# Topology/Morphisms

 Topology ← Vector Spaces Morphisms Convexity →

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 ${\displaystyle f:V\to W}$ where ${\displaystyle V,W}$ are vector spaces over a field F. Such that for all ${\displaystyle v,w\in V}$

1. ${\displaystyle f(v+w)=f(v)+f(w)}$

2. ${\displaystyle f(\alpha v)=\alpha f(v)}$

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 ${\displaystyle dim(Im(f))}$. 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 ${\displaystyle dim(Ker(f))}$.

The Rank-Nullity Theorem

For any linear map ${\displaystyle f:V\to W}$

${\displaystyle dim(Ker(f))+dim(Im(f))=dim(V)}$

 Topology ← Vector Spaces Morphisms Convexity →