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leon8179
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I know that the set of real numbers over the field of rational numbers is an infinite dimensional vector space. BUT I don't quite understand why the basis of that vector space is not countable. Can someone help me?
A real number vector space is a mathematical concept that describes a set of objects, called vectors, that can be added together and multiplied by real numbers. This set must also follow certain rules, such as closure under addition and scalar multiplication, to be considered a vector space.
A basis is a set of vectors that can be used to represent any vector in a vector space through linear combinations. If a basis is countable, it means that the set has a finite or infinite number of elements that can be counted in a one-to-one correspondence with the natural numbers.
If the basis is finite, then it is automatically countable. To determine if an infinite basis is countable, we can use the concept of cardinality. If the basis has the same cardinality as the set of natural numbers, then it is countable.
A countable basis in a real number vector space means that the vector space has a finite or countably infinite number of dimensions. This is important because it allows us to understand and analyze the properties and behavior of the vector space more easily.
Yes, a basis for a real number vector space can be uncountable. This means that the vector space has an uncountably infinite number of dimensions. However, such vector spaces are more complex and difficult to analyze compared to those with a countable basis.