Journal of Formalized Mathematics
Volume 7, 1995
University of Bialystok
Copyright (c) 1995 Association of Mizar Users

## The Steinitz Theorem and the Dimension of a Vector Space

Mariusz Zynel
Warsaw University, Bialystok

### 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.

#### MML Identifier: VECTSP_9

The terminology and notation used in this paper have been introduced in the following articles [10] [18] [11] [2] [19] [4] [5] [1] [6] [3] [16] [7] [12] [8] [17] [14] [15] [13] [9]

#### Contents (PDF format)

1. Preliminaries
2. The Steinitz Theorem
3. Finite-Dimensional Vector Spaces
4. The Dimension of a Vector Space

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