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i know that the set "all real numbers" make up a vector space, but how do you prove that it is so?
A vector space is a mathematical structure that consists of a set of objects called vectors, which can be added together and multiplied by scalars (such as real numbers). It follows a set of axioms and properties that allow for operations such as addition, scalar multiplication, and the existence of a zero vector and additive inverse.
To prove that the set of real numbers is a vector space, we need to show that it satisfies all the axioms and properties of a vector space. This includes closure under addition and scalar multiplication, associativity and commutativity of addition, existence of a zero vector and additive inverse, and distributivity of scalar multiplication over addition.
A subspace is a subset of a vector space that also follows the axioms and properties of a vector space. In other words, it is a smaller space that is still a vector space. A vector space can contain multiple subspaces, but a subspace cannot contain any subspaces.
Yes, we can use linear independence to prove that the set of real numbers is a vector space. If we can show that any set of vectors in the real numbers is linearly independent, then it follows that the set satisfies the axioms and properties of a vector space.
The dimension of a vector space is the minimum number of vectors needed to span the entire space. Since the set of real numbers is an infinite set, it has an infinite dimension. This means that any set of linearly independent vectors in the real numbers can span the entire space, making it a vector space.