Topology/Morphisms

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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.

Linear mapsEdit

Definition of Linear Map

A linear map is a function   where   are vector spaces over a field F. Such that for all  

1.  

2.  

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 ImagesEdit

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  

 


Topology
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