# 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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