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Vector Spaces

In: Basic Modern Algebra with Applications

Author

Listed:
  • Mahima Ranjan Adhikari

    (Institute for Mathematics, Bioinformatics, Information Technology and Computer Science (IMBIC))

  • Avishek Adhikari

    (University of Calcutta, Department of Pure Mathematics)

Abstract

Chapter 8 introduces another algebraic system, called vector spaces (linear spaces) interlinking both internal and external operations. In this chapter vector spaces and closely related fundamental concepts, such as linear independence, basis, dimension, linear transformation and its matrix representation, eigenvalue, inner product space, Hilbert space, quadratic form, Jordan canonical form etc., are studied. Such concepts form an integral part of linear algebra. Vector spaces have multi-faceted applications. Such spaces over finite fields play an important role in computer science, coding theory, design of experiments and combinatorics. Vector spaces over the infinite fields Q of rationals are important in number theory and design of experiments and vector spaces over C are essential for the study of eigenvalues. As the concept of a vector provides a geometric motivation, vector spaces facilitate the study of many areas of mathematics and integrate the abstract algebraic concepts with the geometric ideas.

Suggested Citation

  • Mahima Ranjan Adhikari & Avishek Adhikari, 2014. "Vector Spaces," Springer Books, in: Basic Modern Algebra with Applications, edition 127, chapter 0, pages 273-354, Springer.
    • Handle: RePEc:spr:sprchp:978-81-322-1599-8_8
    DOI: 10.1007/978-81-322-1599-8_8
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