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.
Parents
Children
- The set containing only 0 is a vector space.
- R^n is a vector space.
- C^n is a vector space.
- F^n is a vector space.
- The set of all functions on a set is a vector space.
- The solutions to a homogeneous system of linear equations is a vector space.
- The solutions to a homogeneous linear differential equation is a vector space.
- The set of all polynomials is a vector space.
- The set of all polynomials of degree at most n is a vector space.
- The set of m by n matrices is a vector space.
- The set of all sequences is a vector space.
- The crazy vector space is a vector space.
- The row space of a matrix is a vector space
- The column space of a matrix is a vector space
- The null space of a matrix is a subspace of R^n (or C^n).
- The left null space of a matrix is a subspace of R^m (or C^m).
- The set of linear transformations between two vector spaces is a vector space.