1

Group Theory and Linear Algebra

Linear Algebra

Division of Integers

2

Modular Arithmetic

3

Algebraic Fields

4

Vector Space

5

Linear Independence

6

Linear Transformations

7

Matrix Representations

8

Minimal Polynomials

9

10

Characteristic Polynomials

Jordan Normal Form

11

12

Inner Product Spaces

13

Orthonormal Bases

14

15

Orthogonal Complement

Adjoints

16

17

Spectral Theorem

Matrix Square Roots

18

Group Theory

Group Definitions

Examples of Groups

19

Subgroups

20

21

Isomorphisms

22

Direct Product

Orders of Subgroups

23

Cosets

24

Quotient Groups

25

Homomorphisms

26

Group Action

27

Orbit-Stabalizer Relation

28

Conjugation

29

Sylow Theorems

Isometries

30

Euclidean Isometries

Random Stuff

Not every matrix has a square root, e.g:

Group of roots of unity is cyclic with generator

Every cyclic group is abelian

Vectors in Gram-Schmidt don't need to be orthogonal, just linearly independent

A square matrix is invertible if its determinant is non-zero

has only the trivial solution