## The Steinitz theorem and the dimension of a vector space (1995)

Venue: | Journal of Formalized Mathematics |

Citations: | 7 - 0 self |

### BibTeX

@ARTICLE{Zynel95thesteinitz,

author = {Mariusz Zynel},

title = {The Steinitz theorem and the dimension of a vector space},

journal = {Journal of Formalized Mathematics},

year = {1995},

volume = {7}

}

### OpenURL

### Abstract

Summary. 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.

### Citations

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Citation Context ...of them can be applied in the algebraic representation of several geometries. MML Identifier: VECTSP 9. The terminology and notation used in this paper have been introduced in the following articles: =-=[13]-=-, [23], [12], [8], [2], [6], [24], [4], [5], [22], [1], [7], [3], [17], [19], [9], [21], [15], [10], [20], [16], [18], [14], and [11]. 1. Preliminaries For simplicity we follow the rules: G1 is a fiel... |

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Citation Context ... representation of several geometries. MML Identifier: VECTSP 9. The terminology and notation used in this paper have been introduced in the following articles: [13], [23], [12], [8], [2], [6], [24], =-=[4]-=-, [5], [22], [1], [7], [3], [17], [19], [9], [21], [15], [10], [20], [16], [18], [14], and [11]. 1. Preliminaries For simplicity we follow the rules: G1 is a field, V is a vector space over G1, W is a... |

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