Topology/Morphisms

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 maps

Definition of Linear Map

A linear map is a function $f:V\to W$ where $V,W$ are vector spaces over a field F. Such that for all $v,w\in V$

1. $f(v+w)=f(v)+f(w)$

2. $f(\alpha v)=\alpha f(v)$

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 $dim(Im(f))$. 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 $dim(Ker(f))$.

The Rank-Nullity Theorem

For any linear map $f:V\to W$

$dim(Ker(f))+dim(Im(f))=dim(V)$