Proving that QC is a Vector Space
In this article, we will delve into the mathematical proof to show that the set of rational functions with coefficients in the field of rational numbers, denoted as (mathbb{Q}C), forms a vector space over the field (mathbb{Q}). To establish this, we need to verify all the axioms of a vector space over the specified field.
Definition of a Vector Space
A set (V) is considered a vector space over a field (F) if it satisfies the following properties:
Closure under Addition
(V) is closed under addition, meaning that for any two elements (u, v in V), their sum (u v) is also in (V).
Closure under Scalar Multiplication
(V) is closed under scalar multiplication, implying that for any element (v in V) and any scalar (a in F), the product (a cdot v) is also in (V).
Existence of Additive Identity
There exists an additively neutral element (0 in V) such that for any (v in V), (v 0 v).
Existence of Additive Inverses
For every (v in V), there exists an element (-v in V) such that (v (-v) 0).
Associativity of Addition
For any (u, v, w in V), the operation of addition is associative: ((u v) w u (v w)).
Commutativity of Addition
For any (u, v in V), the operation of addition is commutative: (u v v u).
Distributive Property
The distributive property states that for any (a in F) and (u, v in V), the following holds: (a cdot (u v) a cdot u a cdot v).
Compatibility of Scalar Multiplication
For any (a, b in F) and (v in V), the compatibility property is satisfied: ((ab) cdot v a cdot (b cdot v)).
Identity Element of Scalar Multiplication
The multiplicative identity (1 in F) acts as the identity element for scalar multiplication: (1 cdot v v) for all (v in V).
Verification for (mathbb{Q}C)
Now let's apply these axioms to the set of rational functions with coefficients in the field of rational numbers, (mathbb{Q}C).
Closure under Addition
For any two rational functions (u, v in mathbb{Q}C), their sum is also a rational function. This is because the sum of two fractions is another fraction. Thus, (mathbb{Q}C) is closed under addition.
Closure under Scalar Multiplication
For any rational function (v in mathbb{Q}C) and any rational number (a in mathbb{Q}), the product (a cdot v) is also a rational function. Therefore, (mathbb{Q}C) is closed under scalar multiplication.
Existence of Additive Identity
The zero function (0 in mathbb{Q}C) serves as the additive identity, meaning that for any rational function (v in mathbb{Q}C), (v 0 v).
Existence of Additive Inverses
For any rational function (v in mathbb{Q}C), there exists a rational function (-v in mathbb{Q}C) such that (v (-v) 0).
Associativity of Addition
The operation of addition in rational functions is associative. This follows from the properties of addition in the field of rational numbers.
Commutativity of Addition
The operation of addition in rational functions is commutative. Again, this property is inherited from the field of rational numbers.
Distributive Property
The distributive property of scalar multiplication over addition is satisfied in (mathbb{Q}C). This follows from the distributivity of multiplication over addition in the field of rational numbers.
Compatibility of Scalar Multiplication
For any two rational numbers (a, b in mathbb{Q}) and any rational function (v in mathbb{Q}C), the compatibility property holds: ((ab) cdot v a cdot (b cdot v)).
Identity Element of Scalar Multiplication
The multiplicative identity (1 in mathbb{Q}) acts as the identity element for scalar multiplication in (mathbb{Q}C): (1 cdot v v) for all (v in mathbb{Q}C).
Conclusion
Since the set of rational functions with coefficients in the field of rational numbers, (mathbb{Q}C), satisfies all the axioms of a vector space over the field (mathbb{Q}), we can conclude that (mathbb{Q}C) is indeed a vector space over (mathbb{Q}).