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Abstract Algebra/Vector Spaces
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Definition (Vector Space)
Let F be a field. A set V with two binary operations: + (addition) and (scalar multiplication), is called a Vector Space if it has the following properties:
forms an abelian group
for and
for and
The scalar multiplication is formally defined by , where .
Elements in F are called scalars, while elements in V are called vectors.
Some Properties of Vector Spaces
Proofs:
We want to show that , but
Suppose such that , then
Abstract Algebra