Endomorphisms of vector spaces

Endomorphisms

Endomorphisms form a vector space

Projections

Kernels and images

Representing endomorphisms with matrices

Matrix and vector representation

Commutation

Identity matrix and the Kronecker delta

Matrix addition and multiplication

Basis of an endomorphism

Changing the basis

Transposition and conjugation

Matrix rank

Types of matrices

Automorphisms of vector spaces

Inverse matrices

Degenerate (singular) matrices

Elementary row operations

Gaussian elimination

Eigenvalues and eigenvectors

Eigenvalues and eigenvectors

Spectrum

Eigenvectors as a basis

Calculating eigenvalues and eigenvectors using the characteristic polynomial

Traces

Matrix operations

Matrix powers

Matrix exponentials

Matrix logarithms

Matrix square roots

Matrix decomposition

Similar matrices

Defective and diagonalisable matrices

Diagonalisable matrices and eigen-decomposition

Spectral theorem for finite-dimensional vector spaces

The linear groups

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

Endomorphisms as group actions

Representing finite groups

Representing compact groups