds of "pdf textbooks" on math/phys/engineering?
https://www.reddit.com/r/mathbooks/ is a good place to find free titles.
Re: Free Math Books
#58In 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…
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…
[deleted]
Re: Free Math Books
#59In 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…
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…
I find both the geometric and algebraic definitions quoted here unsatisfying. What is a “volume” spanned by a vector space of polynomials or co-tangent functionals?
*A* determinate function (not the) is simply a skew symmetric n-linear map into the underlying field.
Done. Now we get the volume interpretation when it’s appropriate, the wedge product interpretation, and the generalization to finitely generated projective modules (if a determinate function exists, there are additional conditions needed for the existence.)
Re: Free Math Books
#60Earlier quoted context omitted.
The determinant of n vectors {vi} relative to a particular basis {ei} in an n-dimensional vector space is the scalar-valued ratio: ( v1 ∧ v2 ∧ ··· ∧ vn ) / ( e1 ∧ e2 ∧ ··· ∧ en ) The signed volume per se is just the n-vector: v1 ∧ v2 ∧ ··· ∧ vn Generally working with the wedge product is more pleasant and conceptually clearer than working with determinants. Among other things we don't need to make an arbitrary choice…
The beauty of Halmos' derivation, which is similar but not identical to exterior algebra (wedge product), is that his approach is basis independent. A determinant by his definition is scalar invariant over all bases. It is very geometrical in nature.
The determinant inherently involves a basis (or at the very least a choice of unit n-vector). Or if you like you can think of the determinant as a function of a square matrix (grid of numbers), rather than a function of a collection of vectors.
When you take the basis out, that's the wedge product, which inherently includes the orientation. Conveniently, there is only one degree of freedom for n-vectors in n-dimensional space. When we take the quotient of two n-vectors in n-dimensional space we therefore get a scalar.