5 ms·

And modest too.

by glomph 13y ago

And modest too.

graycat 13y ago
Horn wrote about my work, "Best performance in the course throughout the semester. Knows this material cold." It was just true. Why? It was an 'advanced' course in linear algebra and matrix theory, and I'd never had even a first course in the subject. But long before the course, in total I'd done much more than just such a course. I had had two good courses in abstract algebra, each of which had provided a good foundation for linear algebra and also touched on the subject. I had taken a reading course in theoretical physics based heavily on matrix theory. And I had written my undergraduate honors paper on group representation theory, essentially in linear algebra. I'd worked carefully through two of the best books on linear algebra, one by E. Nearing and the other by P. Halmos. The Halmos book is famous as the crown jewel of books on linear algebra. Why? It can be good to look at linear algebra as a baby version of Hilbert space theory, heavily from von Neumann, and Halmos wrote his book when he was an assistant to von Neumann at the Institute for Advanced Study. So, deliberately, the Halmos book is an introduction to Hilbert space theory. It also has the profound perspective of one of the best mathematicians of all time and the elegance of thought of von Neumann. For applications, in my career and in my own study, I'd worked through stacks of books and papers on applications in numerical linear algebra, multivariate statistics, optimization, curve fitting, signal processing, especially around the fast Fourier transform, differential equations, advanced calculus, deterministic optimal control theory, exterior algebra, etc. At one point I wrote my own notes on much of linear algebra, with black ink on white paper, 100% cotton I got especially for the notes. My work was very careful although not elegant. For review, just before the course, I wrote out from memory 80 pages covering about 2/3rds of the course. As Horn wrote, I knew the subject cold, long before I took his course. Horn's homework, tests, and final exam were all challenging for the other students, and I effortlessly blew them away, by large margins on all three. E.g., I never studied for the tests or either the mid-term or final exam. When doing the homework, I nearly never looked at my notes from class and, instead, just wrote out the solutions based on what I already knew. I didn't belong in the course and had told the department that at the beginning. The department's reaction was to smile and say "Take the course anyway.". So, I did. For me, the course was a huge waste: From a long commute to campus and taking care of my wife, I was short on time, but the Horn homework had me up nearly all night and then exhausted the next day once a week and, thus, ate into a big fraction of the best time I had each week. Some of my other, more important work suffered. When I did the Horn homework, I was usually just exhausted. I'd get the papers back, graded, in about two weeks, recognize my handwriting, but, from having been so tired, actually had no memory of having done the work. Bummer. It was a trial by fire and exhaustion, not education. I was being asked to 'prove myself', not learn something. In the program, I only wanted to learn some good material and had no desire at all to 'compete', certainly not in that linear algebra course beneath me. I didn't know I was blowing away the other students until the grader showed me the scores on homework and tests, all except the final exam. In many ways Horn was a very good teacher, but, in total, he blew away nearly all the students. If I hadn't already known the material coming into the course, then I couldn't have learned it in the course. The way I had learned the material was from some slow, careful, thoughtful study and a lot of work with applications, and the course offered neither.
- Arnor 13y ago
  Would you mind dropping the titles of those books. I think I found the P Halmos (http://www.amazon.com/Algebra-Problem-Dolciani-Mathematical-Expositions/dp/0883853221 http://www.amazon.com/Algebra-Problem-Dolciani-Mathematical-...), but I'm having some trouble finding the Nearing. Also, what background do you think is necessary for these texts?
  - graycat 13y ago
    Two of the biggest 'branches' of mathematics are analysis and algebra. Essentially the first course in 'analysis' is calculus. As in Simmons (in this list), the two most important pillars of analysis are continuity and linearity. The first place to learn linearity well is 'linear algebra' -- or, much the same thing, matrix theory or finite dimensional vector spaces. Then the algebraic aspects can be emphasized and can lead to applications in, say, the theory of error correcting codes. For the analysis aspects, can continue with Hilbert and Banach spaces and more and also various applications. Much of the mathematics in applied mathematics, science, engineering, and technology is linear algebra. Here is a list of books relevant to linear algebra and/or to the course I took from Roger Horn. This list also contains maybe 60% of what I'd worked from before Horn's course that let me do well in that course. ======================================== Linear Algebra ---------------------------------------- Horn's course was close to Roger A. Horn, Charles R. Johnson, 'Matrix Analysis', 0-521-38632-2, Cambridge University Press, 1990. with also a few topics from Roger A. Horn, Charles R. Johnson, 'Topics in Matrix Analysis', 0-521-46713-6, Cambridge University Press, 1994. ---------------------------------------- The course also used for reference and some topics and exercises Richard Bellman, 'Introduction to Matrix Analysis: Second Edition', McGraw-Hill, New York, 1970. This book is just packed with little results; at some point can get the impression that the author went on and on ... writing. Bellman was a very bright guy in mathematics, engineering, and medicine. ---------------------------------------- Relatively easy to read and relatively close to applications, and another book Horn's course used as a reference, is Ben Noble, 'Applied Linear Algebra', Prentice-Hall, Englewood Cliffs, NJ, 1969. Some edition of this book may be a good place to start for a student interested in applications now. ---------------------------------------- About the easiest reading in this list, and my first text on linear algebra, was D. C. Murdoch, 'Linear Algebra for Undergraduates', John Wiley and Sons, New York, 1957. This book is awfully old, but what it has are still the basics. ---------------------------------------- For my undergraduate honors paper I made some use of, and later read carefully nearly all of, Evar D. Nering, 'Linear Algebra and Matrix Theory', John Wiley and Sons, New York, 1964. The main part of this book is a relatively solid start, maybe a bit terse and advanced for a first text. The book also has in the back a collection of advanced topics, some of which might be quite good to know at some point and difficult to get elsewhere. One of the topics in the back is linear programming, and for that I'd recommend something else, e.g., Chv'atal and/or Bazaraa and Jarvis in this list. ---------------------------------------- Likely the crown jewel of books on linear algebra is Paul R. Halmos, 'Finite-Dimensional Vector Spaces, Second Edition', D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958. Halmos wrote this in about 1942 when he was an assistant to von Neumann at the Institute for Advanced Study. The book is intended to be a finite dimensional introduction to Hilbert space theory, or how to do linear algebra using mostly only what also works in Hilbert space. It's likely fair to credit von Neumann with Hilbert space. The book is elegant. Apparently at one time Harvard's course Math 55, with a colorful description at, http://www.american.com/archive/2008/march-april-magazine-contents/why-can2019t-a-woman-be-more-like-a-man/?searchterm-Sommers http://www.american.com/archive/2008/march-april-magazine-co... used this text by Halmos and also, as also in this list, Rudin's 'Principles' and Spivak. ---------------------------------------- Long highly regarded as a linear algebra text is Hoffman and Kunze, 'Linear Algebra, Second Edition', Prentice-Hall, Englewood Cliffs, New Jersey, 1971. ======================================== Numerical Methods ---------------------------------------- If want to take numerical computations in linear algebra seriously, then consider the next book or something better if can find it George E. Forsythe and Cleve B. Moler, 'Computer Solution of Linear Algebraic Systems', Prentice-Hall, Englewood Cliffs, 1967. ======================================== Multivariate Statistics My main start with multivariate statistics was N. R. Draper and H. Smith, 'Applied Regression Analysis', John Wiley and Sons, New York, 1968. Apparently later editions of this book remain of interest. ---------------------------------------- A relatively serious book on 'regression analysis' is C. Radhakrishna Rao, 'Linear Statistical Inference and Its Applications: Second Edition', ISBN 0-471-70823-2, John Wiley and Sons, New York, 1967. ---------------------------------------- Three famous, general books on multivariate statistics are Maurice M. Tatsuoka, 'Multivariate Analysis: Techniques for Educational and Psychological Research', John Wiley and Sons, 1971. William W. Cooley and Paul R. Lohnes, 'Multivariate Data Analysis', John Wiley and Sons, New York, 1971. Donald F. Morrison, 'Multivariate Statistical Methods: Second Edition', ISBN 0-07-043186-8, McGraw-Hill, New York, 1976. ======================================== Analysis of Variance A highly regarded first book on analsis of variance and experimental design is George W. Snedecor and William G. Cochran, 'Statistical Methods, Sixth Edition', ISBN 0-8138-1560-6, The Iowa State University Press, Ames, Iowa, 1971. and a famous, more mathematical, book is Henry Scheff'e, 'Analysis of Variance', John Wiley and Sons, New York, 1967. ======================================== Linear Optimization A highly polished book on linear programming is Vav sek Chv'atal, 'Linear Programming', ISBN 0-7167-1587-2, W. H. Freeman, New York, 1983. ---------------------------------------- Nicely written and with more emphasis on the important special case of network flows is Mokhtar S. Bazaraa and John J. Jarvis, 'Linear Programming and Network Flows', ISBN 0-471-06015-1, John Wiley and Sons, New York, 1977. ---------------------------------------- A grand applied mathematics dessert buffet, based on Banach space and the Hahn-Banach theorem is David G. Luenberger, 'Optimization by Vector Space Methods', John Wiley and Sons, Inc., New York, 1969. ======================================== Mathematical Analysis Relevant to Understanding Linearity ---------------------------------------- Long the first place a math student gets a fully serious encounter with calculus and closely related topics has been Walter Rudin, 'Principles of Mathematical Analysis, Third Edition', McGraw-Hill, New York, 1964. The first chapters of this book do well as an introduction to metric spaces, and that work applies fully to vector spaces. ---------------------------------------- A nice place to get comfortable doing mathematics in several dimensions is Wendell H. Fleming, 'Functions of Several Variables', Addison-Wesley, Reading, Massachusetts, 1965. Some of the material here is also good for optimization. ---------------------------------------- Another place to get comfortable doing mathematics in several dimensions is Michael Spivak, {\it Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus,\/} W.\ A.\ Benjamin, New York, 1965.\ \ ---------------------------------------- The first half, the 'real' half of the next book has polished introductions to Hilbert and Banach spaces which are some of the most important vector spaces Walter Rudin, 'Real and Complex Analysis', ISBN 07-054232-5, McGraw-Hill, New York, 1966. ---------------------------------------- An elegant introduction to how to get comfortable in metric space is George F. Simmons, 'Introduction to Topology and Modern Analysis', McGraw Hill, New York, 1963. ======================================== Ordinary Differential Equations ---------------------------------------- Linear algebra is important, as some points crucial, for ordinary differential equations, a polished introduction from a world expert is Earl A. Coddington, 'An Introduction to Ordinary Differential Equations', Prentice-Hall, Englewood Cliffs, NJ, 1961.

And modest too.

by glomph 13y ago

And modest too.

Horn wrote about my work, "Best performance in the course throughout the semester. Knows this material cold." It was just true. Why? It was an 'advanced' course in linear algebra and matrix theory, and I'd never had even a first course in the subject. But long before the course, in total I'd done much more than just such a course. I had had two good courses in abstract algebra, each of which had provided a good foundation for linear algebra and also touched on the subject. I had taken a reading course in theoretical physics based heavily on matrix theory. And I had written my undergraduate honors paper on group representation theory, essentially in linear algebra. I'd worked carefully through two of the best books on linear algebra, one by E. Nearing and the other by P. Halmos. The Halmos book is famous as the crown jewel of books on linear algebra. Why? It can be good to look at linear algebra as a baby version of Hilbert space theory, heavily from von Neumann, and Halmos wrote his book when he was an assistant to von Neumann at the Institute for Advanced Study. So, deliberately, the Halmos book is an introduction to Hilbert space theory. It also has the profound perspective of one of the best mathematicians of all time and the elegance of thought of von Neumann. For applications, in my career and in my own study, I'd worked through stacks of books and papers on applications in numerical linear algebra, multivariate statistics, optimization, curve fitting, signal processing, especially around the fast Fourier transform, differential equations, advanced calculus, deterministic optimal control theory, exterior algebra, etc. At one point I wrote my own notes on much of linear algebra, with black ink on white paper, 100% cotton I got especially for the notes. My work was very careful although not elegant. For review, just before the course, I wrote out from memory 80 pages covering about 2/3rds of the course. As Horn wrote, I knew the subject cold, long before I took his course. Horn's homework, tests, and final exam were all challenging for the other students, and I effortlessly blew them away, by large margins on all three. E.g., I never studied for the tests or either the mid-term or final exam. When doing the homework, I nearly never looked at my notes from class and, instead, just wrote out the solutions based on what I already knew. I didn't belong in the course and had told the department that at the beginning. The department's reaction was to smile and say "Take the course anyway.". So, I did. For me, the course was a huge waste: From a long commute to campus and taking care of my wife, I was short on time, but the Horn homework had me up nearly all night and then exhausted the next day once a week and, thus, ate into a big fraction of the best time I had each week. Some of my other, more important work suffered. When I did the Horn homework, I was usually just exhausted. I'd get the papers back, graded, in about two weeks, recognize my handwriting, but, from having been so tired, actually had no memory of having done the work. Bummer. It was a trial by fire and exhaustion, not education. I was being asked to 'prove myself', not learn something. In the program, I only wanted to learn some good material and had no desire at all to 'compete', certainly not in that linear algebra course beneath me. I didn't know I was blowing away the other students until the grader showed me the scores on homework and tests, all except the final exam. In many ways Horn was a very good teacher, but, in total, he blew away nearly all the students. If I hadn't already known the material coming into the course, then I couldn't have learned it in the course. The way I had learned the material was from some slow, careful, thoughtful study and a lot of work with applications, and the course offered neither.

Arnor 13y ago
Would you mind dropping the titles of those books. I think I found the P Halmos (http://www.amazon.com/Algebra-Problem-Dolciani-Mathematical-Expositions/dp/0883853221 http://www.amazon.com/Algebra-Problem-Dolciani-Mathematical-...), but I'm having some trouble finding the Nearing. Also, what background do you think is necessary for these texts?
- graycat 13y ago
  Two of the biggest 'branches' of mathematics are analysis and algebra. Essentially the first course in 'analysis' is calculus. As in Simmons (in this list), the two most important pillars of analysis are continuity and linearity. The first place to learn linearity well is 'linear algebra' -- or, much the same thing, matrix theory or finite dimensional vector spaces. Then the algebraic aspects can be emphasized and can lead to applications in, say, the theory of error correcting codes. For the analysis aspects, can continue with Hilbert and Banach spaces and more and also various applications. Much of the mathematics in applied mathematics, science, engineering, and technology is linear algebra. Here is a list of books relevant to linear algebra and/or to the course I took from Roger Horn. This list also contains maybe 60% of what I'd worked from before Horn's course that let me do well in that course. ======================================== Linear Algebra ---------------------------------------- Horn's course was close to Roger A. Horn, Charles R. Johnson, 'Matrix Analysis', 0-521-38632-2, Cambridge University Press, 1990. with also a few topics from Roger A. Horn, Charles R. Johnson, 'Topics in Matrix Analysis', 0-521-46713-6, Cambridge University Press, 1994. ---------------------------------------- The course also used for reference and some topics and exercises Richard Bellman, 'Introduction to Matrix Analysis: Second Edition', McGraw-Hill, New York, 1970. This book is just packed with little results; at some point can get the impression that the author went on and on ... writing. Bellman was a very bright guy in mathematics, engineering, and medicine. ---------------------------------------- Relatively easy to read and relatively close to applications, and another book Horn's course used as a reference, is Ben Noble, 'Applied Linear Algebra', Prentice-Hall, Englewood Cliffs, NJ, 1969. Some edition of this book may be a good place to start for a student interested in applications now. ---------------------------------------- About the easiest reading in this list, and my first text on linear algebra, was D. C. Murdoch, 'Linear Algebra for Undergraduates', John Wiley and Sons, New York, 1957. This book is awfully old, but what it has are still the basics. ---------------------------------------- For my undergraduate honors paper I made some use of, and later read carefully nearly all of, Evar D. Nering, 'Linear Algebra and Matrix Theory', John Wiley and Sons, New York, 1964. The main part of this book is a relatively solid start, maybe a bit terse and advanced for a first text. The book also has in the back a collection of advanced topics, some of which might be quite good to know at some point and difficult to get elsewhere. One of the topics in the back is linear programming, and for that I'd recommend something else, e.g., Chv'atal and/or Bazaraa and Jarvis in this list. ---------------------------------------- Likely the crown jewel of books on linear algebra is Paul R. Halmos, 'Finite-Dimensional Vector Spaces, Second Edition', D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958. Halmos wrote this in about 1942 when he was an assistant to von Neumann at the Institute for Advanced Study. The book is intended to be a finite dimensional introduction to Hilbert space theory, or how to do linear algebra using mostly only what also works in Hilbert space. It's likely fair to credit von Neumann with Hilbert space. The book is elegant. Apparently at one time Harvard's course Math 55, with a colorful description at, http://www.american.com/archive/2008/march-april-magazine-contents/why-can2019t-a-woman-be-more-like-a-man/?searchterm-Sommers http://www.american.com/archive/2008/march-april-magazine-co... used this text by Halmos and also, as also in this list, Rudin's 'Principles' and Spivak. ---------------------------------------- Long highly regarded as a linear algebra text is Hoffman and Kunze, 'Linear Algebra, Second Edition', Prentice-Hall, Englewood Cliffs, New Jersey, 1971. ======================================== Numerical Methods ---------------------------------------- If want to take numerical computations in linear algebra seriously, then consider the next book or something better if can find it George E. Forsythe and Cleve B. Moler, 'Computer Solution of Linear Algebraic Systems', Prentice-Hall, Englewood Cliffs, 1967. ======================================== Multivariate Statistics My main start with multivariate statistics was N. R. Draper and H. Smith, 'Applied Regression Analysis', John Wiley and Sons, New York, 1968. Apparently later editions of this book remain of interest. ---------------------------------------- A relatively serious book on 'regression analysis' is C. Radhakrishna Rao, 'Linear Statistical Inference and Its Applications: Second Edition', ISBN 0-471-70823-2, John Wiley and Sons, New York, 1967. ---------------------------------------- Three famous, general books on multivariate statistics are Maurice M. Tatsuoka, 'Multivariate Analysis: Techniques for Educational and Psychological Research', John Wiley and Sons, 1971. William W. Cooley and Paul R. Lohnes, 'Multivariate Data Analysis', John Wiley and Sons, New York, 1971. Donald F. Morrison, 'Multivariate Statistical Methods: Second Edition', ISBN 0-07-043186-8, McGraw-Hill, New York, 1976. ======================================== Analysis of Variance A highly regarded first book on analsis of variance and experimental design is George W. Snedecor and William G. Cochran, 'Statistical Methods, Sixth Edition', ISBN 0-8138-1560-6, The Iowa State University Press, Ames, Iowa, 1971. and a famous, more mathematical, book is Henry Scheff'e, 'Analysis of Variance', John Wiley and Sons, New York, 1967. ======================================== Linear Optimization A highly polished book on linear programming is Vav sek Chv'atal, 'Linear Programming', ISBN 0-7167-1587-2, W. H. Freeman, New York, 1983. ---------------------------------------- Nicely written and with more emphasis on the important special case of network flows is Mokhtar S. Bazaraa and John J. Jarvis, 'Linear Programming and Network Flows', ISBN 0-471-06015-1, John Wiley and Sons, New York, 1977. ---------------------------------------- A grand applied mathematics dessert buffet, based on Banach space and the Hahn-Banach theorem is David G. Luenberger, 'Optimization by Vector Space Methods', John Wiley and Sons, Inc., New York, 1969. ======================================== Mathematical Analysis Relevant to Understanding Linearity ---------------------------------------- Long the first place a math student gets a fully serious encounter with calculus and closely related topics has been Walter Rudin, 'Principles of Mathematical Analysis, Third Edition', McGraw-Hill, New York, 1964. The first chapters of this book do well as an introduction to metric spaces, and that work applies fully to vector spaces. ---------------------------------------- A nice place to get comfortable doing mathematics in several dimensions is Wendell H. Fleming, 'Functions of Several Variables', Addison-Wesley, Reading, Massachusetts, 1965. Some of the material here is also good for optimization. ---------------------------------------- Another place to get comfortable doing mathematics in several dimensions is Michael Spivak, {\it Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus,\/} W.\ A.\ Benjamin, New York, 1965.\ \ ---------------------------------------- The first half, the 'real' half of the next book has polished introductions to Hilbert and Banach spaces which are some of the most important vector spaces Walter Rudin, 'Real and Complex Analysis', ISBN 07-054232-5, McGraw-Hill, New York, 1966. ---------------------------------------- An elegant introduction to how to get comfortable in metric space is George F. Simmons, 'Introduction to Topology and Modern Analysis', McGraw Hill, New York, 1963. ======================================== Ordinary Differential Equations ---------------------------------------- Linear algebra is important, as some points crucial, for ordinary differential equations, a polished introduction from a world expert is Earl A. Coddington, 'An Introduction to Ordinary Differential Equations', Prentice-Hall, Englewood Cliffs, NJ, 1961.