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Vector Spaces

  • David A. Towers
Chapter
Part of the Macmillan Mathematical Guides book series (MG)

Abstract

Let us start reviewing the situation we studied in Chapter 1. We were concerned with two sets: a set V of vectors and a set F of scalars. We defined a means of adding vectors and of multiplying vectors by scalars, and found that these two operations satisfied the following axioms, or laws:
  • V1 \(\left( {b{\text{ + }}c} \right)\) for all a, b, cϵV;

  • V2 there is a vector 0ϵV with the property that

    \(a{\text{ + 0 = 0 + }}a{\text{ = }}a\) for all aϵV;

  • V3 for each aϵV there is a corresponding vector −aϵV such that

    \(a{\text{ + (}} - {\text{ }}a{\text{)}} {\text{ = }} {\text{(}} - {\text{ }}a{\text{) + }}a{\text{ = }}0\);

  • V4 \({\text{a + b = b + a}}\) for all a, bϵV;

  • V5 \(\alpha \left( {{\text{a + b}}} \right) = \alpha {\text{a}} + \alpha {\text{b}}\) for all αϵF, and all a, bϵV;

  • V6 \(\left( {\alpha + \beta } \right){\text{a}} = \alpha {\text{a}} + \beta {\text{a}}\) for all α, βϵF, and all aϵV;

  • V7 \(\left( {\alpha \beta } \right){\text{a}} = \alpha \left( {\beta {\text{a}}} \right)\) for all α, βϵF, and all aϵV;

  • V8 \(1{\text{a}} = {\text{a}}\) for all aϵV;

  • V9 \(0{\text{a}} = 0\) for all aϵV.

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Copyright information

© David A. Towers 1988

Authors and Affiliations

  • David A. Towers
    • 1
  1. 1.University of LancasterUK

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