An annihilator of a subspace of a vector space is a linear functional that annihilates every vector in the subspace. In other words, it is a linear function...
An annihilator of a subspace of a vector space is a linear functional that annihilates every vector in the subspace. In other words, it is a linear functional that evaluates to zero for all vectors in the subspace.
The annihilator of a subspace is the dual space of that subspace. This means that the annihilator of a subspace is a linear functional that takes a vector in the vector space and returns a vector in the dual space.
The annihilator of a subspace is also called the annihilator of the subspace. It is an important tool in linear algebra, as it can be used to characterize subspaces and determine if a subspace is closed.
Examples:
The annihilator of the subspace of R³ consisting of vectors with y = 0 is the linear functional that maps a vector to its y-coordinate.
The annihilator of the subspace of C² consisting of vectors with y = 0 and z = 0 is the linear functional that maps a vector to its z-coordinate.
The annihilator of the subspace of R³ consisting of vectors with x = y = z is the linear functional that maps a vector to the vector (0, 0, 0)