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Did fields other than R and C eat the authors' children or something?

jacobolus 9y ago
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 number), in which case we have a K-vector space (in (V3), ∗ denotes multiplication in the field K). In most cases, the field K will be the field R of reals.
- catnaroek 9y ago
  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.
  - jacobolus 9y ago
    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?
    - catnaroek 9y ago
      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.
      - jacobolus 9y ago
        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.

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 number), in which case we have a K-vector space (in (V3), ∗ denotes multiplication in the field K). In most cases, the field K will be the field R of reals.

catnaroek 9y ago
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.
- jacobolus 9y ago
  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?
  - catnaroek 9y ago
    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.
    - jacobolus 9y ago
      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.