Finite-dimensional vector spaces

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1. graycatJul 2025
   "Functions Are Vectors"?

   Let's see: Let A be a non-empty set, N the set of positive whole numbers, and X the set of all functions f

   f: A --> N

   with usual notation.

   Assume as is common, the scalars are the set of real numbers, but the set of complex numbers will also do.

   So, is X a vector space and, thus, each f in X a vector?

   No, since -f is not in X. Neither is (1/2)f.

   Some references (with TeX markup):

   Paul R.\ Halmos, {\it Finite-Dimensional Vector Spaces, Second Edition\/}

   linear algebra treated as functional analysis.

   Walter Rudin, {\it Real and Complex Analysis\/}

   with Lebesgue integration and, then, Banach and Hilbert vector spaces.

   Walter Rudin, {\it Functional Analysis\/}

   with Fourier theory.

   Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability\/}

   with random variables, that is, functions from a probability space to, usually, the set of real numbers with convergence results, building on the work A. Kolmogorov building on the work of H. Lebesgue.

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2. nextosMar 2025
   Richard Zach et al. logic textbooks are a work of art: https://slc.openlogicproject.org.

   Aside, I'd be happy to pay for great classic math textbooks in a well typeset edition.

   Finite Dimensional Vector Spaces is one such example. The text is fantastic, but everything is too cramped and hard to read compared to a modern book from the 1970s onward.

   A newer edition typeset in LaTeX would be great.

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3. smatijaFeb 2025
   Linear algebra doesn't have many preconditions - so being used to math texts is much more important for it than having some pre-knowledge.

   Easiest to start will be either Linear Algebra done right by Axler or Linear algebra done wrong by Treil - try both, they take very different approaches to introducing the subject, so one of them may be compatible with you.

   After finishing either of these I heartily recommend you Finite-Dimensional Vector Spaces by Halmos - it's way more abstract, so I cannot recommend it as first introduction to LA, but it proves its results in very beautiful and a bit uncommon way. It's probably my favourite book from undergrad.

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4. graycatDec 2024
   This thread seems to be about good writing for math.

   Okay, for some decades, I've read, written, taught, applied, and published, in total, quite a lot of math. Got a Ph.D. in applied math.

   Yes, there are problems in writing math, that is, some math is poorly written.

   But, some math is quite nicely written. (1) Of course, at least define every symbol before using it. (2) It helps to motivate some math before presenting it. (3) Sometimes intuitive statments can help.

   For more, carefully reading some well-written math can help learning how to write math well:

   Paul R.\ Halmos, {\it Finite-Dimensional Vector Spaces, Second Edition,\/} D.\ Van Nostrand Company, Inc., Princeton, New Jersey, 1958.\ \

   R.\ Creighton Buck, {\it Advanced Calculus,\/} McGraw-Hill, New York, 1956.\ \

   Tom M.\ Apostol, {\it Mathematical Analysis: Second Edition,\/} ISBN 0-201-00288-4, Addison-Wesley, Reading, Massachusetts, 1974.\ \

   H.\ L.\ Royden, {\it Real Analysis: Second Edition,\/} Macmillan, New York, 1971.\ \

   Walter Rudin, {\it Real and Complex Analysis,\/} ISBN 07-054232-5, McGraw-Hill, New York, 1966.\ \

   Leo Breiman, {\it Probability,\/} ISBN 0-89871-296-3, SIAM, Philadelphia, 1992.\ \

   Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability,\/} Holden-Day, San Francisco, 1965.\ \

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5. graycatJul 2024
   Yes,

   > The Greatest Educational Life Hack: Learning Math Ahead of Time (justinmath.com)

   worked for me, can work for a lot of people, and is a good idea.

   Partly:

   (1) One way to win a 100 yard dash is to start running half way to the finish line and have no one object. The US educational system will usually overlook something like that starting half way to the finish line.

   (2) Reasons: (a) The system assumes that their teaching is crucial and that no student really can learn on their own, i.e., the student didn't actually start half way to the finish line. (b) The system so wants more good students that they will overlook the evidence that the student was ahead at the start of the class. But, research in math mostly requires working alone directly from original papers, and working from a highly polished text is usually much easier -- so profs learn on their own, and students can too.

   (3) Generally in math, independent study can work well. Basically for each lesson, (a) study the text, (b) work most of the exercises, especially the more challenging ones, and check the answers in the back of the text (need a suitable text or just get the book for teachers), and (c) in a quiet room, lean back, relax, and think a little about what the value, purpose, content of the lesson was, say, be able to explain it to someone who never studied math.

   (3) So, take calculus in high school. And visit, call, whatever, and see what the popular college calculus texts are, get one or two of those (used can be a lot cheaper), and before college have worked hard on both the high school course and the college text(s). Then in college, right, take calculus, likely from a text have already done well in. So, will likely be one of the best students in the class. Then will get a good reputation that can be valuable.

   (4) Will be ahead, so continue this way and stay ahead.

   (5) Next math course, say, modern, abstract algebra, i.e., set theory, groups, fields, Galois theory, elementary number theory, maybe a start on linear algebra.

   Next, linear algebra, maybe the most important and useful course. Work through a popular text that is relatively easy. Then work carefully through one or two of the classics, e.g., Halmos, Finite Dimensional Vector Spaces.

   Likely next, "Baby Rudin", W. Rudin, Principles of Mathematical Analysis, calculus and somewhat more done with depth and precision. See the roles of open and closed sets, closed and bounded sets, i.e., compact sets, continuous functions, the powerful results of continuous functions on compact sets, Fourier series.

   Advanced calculus, i.e., partial derivatives and Stokes formula.

   Analysis, e.g., the real part of W. Rudin, Real and Complex Analysis, Lebesgue's alternate and nicer way to define integration (in short, partition on the Y axis instead of the X axis), the Fourier integral, Banach space, Hilbert space, the Radon-Nikodym theorem (can be used for grand approaches to information, Bayes Rule, the Neyman-Pearson result in best statistical hypothesis testing, and with von Neumann's proof based just on polynomials is charming, ...).

   More, e.g., differential equations, probability, statistics, stochastic processes, optimization, complex analysis, number theory, whatever.

   One consequence: Will learn how to write math. Too often people who don't know advertise that they don't know much math.

   At some point in business, some of that math might be valuable. E.g., current AI uses steepest descent via calculus and optimization, linear algebra, and hypothesis testing.

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6. graycatApr 2024
   Consider ordinary 3 dimensional space: Go to a corner of the room, where the floor and two walls meet, and see three lines, each line the intersection of the two walls or a wall and the floor. In usual building architecture, these three lines are mutually perpendicular, i.e., orthogonal.

   Regard each of these three lines as an axis of a coordinate system. In particular, pick a point on each of the three axes to give us points with coordinates:

   (1,0,0) -- the X axis

   (0,1,0) -- the Y axis

   (0,0,1) -- the Z axis

   Soooo, now we have a standard coordinate system for ordinary 3 dimensional space.

   Now consider a point P in the room (in my case, the end of the pony tail of that sweet, gorgeous blond I keep looking at!) with coordinates

   (x,y,z)

   for some numbers x, y, z.

   For the X coordinate of point P we make use of the X axis and our point P: We take the dot (inner) product of

   (1,0,0) and (x,y,z)

   and get, presto, bingo, wonder of wonders, no doubt, x.

   Similarly for y and z.

   Or, for each axis, our inner product projects point P on that axis. And from those projections we can reconstruct the coordinates of point P. Beginning to sound like Fourier theory????

   This stuff of projections onto orthogonal and then reconstructions from that generalizes.

   In Fourier theory, those sine waves are mutually perpendicular -- project one on another and get 0. That summation is the inner product for that projection.

   And if not a summation, then an integration -- same stuff, use an inner product to do projections onto mutually perpendicular axes.

   For the finite stuff, and just summation signs, it's fairly elementary.

   For the integral stuff, with the Riemann integral of calculus, can see Baby Rudin, Principles of Mathematical Analysis and Fourier series, over finite lengths of time. For the Fourier integral, over the whole real axis, see the Lebesgue integral (partition on the Y axis instead of the X axis and get nicer properties) in Rudin's Real and Complex Analysis. For a version that uses Schwartz distributions, there Rudin's Functional Analysis.

   Net, it's all about inner products and projections as we saw in the corner of a room.

   Soooo, for some of the generalization, are working in an inner product space. A nice introduction is, of course, Halmos, Finite Dimensional Vector Spaces: Soooo, that's ordinary linear algebra, finite dimensional, but done with techniques that mostly generalize to infinite dimensions, e.g., as in Fourier series, integrals, distributions.

   Rudin's Real and Complex Analysis also does Fubini's theorem that helps with doing Fourier theory on surfaces, e.g., the ocean.

   History: Halmos had gotten his Ph.D. under Doob, of Stochastic Processes (e.g., the stock market, weather, medical EGK, ...), and got a position under John von Neumann at the Institute for Advanced Study in Princeton; so, apparently Halmos wrote his finite dimensional book borrowing from von Neumann's work on the infinite dimensional case, e.g., as in von Neumann's Quantum Mechanics.

   Right, a complete inner product space is a Hilbert space -- the complete is essentially as also in the real numbers, sequences that appear to converge in the distance (from the inner product) used always do have something to converge to.

   Sooo, Hilbert space has a lot, asks a lot, making the examples restricted -- Rudin has a nice theorem that shows that really there's only one Hilbert space!

   Halmos is my favorite author. Rudin is a close second.

   Education Technique: While starting my career in computing, I studied both Halmos and Rudin carefully, independently. In linear algebra, also E. Nering. At the time, did a lot with the FFT (fast Fourier transform, mostly of J. Tukey), and, thus, at one point got "sole source" on some work for the US Navy. Then in grad school, the backgrounds in Halmos, Nering, Rudin, and the computing got me the best in the class on the three Ph.D. qualifying exams -- linear algebra, analysis, computing.

   As I recall, Nering was a student of E. Artin, at Princeton. Nering's book also considers the case of finite fields, that is, not only the rational, real, complex numbers. That helped in a course in error correcting codes.

   Why is the FFT "fast"? Well, if all you want is some one Fourier coefficient, it's not "fast". If you want all the coefficients, then just write a Do-loop and find them all one at a time -- easy enough to do. Get the discrete Fourier transform. But if go for all the coefficients this way very often on fairly long streams of data, then can slow down even a fast computer. BUT: If the number of points in time is a power of 2, commonly 1024 = 2^10, and look carefully at elementary trigonometry, can see how to do a little on all the computing, then a little more on all of it, ..., for ~10 such passes, and, presto, bingo, be really a LOT faster, essentially n log(n) instead of n^2.

   There's more at

   https://www.youtube.com/watch?v=nmgFG7PUHfo

   with the role of Richard Garwin working to do Fourier analysis on seismic data as part of detection of underground nuclear tests.

   Much of the faculty was really, REALLY torqued that (1) I didn't fail the exams and (2) led the class. REALLY torqued!

   Halmos, Rudin -- can be good stuff.

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