Systems of linear equations:

- matrix form, elementary row operations, row echelon form

- Gaussian elimination

- Gauss-Jordan elimination

Matrices:

- matrix operations, basic types of matrices

- nonsingular matrix, inverse of a matrix

Algebraic structures:

- groups, subgroups, permutations

- fields and finite fields in particular

Vector spaces:

- linear span, linear combination, linear dependence and independence

- basis and its existence, coordinates

- Steinitz' replacement theorem

- dimension, dimensions of sum and intersection of subspaces

- fundamental matrix subspaces (row space, column space, kernel)

- rank-nullity theorem

Linear maps:

- examples, image, kernel

- injective linear maps

- matrix representations, transition matrix, composition of linear maps

- isomorphism of vector spaces

Topics on expansion:

- introduction to affine spaces and relation to linear equations

- LU decomposition

Basics of linear algebra (vector spaces and linear maps, solutions of linear equations, matrices).