# Topology/Morphisms

< Topology

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