As I’m a bit further on in this path, so the details of the beginnings (introductions to proofs and logic) are a bit blurred to me, but I’ll present a few books that I think are good for moving from basics to graduate level topics.
For linear algebra I’d recommend Axler’s book, for complex analysis I liked Churchill and Brown in that I could basically read through and do the exercises very quickly as an undergrad. I didn’t have a great time with any of my real analysis texts so I don’t think I can give a good recommendation for that.
Of course your mileage may vary but these are the books that I’ve enjoyed the most, in that they stay grounded in computations, give decent intuition with good geometric pictures, and tend to have approachable exercises compared with other books I’ve read.
Also: learn to program in python, and use SAGE to solve project Euler problems.
There is a great 1st real analysis book that would work from HS level: “S. Abbott (2001). Understanding Analysis. Undergraduate Texts in Mathematics.” (For comparison, Baby Rudin would be way more advanced than that, I’d schedule it even after Axler’s Linear Algebra, which itself should go after a more matrix-y introduction to linear algebra, like Strang.)
Axler’s Linear Algebra, which itself should go after a more matrix-y introduction to linear algebra, like Strang.)
Note however that this type of statement is strongly dependent on individual personality. For me, the correct order was definitely Axler first, then matrix-y later. (A position not to be confused, by the way, with “Axler only”, which would indeed be a mistake, even for me.)
Don’t know about matrix algebra books in general, but Strang is mostly elementary and incomplete, there are few proofs and the focus is on simple examples. It builds intuition for basic things like bases, rows/columns in matrix multiplication, subspace, kernel/image of a matrix and its transpose (not even thought of as adjoint, and the complex case is reduced to a syntactic analogy with real one), action of elementary transformations and change of basis. Axler then provides a detailed explanation of what’s really going on (which builds on the intuition formed by Strang) and extends the picture (enough to train intuition about things like the complex case, invariant subspaces, polar decomposition, generalized eigenvectors).
This seems like a natural order to me. How does the other way around work (taboo “personality”)?
I prefer to be told “what’s really going on” before practicing computations; this is both more intrinsically pleasant (I find) and aids in memory. See my post on Bayes’ theorem, where I contrast my abstract approach with the usual one, which starts with concrete examples.
Perhaps an even more extreme example would be multivariable calculus: I was never able to properly remember, let alone understand or apply, the theorems of Gauss, Green, and Stokes until I had learned the formalism of differential forms and the generalized theorem (itself arbitrarily called Stokes’ theorem, although either of the other names could also have been used). Presumably this is avoided in a first course because the idea of a multilinear map varying from point to point is considered too abstract—a completely spurious reason in my case.
There is a widespread false assumption that computational facility is a prerequisite for theoretical understanding, when in reality they are independent skills. An unfortunate consequence of believing this falsehood in the case of someone to whom theoretical understanding comes easily (e.g. me) is that you can get the impression that you don’t need to bother training computational facility (after all, you’re “already” doing the thing for which it is supposedly a prerequisite), and even eventually end up thinking that doing so would be beneath your status.
What would be ideal would be a larger version of Axler that supplemented the theoretical exercises with large numbers of computational ones (Axler itself contains a few, but not enough).
I think learning styles differ here. The thing that helped me understand multivariable calculus was taking it concurrently with electromagnetism. I really liked having a concrete example to have in mind whenever we talked about vector fields—most of the theorems about general vector fields have specific physical consequences, and I find remembering the physics helps me remember the theorem.
As I’m a bit further on in this path, so the details of the beginnings (introductions to proofs and logic) are a bit blurred to me, but I’ll present a few books that I think are good for moving from basics to graduate level topics.
In number theory to move from basics to graduate level topics would be Hatcher’s Topology of Numbers into Silverman and Tate into Koblitz.
For linear algebra I’d recommend Axler’s book, for complex analysis I liked Churchill and Brown in that I could basically read through and do the exercises very quickly as an undergrad. I didn’t have a great time with any of my real analysis texts so I don’t think I can give a good recommendation for that.
Of course your mileage may vary but these are the books that I’ve enjoyed the most, in that they stay grounded in computations, give decent intuition with good geometric pictures, and tend to have approachable exercises compared with other books I’ve read.
Also: learn to program in python, and use SAGE to solve project Euler problems.
There is a great 1st real analysis book that would work from HS level: “S. Abbott (2001). Understanding Analysis. Undergraduate Texts in Mathematics.” (For comparison, Baby Rudin would be way more advanced than that, I’d schedule it even after Axler’s Linear Algebra, which itself should go after a more matrix-y introduction to linear algebra, like Strang.)
Note however that this type of statement is strongly dependent on individual personality. For me, the correct order was definitely Axler first, then matrix-y later. (A position not to be confused, by the way, with “Axler only”, which would indeed be a mistake, even for me.)
Don’t know about matrix algebra books in general, but Strang is mostly elementary and incomplete, there are few proofs and the focus is on simple examples. It builds intuition for basic things like bases, rows/columns in matrix multiplication, subspace, kernel/image of a matrix and its transpose (not even thought of as adjoint, and the complex case is reduced to a syntactic analogy with real one), action of elementary transformations and change of basis. Axler then provides a detailed explanation of what’s really going on (which builds on the intuition formed by Strang) and extends the picture (enough to train intuition about things like the complex case, invariant subspaces, polar decomposition, generalized eigenvectors).
This seems like a natural order to me. How does the other way around work (taboo “personality”)?
I prefer to be told “what’s really going on” before practicing computations; this is both more intrinsically pleasant (I find) and aids in memory. See my post on Bayes’ theorem, where I contrast my abstract approach with the usual one, which starts with concrete examples.
Perhaps an even more extreme example would be multivariable calculus: I was never able to properly remember, let alone understand or apply, the theorems of Gauss, Green, and Stokes until I had learned the formalism of differential forms and the generalized theorem (itself arbitrarily called Stokes’ theorem, although either of the other names could also have been used). Presumably this is avoided in a first course because the idea of a multilinear map varying from point to point is considered too abstract—a completely spurious reason in my case.
There is a widespread false assumption that computational facility is a prerequisite for theoretical understanding, when in reality they are independent skills. An unfortunate consequence of believing this falsehood in the case of someone to whom theoretical understanding comes easily (e.g. me) is that you can get the impression that you don’t need to bother training computational facility (after all, you’re “already” doing the thing for which it is supposedly a prerequisite), and even eventually end up thinking that doing so would be beneath your status.
What would be ideal would be a larger version of Axler that supplemented the theoretical exercises with large numbers of computational ones (Axler itself contains a few, but not enough).
Hmm.
I think learning styles differ here. The thing that helped me understand multivariable calculus was taking it concurrently with electromagnetism. I really liked having a concrete example to have in mind whenever we talked about vector fields—most of the theorems about general vector fields have specific physical consequences, and I find remembering the physics helps me remember the theorem.