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Other The Should I Become a Mathematician? Thread

  1. Dec 17, 2016 #3726
    Might need a separate thread,but I'll make 2 quick points:
    1) I think mathematics has something to offer anyone at any level
    2) we may need to discuss what we mean by working really really hard
     
  2. May 31, 2017 #3727

    mathwonk

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    Recently I rewrote my linear algebra notes. Hoping to achieve a result that someone could actually learn from, I expanded the explanation from under 15 pages to over 125 pages., Also I felt as if I actually understood the topic at last. The summary of what I learned is in the new 2 page introduction. If anyone enjoys and/or benefits from these, I am happy. One fun thing I learned from writing these is a cell decomposition of the grassmannian is given by the row reduced echelon form! who knew? (probably many of you, but not me.) (Would this qualify as an insight article? If so feel free to post it as one.)

    http://alpha.math.uga.edu/~roy/laprimexp.pdf
     
    Last edited: Jun 1, 2017
  3. May 31, 2017 #3728

    symbolipoint

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    Point #2 seems like it is not too precise, but honestly, WE KNOW WHAT IT MEANS! We do not need overly academic psycho games here.

    Truly, some people do not know how to study too well, but other people do. One can typically not expect to study algebra 1 or 2 for just an hour per day, three times per week, and earn at least a C. Not enough effort. Not enough focus. Not enough study session length for good development. Without my trying to describe the details, filling the effort upward from that description, the study effort comes increasingly closer to "STUDYING REALLY HARD".
     
  4. Jun 1, 2017 #3729

    mathwonk

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    In relation to the discussion of how hard one needs to work to understand something, I would observe that the linear algebra notes I just posted have been rewritten many times over several years, and I only lately feel I understood the topic. That is also in addition to having taught the subject at various levels, from undergrad to graduate algebra many times over several decades, and to have written a graduate level algebra book including a thorough treatment of linear algebra. I have also read treatments by Lang, E. Artin, M. Artin, Hoffman and Kunze, etc etc.... and watched videos by Strang.

    Among the newer insights I have acquired is the fact that such topics as jordan normal form and diagonalization are usually emphasized, although as I recently appreciated, they are completely impractical for actual calculations in any even mildly general problem. This is due to the fact, usually ignored in calculus and other courses, that one generally cannot actually factor a randomly given polynomial into irreducible factors, say over the rationals, much less the reals. Hence all problems that we give students in calculus and linear algebra are carefully rigged to be easily doable, and there is no guarantee at all that the problems they encounter in real life practice, or even make up for themselves, will be even remotely doable by hand. As a professor, working from a book with cooked problem sets already included, I had the luxury of ignoring this inconvenient truth, and remained unaware of how useless the skills were that I was offering my students.

    relatively few books explain to students the one actually feasible technique that they can always use in actual calculations, namely diagonalization of the characteristic matrix, by row and column operations within the ring of polynomials. This always yields the determinant and the invariant factors, hence also the rational canonical form. Then in those rare cases where these factors can be split further into irreducibles, it also may be refined to the jordan form.

    most books also ignore explaining the geometric meaning of the reduced row echelon form, including the nice fact that it allows one to put coordinates on the grassman manifold of subspaces of a given vector space, and even gives a nice cell decomposition that easily yields the homology of the grassmannian. uniqueness of this reduced row echelon form is also usually omitted although there are many rather elementary and easy, as well as enlightening arguments. It has taken me years to appreciate all of these things, so I would just tell any young or new student that math just repays lots and lots of careful and repeated consideration. It also helps to try to explain it to someone else, which is my main reason for writing so many math essays and books or booklets. Of course you always hope someone else will benefit but they seldom seem to attract many readers. In this last case though I can say that envisioning a particular audience helped me focus my explanation by constantly aiming it at what I thought would be clear to that audience. So it helps in writing to imagine who you are writing for.
     
  5. Jun 1, 2017 #3730

    mathwonk

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    To summarize what I have learned, the basic tool in finite dimensional linear algebra and matrix theory is row and column operations. This is a computational way of changing your space by isomorphisms until the structure of your operator or matrix has been simplified enough to be visibly understandable.

    In particular, two matrices are row equivalent, or left equivalent, if and only if the linear operators they represent become equal after changing the target space by an isomorphism, if and only if they have the same kernel, if and only if the matrices have the same row space, iff one can be transformed into the other by row operations, or iff they become equal after left multiplication by an invertible matrix, and finally iff they have the same row reduced echelon form.

    The reduced row echelon form of a matrix is a matrix whose rows consist of a particularly nice basis for the common row space of matrices in the equivalence class; this is the unique basis that projects to the standard basis of the coordinate subspace spanned by the “pivot” coordinates. This provides a canonical representative for the left equivalence class. Finding solutions to the equation AX=0, i.e. finding a basis of the kernel of A, is easily done by row reducing A, since the reduced form has the same kernel, and one which is more easily found.

    Row reduced r by n echelon matrices of rank r allow one to decompose the Grassmannian manifold of all r dimensional subspaces of k^n into “n choose r” cells, where each cell corresponds to the location of the r pivot columns. The largest cell is the one with the first r columns as pivots, hence the manifold has dimension r.(n-r), the number of free entries in such a row reduced matrix. They also allow one to put local coordinate charts on this manifold if we relax the definition of reduced to allow each of the n-r non pivot columns to have r entries, even if they are not the last n-r columns. Unlike the cells, these charts of course overlap.

    Two matrices are right equivalent iff they have the same column space, iff the linear operators they represent have the same image, iff they can be transformed into one another by column operations, iff they become equal after right multiplication by an invertible matrix, i.e. they become equal as linear operators after an isomorphism of the source space. A canonical representative of this class is obtained by row reducing the transpose and then transposing it back. This has as columns a natural basis of the column space, analogous to the case above.

    Two matrices A,B are (2 - sided) equivalent iff they can be transformed into one another by a combination of row and column operations, iff they become equal after (possibly) different isomorphisms of both source and target space, iff B = QAP where Q,P are invertible, iff A,B have the same rank r. A canonical representative for this class is the diagonal matrix whose first r diagonal entries are ones and the rest zeroes.

    Two square matrices A,B are similar iff they become the same after performing conjugation by some invertible matrix, i.e. iff B = CAC^(-1) for some invertible C, i.e. they become equal as operators after performing a single isomorphism of the common source and target space. This equivalence can be determined by row and column operations performed on the associated “characteristic matrix”. If A is a square matrix, its associated characteristic matrix is the matrix [Id.X-A] with polynomial entries. This matrix can be diagonalized by row and column operations in the ring of polynomials, using the Euclidean algorithm. This can be done in a unique way so that the diagonal entries successively divide one another. Two square matrices A,B of the same size, are similar iff their characteristic matrices have the same diagonalized form. The non constant entries on the diagonal, which characterize the similarity class, are called the invariant factors of the (similarity class of the) matrix. Thus two n by n matrices are similar iff they have the same invariant factors.

    If the invariant factors of A are f1,...,fn, then the linear operator represented by the original matrix is similar to the operation of multiplication by X on the product space k[X]/(f1) x ... x k[X]/(fn). The matrix of that multiplication operator, in the standard bases {1, X, X^2,....} for these factor spaces, is called the rational canonical form of the original matrix A.

    Since multiplication by X satisfies the minimal polynomial f on the factor space k[X]/(f), it follows that the largest of the invariant factors of A is the minimal polynomial of the matrix A. In case one can factor this polynomial into irreducible factors over the field k, one can decompose the product decomposition further into a product of space of form k[X]/(g) where each polynomial g is a power of an irreducible factor of the minimal polynomial. This decomposition then gives rise to the jordan canonical form, after a slight tweak of the usual choice of basis. Since multiplication by X carries each basis vector in the standard basis {1,X,X^2,...} into the next one, except for the last, a decomposition into a product of spaces like k[X]/(f) is called a “cyclic” decomposition. The rational canonical decomposition is the cyclic decomposition with the fewest number of factors, while the Jordan decomposition is the one with the largest number of factors.

    The nicest jordan form occurs when the irreducible factors of the minimal polynomial are all linear, and all occur to the first power in the minimal polynomial. Then the jordan form is diagonal. Even though one may not be able to compute this diagonal form, when working over the real number field this case always occurs when the original matrix A equals its transpose. Moreover in this nice case, the basis vectors making the matrix diagonal can even be chosen as mutually orthogonal, which is nice for doing geometry.

    One can deduce from all this that the characteristic polynomial of A, which equals det[Id.X-A], is the product of the invariant factors of A, and its constant term is the determinant of A, and that this term is non zero if and only if A is invertible. One can actually compute the inverse of A by row reducing the matrix [A , Id].

    that’s all folks. I guess the main difference between my old and my new point of view is that I like to focus now more on actually computable techniques, rather than the ideally simplest types of matrices (diagonal) which are impractical to compute,
     
    Last edited: Jul 29, 2017
  6. Jun 1, 2017 #3731

    jedishrfu

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    Why not put this together as an insight article on linear algebra?

    Also theres an interesting set of videos on youtube by 3blue1brown called the Essence of Linear Algebra which are quite good.
     
  7. Jun 1, 2017 #3732

    mathwonk

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    can they just download my notes (linked in post 3727) from my webpage as an insight article? or do i need to write a new one (maybe ≈ 125 pages is too long?). I am not quite up to doing that right this second, having finished this project to my own satisfaction, over many years. (I first posted or linked to the 15 page version here over 12 years ago.)
     
  8. Jun 1, 2017 #3733

    jedishrfu

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    Insight articles are a page or two. Check the PF insights tab to see examples.
     
    Last edited: Aug 7, 2017
  9. Aug 7, 2017 #3734
    I'm a prospective math major. While I see myself likely going to grad school, I'd like to have the option of earning a living straight out of undergrad—you never know how circumstances and interests will change in four years.

    So, how can I manouever myself into a position where I can either continue into higher education or land a well-paying job? I plan on taking a few courses in computer science, economics, and physics. Is it wise to cultivate knowledge in a field where I can apply math skills?
     
  10. Aug 7, 2017 #3735

    symbolipoint

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    Yes computer science and programming, good, important things to include. Engineering courses can add to your value too.
     
  11. Jan 22, 2018 #3736

    mathwonk

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    here is a link to my latest version of a condensed, advanced, undergraduate/graduate linear algebra book. It is aimed at someone who has already had at least one course in the topic and wants a beeline treatment of the main classification results on linear transformations up to various equivalences such as similarity. In particular both rational canonical form and Jordan form are discussed to some extent, plus spectral theorems. If anyone has the stamina to read some of it, I would benefit from some feedback.

    I am aware it takes time to read something like this, since I myself have been reading Mumford's redbook of algebraic geometry since June and am only up to page 58. A reader of my notes might be someone who possibly already understands linear algebra, and is interested in seeing whether this summary agrees with his/her understanding. Thanks in advance.

    http://alpha.math.uga.edu/~roy/laprimexp.pdf
     
    Last edited: Feb 4, 2018
  12. Feb 4, 2018 #3737

    mathwonk

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    By the way, even for those who do not wish to spend a lot of time reading those notes, there is a 3 page introduction summarizing the whole document which I recommend to anyone who wants to know briefly what the subject is about. Indeed even reading the first page of this introduction will tell you exactly what a general linear transformation on a finite dimensional space looks like, up to isomorphism ("similarity").

    The reason I recommend the introduction is that I wrote it only after completely finishing the notes, i.e. after I myself actually felt I understood the topic as well as I ever would.

    Basically the answer is that every linear operator breaks up into a sum of "cyclic" operators. These cyclic operators can have different degrees. The simplest cyclic operator, of degree one, just takes a basis vector u to a scalar multiple of itself, like au. A cyclic operator of degree 2, operates on a sequence of 2 basis vectors, u,v, and takes u to v, and takes v to some linear combination of u and v, say au+bv. A cyclic operator of degree 3 operates on a sequence of 3 basis vectors u,v,w, and takes u to v, takes v to w, and takes w to some linear combination of u,v,w, say au+bv+cw. And so on. Thus to understand a cyclic operator you just need to know its degree and the coefficients of the last linear combination, which can be conveniently expressed as the coefficients of a polynomial, the "minimal polynomial" of the cyclic operator.

    An operator which is a sum of operators all of degree one, is called diagonalizable, and most books emphasize this case. Unfortunately they often do not tell you that in practice it is usually impossible to decide whether this happens, and even when it does it is usually impossible to find those degree one operators. It is always possible however to decompose any operator into cyclic ones of higher degree, by finding their minimal polynomials, and hence this is emphasized first in my notes, before discussing the diagonalizable special case. The point is that although it is usually impossible in practice to diagonalize a given scalar matrix, it is always possible to diagonalize the associated "characteristic matrix" of polynomials, and this lets you find the minimal polynomials of a sequence of cyclic operators that decompose your original matrix.

    I see I am repeating myself somewhat, but at least this version is more elementary than the ones above, and hopefully thus more clear.
     
  13. Feb 24, 2018 #3738
    I'm about to bore young students with some simple advice, but sometimes we need to repeat the simple stuff.

    If you want to excel at anything you have to go above and beyond what is expected of you. Doing things like, working through problems that weren't assigned in class, and going over old exam papers, things like that I assume are standard practice. You need to be doing more than that.

    When I was a young student, you're so impressed or even overwhelmed by your surroundings, your fellow studends, the professors, etc., that you take the education you're given as the one true gospel. Well there are many short-comings with your education system, that people have been thinking about since the beginning of time. For example, the topics you cover and the forms and types of the courses you take will be standard, and have been for several decades now (in Maths anyway), when they may to be the best anymore; also, they try to cram a lot into a 4 year degree, but there's still a lot they have to leave out; your lecturers are probably trying to balance a research career with teaching, etc.

    The one time I came top of a class was in Real Analysis. Some of the students I beat in that course went on to get PhDs at top international universities. The only reason I came first that was because in the vacation before the course, I got a book on analysis out of the library and went through the first few chapters of it myself, because I felt this was a subject I was weak in, but I was intrigued by it nonetheless.

    I think it goes without saying, you have to be using your vacations to do some additional studying; whether it's revising old coursework, trying to solve problems you couldn't do before, preparing for new courses, or studying some other topic on your own.

    But besides that, you should be studying outside of what assigned to you by your lecturers. If your course only covers 7 chapters in a 10 chapter book, make a plan to study the others at some other time. Go to the library, find other books on the subject, and if they catch your fancy, work through that as well. Don't take what's given to you in your coursework as the be-all and end-all of your education. Only you can really educate yourself, and you have to take control of your education. Go to the library and try to read old papers on the subject you're studying. If they're too advanced, try to find older papers and try again. Study topics outside those you will do in your degree. For example, often Number Theory isn't taught in universities, but you may want to study that on your own. Etc.

    There's a lot of advice on PF and elsewhere on the web on how to take control of your own mathematics education. Don't be afraid to go further than what's set out for you in courses.
     
  14. Feb 24, 2018 #3739
    This will be glossed over, but I feel an interesting piece of advice is: getting things wrong is more important than getting things right in mathematics.
     
  15. Feb 24, 2018 #3740

    symbolipoint

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    Notable comment. It reminds me of some saying from some obscure source which remarked, ",...wisdom is the result of bad experience."
     
  16. Feb 25, 2018 #3741

    mathwonk

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    Another piece of basic math advice: I have been struggling at length to make a proof today and have just realized I have not yet used the hypothesis!


    the advice of qspeechc reminds me of a course I took in which the lecturer omitted a few basic proofs which I learned on my own. On the final I gave those proofs and it seemed to make an impression. This is quite useful when requesting a letter of recommendation.
     
  17. May 10, 2018 #3742
    How important is Graph Theory and Combinatorics to a mathematician? I'm sure it depends on the field, but let's say the main fields, algebra, analysis, geometry, number theory, etc.?
     
  18. May 10, 2018 #3743

    mathwonk

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    I have myself never used either in my career as algebraic geometer. Of course I don't know anything about them either which makes it hard to use them. On the other hand one of my more knowledgable friends, did use graph theory in his proof that surfaces in P^3 can have a certain number of singular points.

    Beauville, A. ``Sur le nombre maximum de points doubles d'une surface dans o_847.gif ( o_842.gif ).'' Journées de géométrie algébrique d'Angers (1979). Sijthoff & Noordhoff, pp. 207-215, 1980

    I was amazed and impressed at the time, but not enough to study graph theory. I tend to study things not so much because they may be useful but because they appeal to me.

    edit: I seem to be confusing graph theory with coding theory, but they may be related, or at least one may be used in the other. I seem to recall my friend utilized the concept of a "Hamming code" in his proof, but i no longer have the reference in my library.
     
    Last edited: May 11, 2018
  19. May 10, 2018 #3744

    jedishrfu

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  20. May 11, 2018 #3745

    mathwonk

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    i had to google "hamming codes/graph theory" to get a connection.
     
  21. May 11, 2018 #3746

    symbolipoint

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    mathwonk, I am curious and if you have a response maybe you do not want to give it publicly but if you could do your education and choices over again, would you choose something other than PhD in Mathematics?

    You can just ignore the question, respond on the forum, or respond privately.
     
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