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 3.0.1 A vector space is an Abelian group of “vectors” satisfying the axioms of an Abelian
the commutative law of addition,
the associative law for addition,
the existence of an additive identity,
the existence of an additive inverse, along with a field of “scalars” F which are allowed to multiply the
vectors according to the following rules. (The Greek letters denote scalars.)
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
Proposition 3.0.2 In any vector space, 0 is unique, −x is unique, 0x = 0, and
Proof: Suppose 0′ is also an additive identity. Then for 0 the additive identity in the axioms,
Next suppose x+y=0. Then add −x to both sides.
Thus if y acts like the additive inverse, it is the additive inverse.
Now add −0x to both sides. This gives 0 = 0x. Finally,
By the uniqueness of the additive inverse shown earlier,
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
Example 3.0.3 Let Ω be a nonempty set and define V to be the set of functions defined on Ω. Letting
a,b,c be scalars and f,g,h functions, the vector operations are defined as Then this is an example of a vector space. Note that the set where the functions have their values can be
any vector space.
To verify this, we check the axioms.
Since x is arbitrary, f + g = g + f.
Let 0 denote the function which is given by 0
Then this is an
additive identity because
and so f + 0 = f. Let −f be the function which satisfies
Hence f +
and so a
As above, F will be a field.
Definition 3.0.4 Define Fn ≡
if and only if for all j = 1,
= yj. When
∈ Fn, it is conventional to denote
by the single bold face letter x. The numbers, xj are called the coordinates. Elements in Fn are
called vectors. The set
for t in the ith slot is called the ith coordinate axis. The point 0 ≡
is called the origin. Note that
this can be considered as the set of F valued functions defined on
. When the ordered list
is considered, it is just a way to say that f
x2 and so forth. Thus it is a case
of the typical example of a vector space mentioned above.