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Other The Should I Become a Mathematician? Thread

  1. Dec 15, 2016 #3721
    I know you are being hypothetical, but most of the time when I've seen a student fail a class multiple times the problem was in the approach, or they weren't giving it enough time in the *right* way, or they had other classes occupying their grey matter, or some attitude towards mathematics they picked up from somewhere, or some otherwise emotional blockage.

    But OK, instead of going down that road I'll let the rest of your scenario play out, because I don't think that was the point.

    The following was posted by Field's Medalist Timothy Gowers on Google+ about a year ago:

    "What is it like to do maths?

    About 99% of the time it's like this. "

    GOWERS.JPG


    My wife, who supported me through my degree, and who (rightfully) was keen on me not wasting my time, would often ask me how my homework/study was going.

    I would often respond with something like, "Well, I didn't get any of the problems done, and I don't feel like I really understand that much yet. But I spent a lot of good time thinking about the concepts."

    It took a bit of explaining to assure her that this was actually, in my opinion, productive time. Obviously I had no evidence of this fact unless ultimately I ended up producing something (like a completed homework, or a decent test grade).

    So, I think that if you are at least giving a sincere effort to do mathematics, there will always be some benefit, even if you don't seem to have much to show for it. I did terribly in Graduate Algebra, but having gone through the class made other classes and subjects seem a lot easier.

    -Dave K
     
  2. Dec 15, 2016 #3722
    I attended good lectures where I enjoyed the professor's teaching style and found it elucidating. If the professor was not very good, I would often not attend unless I was stuck at a certain point, then I would go in so that I could hear him discuss it and ask questions about the parts I didn't understand.

    -Dave K
     
  3. Dec 15, 2016 #3723

    symbolipoint

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    dkotschessaa,

    What I described was the pattern of repeated fail/non-passage of ONE COURSE AT A TIME. The finer detail is
    1. Take course
    2. NOT pass course
    3. Repeat course one time
    4. Pass course successfully
    5. Next Course of a sequence of courses --- Repeat at #1 with the incremented course
    Meaning, each course passed but always needed to be repeated ONE TIME.
     
  4. Dec 16, 2016 #3724
    Oh.

    In that case, I can't even say. It seems like a pretty unlikely scenario. Most people who fail courses fail one or two and drop out if they fail a whole bunch more. Sometimes they leave and come back much later. Strictly speaking the time it took me to get my bachelors was 18 years, if you count the first time I tried to go to college in 1996. I also repeated pre-calculus and two semesters of calculus, though this was due to time passed rather than failure.

    -Dave K
     
    Last edited: Dec 16, 2016
  5. Dec 16, 2016 #3725

    symbolipoint

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    Many people will give up when they fail a couple of Math (even Calculus) courses, and then pick or find something much less mathematics-intensive. Then there are some, maybe only a few, who will persist and keeping working at the needed mathematics courses until passing because these people stay dedicated to whatever math-intensive field they have chosen. Not sure which is the smarter way to go. Fail a couple of courses and change direction; or keep at it until passing each of the needed mathematics courses.

    Something worth knowing is that if a student really works hard to learn a course the first time, does not pass it, and then repeats the course and again REALLY WORKS HARD the second time too, the course really does become easier to learn and understand.
     
  6. 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
     
  7. 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
  8. 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".
     
  9. 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.
     
  10. 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
  11. 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.
     
  12. 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.)
     
  13. 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
  14. 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?
     
  15. 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.
     
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