Volume 7, 1995

University of Bialystok

Copyright (c) 1995 Association of Mizar Users

**Mariusz Zynel**- Warsaw University, Bialystok

- The main purpose of the paper is to define the dimension of an abstract vector space. The dimension of a finite-dimensional vector space is, by the most common definition, the number of vectors in a basis. Obviously, each basis contains the same number of vectors. We prove the Steinitz Theorem together with Exchange Lemma in the second section. The Steinitz Theorem says that each linearly-independent subset of a vector space has cardinality less than any subset that generates the space, moreover it can be extended to a basis. Further we review some of the standard facts involving the dimension of a vector space. Additionally, in the last section, we introduce two notions: the family of subspaces of a fixed dimension and the pencil of subspaces. Both of them can be applied in the algebraic representation of several geometries.

- Preliminaries
- The Steinitz Theorem
- Finite-Dimensional Vector Spaces
- The Dimension of a Vector Space

- [1]
Grzegorz Bancerek.
Cardinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Jaroslaw Kotowicz.
Functions and finite sequences of real numbers.
*Journal of Formalized Mathematics*, 5, 1993. - [8]
Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski.
Abelian groups, fields and vector spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [9]
Robert Milewski.
Associated matrix of linear map.
*Journal of Formalized Mathematics*, 7, 1995. - [10]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [11]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [12]
Wojciech A. Trybulec.
Vectors in real linear space.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Wojciech A. Trybulec.
Basis of vector space.
*Journal of Formalized Mathematics*, 2, 1990. - [14]
Wojciech A. Trybulec.
Linear combinations in vector space.
*Journal of Formalized Mathematics*, 2, 1990. - [15]
Wojciech A. Trybulec.
Operations on subspaces in vector space.
*Journal of Formalized Mathematics*, 2, 1990. - [16]
Wojciech A. Trybulec.
Pigeon hole principle.
*Journal of Formalized Mathematics*, 2, 1990. - [17]
Wojciech A. Trybulec.
Subspaces and cosets of subspaces in vector space.
*Journal of Formalized Mathematics*, 2, 1990. - [18]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [19]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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