Is the Set of Real Numbers a Vector Space over the Field of Rational Numbers?
Mathematically, to determine if the set of real numbers mathbb{R} is a vector space over the field of rational numbers mathbb{Q} involves verifying whether mathbb{R} meets the vector space axioms under the operations of addition and scalar multiplication defined in mathbb{R}.
Definitions and Axioms of a Vector Space
A vector space over a field mathbb{F} in this case mathbb{Q} must adhere to several key axioms:
Closure Under Addition
For any mathbb{R} times; mathbb{R} where both numbers are in mathbb{R}, their sum mathbb{R} mathbb{R} is also in mathbb{R}.
Closure Under Scalar Multiplication
For any mathbb{R} in mathbb{R} and any mathbb{Q} in mathbb{Q}, the product mathbb{Q} * mathbb{R} remains in mathbb{R}.
Associativity of Addition
For all mathbb{R} times; mathbb{R} times; mathbb{R}, mathbb{R} (mathbb{R} mathbb{R}) (mathbb{R} mathbb{R}) mathbb{R}.
Commutativity of Addition
For all mathbb{R} times; mathbb{R}, mathbb{R} mathbb{R} mathbb{R} mathbb{R}.
Existence of Additive Identity
There exists an element 0 in mathbb{R} such that for any mathbb{R} in mathbb{R}, mathbb{R} 0 mathbb{R}.
Existence of Additive Inverses
For each mathbb{R} in mathbb{R}, there exists an element -mathbb{R} in mathbb{R} such that mathbb{R} (-mathbb{R}) 0.
Distributive Properties
- Distributive over scalar multiplication: mathbb{Q}(. mathbb{R} mathbb{R}) mathbb{Q} * mathbb{R} mathbb{Q} * mathbb{R}. - Distributive over vector addition: mathbb{Q} * (mathbb{R} mathbb{R}) mathbb{Q} * mathbb{R} mathbb{Q} * mathbb{R}, for any mathbb{Q} in mathbb{Q} and mathbb{R} times; mathbb{R}. - Associativity: mathbb{Q} * (mathbb{Q} * mathbb{R}) (mathbb{Q} * mathbb{Q}) * mathbb{R}, for any mathbb{Q}, mathbb{Q} in mathbb{Q} and mathbb{R} in mathbb{R}. - Identity element of scalar multiplication: 1 * mathbb{R} mathbb{R} for any mathbb{R} in mathbb{R}.
Verification
Closure Under Addition
True, as the sum of any two real numbers is still a real number mathbb{R} mathbb{R} mathbb{R}.
Closure Under Scalar Multiplication
True, as multiplying a real number by a rational number yields a real number mathbb{Q} * mathbb{R} mathbb{R}.
Associativity of Addition
True for real numbers as mathbb{R} (mathbb{R} mathbb{R}) (mathbb{R} mathbb{R}) mathbb{R}.
Commutativity of Addition
True for real numbers as mathbb{R} mathbb{R} mathbb{R} mathbb{R}.
Existence of Additive Identity
The number 0 is in mathbb{R} and serves as the additive identity: mathbb{R} 0 mathbb{R}.
Existence of Additive Inverses
For every mathbb{R} in mathbb{R}, the element -mathbb{R} is also in mathbb{R} satisfying mathbb{R} (-mathbb{R}) 0.
Distributive Properties
Both properties hold for real numbers and rational scalars as mathbb{Q}(. mathbb{R} mathbb{R}) mathbb{Q} * mathbb{R} mathbb{Q} * mathbb{R} and mathbb{Q} * (mathbb{R} mathbb{R}) mathbb{Q} * mathbb{R} mathbb{Q} * mathbb{R}.
Associativity of Scalar Multiplication
Holds true as mathbb{Q} * (mathbb{Q} * mathbb{R}) (mathbb{Q} * mathbb{Q}) * mathbb{R}.
Identity Element of Scalar Multiplication
True as 1 * mathbb{R} mathbb{R} for any mathbb{R} in mathbb{R}.
Conclusion
Since mathbb{R} satisfies all the axioms of a vector space over mathbb{Q}, we can conclude that mathbb{R} is indeed a vector space over mathbb{Q}.