Annihilator (mathematics)
There are two formulations of mathematics , which are denoted with the word annullator (or annihilator ).
Annulator in the context of forms
The cancellation space is a generalization of the orthogonal complement to vector spaces, in which the dual space cannot be identified with the space itself via a scalar product .
definition
Let be a vector space , the corresponding dual space and a subset of . Then is called
the canceler of .
Features of the canceler
- is a subspace of the dual space . That is why one speaks of the cancellation room .
- , where is the subspace generated by .
- Is so is .
- If is finite-dimensional and a subspace of , then we have . In this case and the dual space are canonically isomorphic and it applies , where and have been identified with one another.
Cancellation of a module
There was a ring and a - module . Then the cancellation is from
The annulator can also be described as the core of the structure mapping
- , where the left multiplication is with .
The canceler is an ideal in .
literature
- Gerd Fischer : Linear Algebra. 14th, revised edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03217-0 .