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Evar D. Nering, Linear Algebra and Matrix Linear Algebra and Matrix Theory* Kenneth Hoffmann And Ray Kunze. Linear Algebra, 2nd Edition, Prentice-Hall, Englew

Evar D. Nering, Linear Algebra and Matrix Linear Algebra and Matrix Theory* Kenneth Hoffmann And Ray Kunze. Linear Algebra, 2nd Edition, Prentice-Hall, Englewood Cliffs, New Jersey, 1971. https://www.zuj.edu.jo/download/linear-algebra-2nd-edition-kenneth-hoffmann-and-ray-kunze-pdf/ https://www.zuj.edu.jo/download/linear-algebra-2nd-edition-k... Halmos, Finite Dimensional Vector Spaces George E. Forsythe and Cleve B. Moler, Computer Solution of Linear Algebraic Systems Paul R. Halmos, Naive Set Theory, Van Nostrand, Princeton, NJ, 1960. More has been done since this book, but this book is a gorgeous introduction to axiomatic set theory. So, even people who want to dig into the latest work would do well to have this as the first book. And for people wanting to read any of the more advanced material here, knowledge of this book will be from good to have to important. https://piazza-resources.s3.amazonaws.com/is25bi6c6o1oh/isv14pknyni2z3/Halmos_NaiveSetTheory.pdf?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=ASIAR6AWVCBX2QKTT5FL%2F20181117%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20181117T065628Z&X-Amz-Expires=3600&X-Amz-SignedHeaders=host&X-Amz-Security-Token=FQoGZXIvYXdzEDcaDD%2BDN9rfglTm8iQUcSK3A9mwFlP1FL3rh9MEaKv5ruNv9Jumkq6iYIs2VOYjuLbwlclIwLwgwc68%2BB8dae8N5wlPOVR9e8GMYBM6G2eRiinQGvPgG0I4ZQa6qZTUbS5p4Lf%2FLNCL0%2FZWPlh9osB5anqQzIUIA6nJ7A22RVzQyPatcty7%2BRoIEPnEd5gPAFoTVZ2wYTxqokLR6FT8%2BAEnHdpt4meHCJa34DwhJ57svoc467DOa2%2BFaHavgWkHf%2BSNBa%2FJlbvtKEzp%2BPxEv0eBmjGhTy8%2BVL3HwNZn4SuvG8ZXGCWgmiOrCMjaePaS79lMTBxlY8OdMwEU%2FAioIdIEFaToS%2FxwHtaIjq8EKHCCFF7hJFkzpQypbp3qzoDt1wAMs%2FQcyCh5P%2BwjFpGq5ZSCo2fqwXSa6v2mCKIUp7b0bo2v2lWPHCsMrbky%2FS5TOJkY%2BCscHyFW1hcRV3UHRYfHGzO2PXrBJsH9O2aZkwEoROZkc4Q4ILNgLbG9rPHr1elkHvqMVAIDvrR1sdLc%2F316CIiDfJpHbpUzXGfWV2Glw8HRSUWFXCRBFeUO8TarPpYgJ%2FSdOR36sxDiaCVHvsGnywX24rgcDGQo0MW%2B3wU%3D&X-Amz-Signature=6a4db0ee5fe2a59b7d7db55665a6a7a440d90d15357db939469b204832ea7594 https://piazza-resources.s3.amazonaws.com/is25bi6c6o1oh/isv1... Walter Rudin, Principles of Mathematical Analysis The third edition is a lot better than the first two. H. L. Royden, Real Analysis: Second Edition Beautifully written, elegant, but maybe don't work way too hard on the exercises about upper/lower semi-continuity, and there is a better summary than Littlewood's three principles. Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis John C. Oxtoby, Measure and Category: A Survey of the Analogies between Topological and Measure Spaces Walter Rudin, Real and Complex Analysis Walter Rudin, Functional Analysis Leo Breiman, Probability Kai Lai Chung, A Course in Probability Theory, Second Edition Jacques Neveu, Mathematical Foundations of the Calculus of Probability Erhan Cinlar, Introduction to Stochastic Processes J. L. Doob, Stochastic Processes I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes I, II Donald E. Knuth, The TeX book Donald E. Knuth, The Art of Computer Programming, Second Edition Leo Breiman, "Statistical Modeling: The Two Cultures," Statistical Science, Vol. 16, No. 3, 199–231, 2001. Paul R. Halmos, "The Theory of Unbiased Estimation", Annals of Mathematical Statistics, Volume 17, Number 1, pages 34-43, 1946. Paul R. Halmos and L. J. Savage, "Application of the Radon-Nikodym Theorem to the Theory of Sufficient Statistics", The Annals of Mathematical Statistics, Volume 20, Number 2 (1949), 225-241. https://projecteuclid.org/download/pdf_1/euclid.aoms/1177730032 https://projecteuclid.org/download/pdf_1/euclid.aoms/1177730...

graycat 8y ago
Here are URLs of PDFs of two of the references above: Leo Breiman, "Statistical Modeling: The Two Cultures," Statistical Science, Vol. 16, No. 3, 199–231, 2001. http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ss/1009213726 http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?vi... Paul R. Halmos, "The Theory of Unbiased Estimation", Annals of Mathematical Statistics, Volume 17, Number 1, pages 34-43, 1946. https://projecteuclid.org/download/pdf_1/euclid.aoms/1177731020 https://projecteuclid.org/download/pdf_1/euclid.aoms/1177731...
- nextos 8y ago
  Finite Dimensional Vector Spaces is a jewel. But I wish the typesetting was updated to something more modern. Same applies to Rudin. I find well typeset TeX a joy to read. Whereas FDVS is a bit cramped and looks antique.
  - graycat 8y ago
    FDVS was written in 1942 when Halmos had just gotten his Ph.D. from J. Doob, author of Stochastic Processes in my list, at U. Illinois, and was an assistant to John von Neumann at the Institute of Advanced Study in Princeton. IIRC Hilbert space was a von Neumann idea: It is first, just a definition -- complete inner product (dot product in much of physics and engineering) space. But the good stuff is (1) importance of the examples and (2) the theorems that show the consequences, e.g., in Fourier theory. Well, the vector spaces of most interest in linear algebra are actually (don't tell anyone) finite dimensional Hilbert spaces. So, one role of FDVS is to provide a text on linear algebra that is also an introduction to Hilbert space, that is, that tries to use ideas that work in any Hilbert space to get the basic results in linear algebra. The treatment of self-adjoint transformations and spectral theory are likely the most influenced by this role. This role is accomplished so well that sometimes physics students starting on quantum mechanics are advised to get at least the start they need on Hilbert space from FDVS. Sure, a better start is the one chapter on Hilbert space in Rudin's Real and Complex Analysis. The chapter there on the Fourier transform is also good, short, all theorems nicely proved, the main, early results made clear. Also a good start on the basic results of self-adjoint matrices are the inverse and implicit function theorems given as nice exercises in the third edition of Rudin's Principles .... And spectral theory is in Rudin's Functional Analysis. Also get a bonus of a nice treatment of distributions, that is, replace the Dirac delta function usage in quantum mechanics. For how to get the eigen value and orthogonal eigen vector results for self-adjoint matrices from the inverse and implicit (these two go together like ice cream and cake) function theorems is in Fleming, Functions of Several Variables. Then you will be off and running on factor analysis, principle components, the polar decomposition, the singular value decomposition, and more.

graycat 8y ago

Here are URLs of PDFs of two of the references above: Leo Breiman, "Statistical Modeling: The Two Cultures," Statistical Science, Vol. 16, No. 3, 199–231, 2001. http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ss/1009213726 http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?vi... Paul R. Halmos, "The Theory of Unbiased Estimation", Annals of Mathematical Statistics, Volume 17, Number 1, pages 34-43, 1946. https://projecteuclid.org/download/pdf_1/euclid.aoms/1177731020 https://projecteuclid.org/download/pdf_1/euclid.aoms/1177731...

nextos 8y ago
Finite Dimensional Vector Spaces is a jewel. But I wish the typesetting was updated to something more modern. Same applies to Rudin. I find well typeset TeX a joy to read. Whereas FDVS is a bit cramped and looks antique.
- graycat 8y ago
  FDVS was written in 1942 when Halmos had just gotten his Ph.D. from J. Doob, author of Stochastic Processes in my list, at U. Illinois, and was an assistant to John von Neumann at the Institute of Advanced Study in Princeton. IIRC Hilbert space was a von Neumann idea: It is first, just a definition -- complete inner product (dot product in much of physics and engineering) space. But the good stuff is (1) importance of the examples and (2) the theorems that show the consequences, e.g., in Fourier theory. Well, the vector spaces of most interest in linear algebra are actually (don't tell anyone) finite dimensional Hilbert spaces. So, one role of FDVS is to provide a text on linear algebra that is also an introduction to Hilbert space, that is, that tries to use ideas that work in any Hilbert space to get the basic results in linear algebra. The treatment of self-adjoint transformations and spectral theory are likely the most influenced by this role. This role is accomplished so well that sometimes physics students starting on quantum mechanics are advised to get at least the start they need on Hilbert space from FDVS. Sure, a better start is the one chapter on Hilbert space in Rudin's Real and Complex Analysis. The chapter there on the Fourier transform is also good, short, all theorems nicely proved, the main, early results made clear. Also a good start on the basic results of self-adjoint matrices are the inverse and implicit function theorems given as nice exercises in the third edition of Rudin's Principles .... And spectral theory is in Rudin's Functional Analysis. Also get a bonus of a nice treatment of distributions, that is, replace the Dirac delta function usage in quantum mechanics. For how to get the eigen value and orthogonal eigen vector results for self-adjoint matrices from the inverse and implicit (these two go together like ice cream and cake) function theorems is in Fleming, Functions of Several Variables. Then you will be off and running on factor analysis, principle components, the polar decomposition, the singular value decomposition, and more.