Abstract
In the third chapter we move to a more abstract and general level. The notions of vector space, subspace, dimension, basis, linear transformations, isomorphism, etc. are introduced and discussed. At the end of this chapter, the notions of dual vector space and forms and polynomials in vectors are considered. Most of the abstract concepts are illustrated with various examples and applications. For instance, the notion of a dual space is accompanied with the idea of “generalized functions” (distributions), and the notions of forms and polynomials in vectors are accompanied with Euler’s identity for homogeneous polynomials.
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Notes
- 1.
Readers who are familiar with the concept of a group will be able to reformulate conditions (a)–(c) in a compact way by saying that with respect to the operation of vector addition, the vectors form an abelian group.
- 2.
For readers familiar with the definition of a field, we can give a general definition: The characteristic of a field \(\mathbb { K}\) is the smallest natural number k such that the k-fold sum kD=D+⋯+D is equal to 0 for every element \(D \in \mathbb { K}\) (as is easily seen, this number k is the same for all D≠0). If no such natural number k exists (as in, for example, the most frequently encountered fields, \(\mathbb { K}=\mathbb {R}\) and \(\mathbb { K}=\mathbb {C}\)), then the characteristic is defined to be zero.
- 3.
Translator’s note: It may be tempting to consider “null space” a possible synonym for the zero space. However, that term is reserved as a synonym for “kernel,” to be introduced below, in Definition 3.67.
- 4.
More precisely, this identification is achieved with the help of the concept of isomorphism of vector spaces, which will be introduced below, in Sect. 3.5.
- 5.
Translator’s note: Another name for kernel that the reader may encounter is null space (since the kernel is the space of all vectors that map to the null vector).
- 6.
Such a generalized function is called a Dirac delta function in honor of the English physicist Paul Adrien Maurice Dirac, who was the first to use generalized functions (toward the end of the 1920s) in his work on quantum mechanics.
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© 2012 Springer-Verlag Berlin Heidelberg
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Shafarevich, I.R., Remizov, A.O. (2012). Vector Spaces. In: Linear Algebra and Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30994-6_3
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DOI: https://doi.org/10.1007/978-3-642-30994-6_3
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