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