@MISC{VECTSP_9.MIZ,
  AUTHOR = {\.Zynel, Mariusz},
  TITLE = {The {S}teinitz Theorem and the Dimension of a Vector Space},
  SECTION1 = {Preliminaries},
  SECTION2 = {The Steinitz Theorem},
  SECTION3 = {Finite-Dimensional Vector Spaces},
  SECTION4 = {The Dimension of a Vector Space},
  DAY = {06},
  MONTH = {October},
  YEAR = {1995},
  ADDRESS1 = {Warsaw University\\Bia{\l}ystok},
  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.}}

