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Examples of vector spaces

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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 all sequences is a vector space.
The crazy vector space is a vector space.
The set containing only 0 is a vector space.
F^n is a vector space.
R^n is a vector space.
C^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 set of m by n matrices 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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The column space of a matrix is a vector space

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The row space of a matrix is a vector spaceClose match
Definition of column space of a matrixClose match
The column space of an m-by-n matrix is a subspace of R^m (or C^m)Close match
Definition of row space of a matrixClose match
Subspaces associated to a matrixClose match
The null space of a matrix is a subspace of R^n (or C^n).Close match
Definition of column rank of a matrixClose match
The row space and the column space of a matrix have the same dimension.Close match
The left null space of a matrix is a subspace of R^m (or C^m).Close match
Row operations do not necessarily preserve the column space.Close match
The set of m by n matrices is a vector space.Close match
Definition of column vectorClose match
The rank of a matrix equals the rank of the linear transformation it represents.Close match
Equivalence theorem for nonsingular matrices: the dimension of the column space of A is n.Close match
Equivalence theorem for nonsingular matrices: the columns of A span R^n (or C^n).Close match
The pivot columns of a matrix are a basis for the column space.Close match
Definition of rank of a matrixClose match
SpansClose match
Vector spacesClose match
The solutions to a homogeneous system of linear equations is a vector space.Close match

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