It is time to consider the idea of an abstract Vector space which is something which has two operations
satisfying the following vector space axioms.
Definition 3.0.1A vector spaceis an Abelian group of “vectors” satisfying the axioms of anAbeliangroup,
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 to multiply thevectors accordingto the following rules. (The Greek letters denote scalars.)
α (v+ w) = αv+ αv, (3.1)
(3.1)
(α + β)v = αv+ βv, (3.2)
(3.2)
α (βv ) = αβ (v), (3.3)
(3.3)
1v = v. (3.4)
(3.4)
The field of scalars is usually ℝ or ℂ and the vector space will be called real or complex depending on
whether the field is ℝ or ℂ. However, other fields are also possible. For example, one could use the field of
rational numbers or even the field of the integers mod p for p a prime. A vector space is also
called a linear space. These axioms do not tell us anything about what is being considered.
Nevertheless, one can prove some fundamental properties just based on these vector space
axioms.
Proposition 3.0.2In 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 the axioms,
′ ′
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
Now add −0x to both sides. This gives 0 = 0x. Finally,
(− 1)x + x = (− 1)x + 1x = (− 1+ 1)x = 0x = 0
By the uniqueness of the additive inverse shown earlier,
(− 1)
x = −x. ■
If you are interested in considering other fields, you should have some examples other than ℂ, ℝ, ℚ.
Some of these are discussed in the following exercises. If you are happy with only considering
ℝ and ℂ, skip these exercises. Here is an important example which gives the typical vector
space.
Example 3.0.3Let Ω be a nonempty set and define V to be the set of functions defined on Ω. Lettinga,b,c be scalars and f,g,h functions, the vector operations are defined as
(f + g)(x) ≡ f (x)+ g (x)
(af)(x) ≡ a(f (x))
Then this is an example of a vector space. Note that the set where the functions have their values can beany vector space.
To verify this, we check the axioms.
(f + g)(x) = f (x)+ g(x) = g(x)+ f (x) = (g+ f)(x)