Chapter 3

Vector SpacesIt is time to consider the idea of an abstract vector space which is something which has twooperations satisfying the following vector space axioms.

Definition 3.0.1 A vector space is an Abelian group of “vectors” satisfying the axioms ofan Abelian group,

v+w = w+ v,

the commutative law of addition,

(v+w)+ z = v+(w+ z) ,

the associative law for addition,v+0 = v,

the existence of an additive identity,

v+(−v) = 0,

the existence of an additive inverse, along with a field of “scalars” F which are allowed tomultiply the vectors according to the following rules. (The Greek letters denote scalars.)

α (v+w) = αv+αv, (3.1)

(α +β )v = αv+βv, (3.2)

α (βv) = αβ (v) , (3.3)

1v = v. (3.4)

For example, any field is a vector space having field of scalars equal to the field itself.The field of scalars is often R or C and the vector space will be called real or complexdepending on whether the field is R or C. However, other fields are also possible. Forexample, one could use the field of rational numbers or even the field of the integers modp for p a prime. A vector space is also called a linear space. These axioms do not tell usanything about what is being considered. Nevertheless, one can prove some fundamentalproperties just based on these vector space axioms.

Proposition 3.0.2 In any vector space, 0 is unique,−x is unique, 0x = 0, and (−1)x =−x.

Proof: Suppose 0′ is also an additive identity. Then for 0 the additive identity in theaxioms,

0′ = 0′+0 = 0

Next suppose x+ y = 0. Then add −x to both sides.

−x =−x+(x+ y) = (−x+ x)+ y = 0+ y = y

Thus if y acts like the additive inverse, it is the additive inverse.

0x = (0+0)x = 0x+0x

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