Linear endomorphisms

Endomorphisms of vector spaces

Endomorphisms

An endomorphism maps a vector space onto itself.

\( end (V)=\hom (V, V)\)

Endomorphisms form a vector space

An endomorphism maps a vector space onto itself.

\( end (V)=\hom (V, V)\)

Need to show that endomorphism is a vector space

Essentially

\(v\in V\)

\(f\in F\)

\(av = f\)

\(bv = g\)

\((a\oplus b)v=f+g\)

\((a\oplus b)v=av + bv\)

so there is some operation we can do on two members of endo

linear in addition. That is, if we have two dual “things”, we can define the addition of functions as the operation which results int he outputs being added.

what about linear in scalar? same approach.

Well we define

\(c\odot a)=cav\)

There is a unique endomorphism which results in two other endomorphisms being added together. define this as addition

Dimension of endomorphisms

\(\dim (end(V))=(\dim V )^2\)

Basis of endomorphisms

Projections

A projection is a linear map which if applied again returns the original result.

A projection can drop a dimension for example.

Kernels and images

The kernel of a linear operator is the set of vectors such that:

\(Mv=0\)

The kernel is also called the nullspace.

This can be shown as \(\ker (M)\)

The image of a linear operator is the set of vectors \(w\) such that:

\(Mv=w\).

This can be shown as \(\Im (M)\)

We also know that:

\(span (M)=\ker (M)+\Im (M)\)

Representing endomorphisms with matrices

Matrix representation

Representing linear maps as matrices

We previously discsussed morphisms on vector spaces. We can write these as matrices.

Matrices represents transformations of vector spaces

Representing vectors as matrices

We can represent vectors as row or column matrices.

\(v=\begin{bmatrix}a_{1} & a_{2}&...&a_{n}\end{bmatrix}\)

\(v=\begin{bmatrix}a_{1}\\a_{2}\\...\\a_{m}\end{bmatrix}\)

Automorphisms of vector spaces

Basis of an endomorphism

Changing the basis

For any two bases, there is a unique linear mapping from of the element vectors to the other.

The linear groups

General linear groups \(GL(n, F)\)

The general linear group, \(GL(n, F)\), contains all \(n\odot n\) invertible matrices \(M\) over field \(F\).

The binary operation is multiplication.

Endomorphisms as group actions

We can view each member of the group \(g\) as a homomorphim on \(s\).

Where \(s\) is a vector space \(V\), the representation on each group member is an invertible square matrix.

If the set we use is the vector space \(V\), then we can represent each group element with a square matrix acting on \(V\).

Faithful means \(a\ne b\) holds for repesentation too.

Representation theory. groups defined by \(ab=c\). if we can match each eleemnt to amatrix where this holds we have represented the matrix.

Representing finite groups

Finite groups can all be represented with square matrices.

Representing compact groups