Examples of vector spaces

This section demonstrates that many different types of mathematical objects satisfy the vector space axioms. Each example below shows a specific set with defined addition and scalar multiplication operations that make it a vector space.

The diversity of examples illustrates the power of the abstract vector space concept: once a theorem is proved for general vector spaces, it automatically applies to coordinate spaces, function spaces, polynomial spaces, matrix spaces, and more.

Each node in this section verifies the vector space axioms for a particular mathematical structure.