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Examples of vector spaces

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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.

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