Vector Spaces: Real Numbers Over Rational Numbers

In summary, the set of real numbers (R) over the set of rational numbers (Q) forms an infinite dimensional vector space with vector and field addition and multiplication having their usual meanings. However, due to the uncountable nature of R and the countable nature of Q, it is impossible to list a basis for this vector space. Theoretically, a function could be used to identify basis numbers, but this has not yet been achieved.
  • #1
arunkp
1
0
Please tell me one of the bases for the infinite dimenional vector space - R (the set of all real numbers) over Q (the set of all rational numbers). The vector addition, field addition and multiplication carry the usual meaning.
 
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  • #2
Why do you think there is one you can describe constructively?
 
  • #3
It's not JUST "infinite dimensional". Since the set of real numbers is uncountable, while the set of rational numbers is countable, any basis for the real numbers, as a vector space over the rational numbers, would have to be uncountable- so it is impossible to list them.

Theoretically, you could set of a function, say over [0, 1], such that f(x) for each x gives a "basis" number. If you figure out how to do that, please let me know!
 

FAQ: Vector Spaces: Real Numbers Over Rational Numbers

What is a vector space?

A vector space is a mathematical structure that consists of a set of elements, called vectors, and two operations, vector addition and scalar multiplication. The vectors in a vector space can be added together and multiplied by real numbers to produce new vectors in the same space.

What are real numbers and rational numbers?

Real numbers are numbers that can be represented on a number line, including both positive and negative numbers, fractions, and decimals. Rational numbers are a subset of real numbers that can be expressed as a ratio of two integers, such as 1/2 or 3/4.

How are real numbers and rational numbers related in a vector space?

In a vector space, the real numbers are used as the scalars for scalar multiplication. This means that a rational number can be multiplied by a vector to produce a new vector in the same space.

What are some examples of vector spaces over rational numbers?

Some examples of vector spaces over rational numbers include the space of all rational numbers, the space of polynomials with rational coefficients, and the space of rational functions.

What are the applications of vector spaces over rational numbers?

Vector spaces over rational numbers have many applications in mathematics, physics, engineering, and computer science. They are used to model and solve problems involving linear transformations, systems of linear equations, and optimization. They are also used in coding theory, cryptography, and data compression.

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