In general, a morphism refers to a function mapping from one space to another that preserves structure. In terms of vector spaces the natural morphism is a linear map.
- Definition of Linear Map
A linear map is a function where are vector spaces over a field F. Such that for all
The image of a linear map is a subspace of the domain. The kernel of a linear map is a subspace of the codomain.
The Importance of Kernels and Images
- Definition of Rank
The rank of a linear map is the dimension of the image of the map . It also can be found using row reduction on the corresponding matrix.
- Definition of Nullity
The nullity of a linear map, or matrix, is the dimension of the kernel of the map .
- The Rank-Nullity Theorem
For any linear map