You had me at Bourbaki.
Fundamentals of Linear Algebra and Optimization [pdf]
41–50 of 63 posts
Re: Fundamentals of Linear Algebra and Optimization [pdf]
#42My test for linear algebra books is how they first present matrices and matrix multiplication. If they define a matrix as an NxM table of numbers with a multiplication operation defined as this complicated formula with a couple of nested sigmas, and then much later a lemma is mentioned that says every linear transformation can be represented as a matrix and then the composition of two transforms is the matrix multipl…
I would teach linear algebra with a book that barely mentions linear transformations. In finite dimensions, linear transformations and matrices are exactly the same object mathematical objects, with very different notations (matrix notation (boxes with numbers inside) vs the linear space/linear transformation notation). I would rather the students to learn deeper mathematics only in matrix notation, rather than to ma…
This is exactly backwards.
Whether you're discussing the Jacobian of a function, or change of basis matrices, learning the matrix formula is a lot less useful than seeing how it falls out of the linear function definition.
The formula is hard to memorize and gives no intuition for why anything is true. But from the linear function definition it is easy to reconstruct the formula.
In fact this is so true that I would say that anyone who only knows the matrix definition does not actually understand linear algebra.
Re: Fundamentals of Linear Algebra and Optimization [pdf]
#43My test for linear algebra books is how they first present matrices and matrix multiplication. If they define a matrix as an NxM table of numbers with a multiplication operation defined as this complicated formula with a couple of nested sigmas, and then much later a lemma is mentioned that says every linear transformation can be represented as a matrix and then the composition of two transforms is the matrix multipl…
I would teach linear algebra with a book that barely mentions linear transformations. In finite dimensions, linear transformations and matrices are exactly the same object mathematical objects, with very different notations (matrix notation (boxes with numbers inside) vs the linear space/linear transformation notation). I would rather the students to learn deeper mathematics only in matrix notation, rather than to ma…
And you would forever cripple your students. As potential users of linear algebra, they would be mentally mutilated beyond all hope of regeneration.
Re: Fundamentals of Linear Algebra and Optimization [pdf]
#44Earlier quoted context omitted.
Often linear algebra tools are applied to Hilbert spaces or Banach spaces, where fields other than R or C don't make much sense. Looking at page 25 of TFA: > In Definition 1.2, the field R may be replaced by the field of complex numbers C, in which case we have a complex vector space. It is even possible to replace R by the field of rational numbers Q or by any other field K (for example Z/pZ, where p is a prime numb…
Yeah, lots of courses pay lip service to the existence of fields of characteristic p during the initial few weeks, then proceed to ignore them completely.
How relevant do you think finite fields are for solving applied optimization problems? What proportion of a course do you think should be devoted to them?
Re: Fundamentals of Linear Algebra and Optimization [pdf]
#45My test for linear algebra books is how they first present matrices and matrix multiplication. If they define a matrix as an NxM table of numbers with a multiplication operation defined as this complicated formula with a couple of nested sigmas, and then much later a lemma is mentioned that says every linear transformation can be represented as a matrix and then the composition of two transforms is the matrix multipl…
I would teach linear algebra with a book that barely mentions linear transformations. In finite dimensions, linear transformations and matrices are exactly the same object mathematical objects, with very different notations (matrix notation (boxes with numbers inside) vs the linear space/linear transformation notation). I would rather the students to learn deeper mathematics only in matrix notation, rather than to ma…
You have succumbed to the coordinate virus, http://geocalc.clas.asu.edu/pdf/MathViruses.pdf
Re: Fundamentals of Linear Algebra and Optimization [pdf]
#46Earlier quoted context omitted.
Yeah, lots of courses pay lip service to the existence of fields of characteristic p during the initial few weeks, then proceed to ignore them completely.
How relevant do you think finite fields are for solving applied optimization problems? What proportion of a course do you think should be devoted to them?
Every ordered field has characteristic 0, so I don't think finite fields are likely to be terribly useful for solving optimization problems. But:
(0) There is way more to linear algebra than linear programming.
(1) A linear algebra course is not the right place for an extensive study of how to solve optimization problems anyway. That belongs in a real analysis course.
Re: Fundamentals of Linear Algebra and Optimization [pdf]
#47Earlier quoted context omitted.
How relevant do you think finite fields are for solving applied optimization problems? What proportion of a course do you think should be devoted to them?
Every ordered field has characteristic 0, so I don't think finite fields are likely to be terribly useful for solving optimization problems. But: (0) There is way more to linear algebra than linear programming. (1) A linear algebra course is not the right place for an extensive study of how to solve optimization problems anyway. That belongs in a real analysis course.
This is a graduate-level course in the department of computer and information science.
> Prerequisite(s): Undergraduate course in linear algebra, calculus
> The goal of this course is to provide firm foundations in linear algebra and optimization techniques that will enable students to analyze and solve problems arising in various areas of computer science, especially computer vision, robotics, machine learning, computer graphics, embedded systems, and market engineering and systems. The students will acquire a firm theoretical knowledge of these concepts and tools. They will also learn how to use these tools in practice by tackling various judiciously chosen projects (from computer vision, etc.). This course will serve as a basis to more advanced courses in computer vision, convex optimization, machine learning, robotics, computer graphics, embedded systems, and market engineering and systems.
Re: Fundamentals of Linear Algebra and Optimization [pdf]
#48My test for linear algebra books is how they first present matrices and matrix multiplication. If they define a matrix as an NxM table of numbers with a multiplication operation defined as this complicated formula with a couple of nested sigmas, and then much later a lemma is mentioned that says every linear transformation can be represented as a matrix and then the composition of two transforms is the matrix multipl…
Starting with an abstract formal description of vector spaces and linear transformations is not a pedagogically useful introduction, and it leaves out a huge amount of the historical/motivational context explaining most of the conventions used in linear algebra, and even the mental/conceptual understanding most working mathematicians have about the meaning of linear models and their use in various parts of mathematic…
And to follow-up, note that this book in particular is for a graduate-level course in the department of computer and information science, which has an undergraduate-level linear algebra course as a prerequisite.
Re: Fundamentals of Linear Algebra and Optimization [pdf]
#49Folks looking for introductory books in this area may also enjoy this new in-progress book: https://web.stanford.edu/~boyd/vmls/ As a follow-up I would recommend Trefethen & Bau, Numerical Linear Algebra .
Boyd and Vandenberghe will have a monopoly in anything optimization related if they keep on...
Re: Fundamentals of Linear Algebra and Optimization [pdf]
#50My test for linear algebra books is how they first present matrices and matrix multiplication. If they define a matrix as an NxM table of numbers with a multiplication operation defined as this complicated formula with a couple of nested sigmas, and then much later a lemma is mentioned that says every linear transformation can be represented as a matrix and then the composition of two transforms is the matrix multipl…
I think the book you're talking about is Axler's "Linear Algebra Done Right." I worked through the problems in this book with some friends, and it is a very good presentation. After doing that book I wanted to see what other books were out there, which ones were the good ones, and sort of classify them. I came up with this list: https://begriffs.com/posts/2016-07-24-best-linear-algebra-bo... I classify the books as G…
> After doing that book I wanted to see what other books were out there, which ones were the good ones, and sort of classify them. I came up with this list: https://begriffs.com/posts/2016-07-24-best-linear-algebra-bo...
Ironically you put "Jänich, K. (1994). Linear algebra. New York: Springer-Verlag." into the "Theoretical" section: In Germany this book (its German original) is not that well-regarded, since it is considered as far to shallow. Much better (and more hard to read) German textbooks are
Gerd Fischer - Lineare Algebra: Eine Einführung für Studienanfänger (Linear Algebra: An Introduction for freshmen)
Siegfried Bosch - Lineare Algebra (and its companion book: Siegfried Bosch - Algebra)