VECTORS IN THREE-DIMENSIONAL SPACE

* Vectors, coordinates and componentwise operations

* Scalar product, length, distance and orthogonality, the Cauchy--Schwartz inequality and angles

* Vector product

PLANES AND LINES

* Planes, lines and their equations

* Mutual positions

THE N-DIMENSIONAL SPACE

* Operations with vectors, scalar product and orthogonality, length and distance, angle between two vectors

MATRICES

* Definition and operations with matrices (sum and product)

* Invertible matrices

* Transpose of a matrix

* Determinant and rank of a matrix

LINEAR SYSTEMS AND MATRICES

* Elementary operations

* Solutions of a system

* Gauss algorithm

* Cramer rule

* Rank and solutions of a linear system

COMPLEX NUMBERS

* Cartesian and exponential form of complex numbers

* Operations with complex numbers (sum, product, power, conjugate)

* Norm of a complex number

VECTOR SPACES

* Definition of vector space and subspaces

* Linear combinations

* Linear dependence and independence

* Bases and coordinates

* Sum, direct sum and Grassmann formula

LINEAR MAPS

* Definition

* Kernel and image of a linear map

* Isomorphisms

* Matrices and linear maps

DIAGONALIZATION OF MAPS AND MATRICES

* Eigenvalues and eigenvectors

* The characteristic polynomial

* Diagonalizable matrices

* Algebraic and geometric multiplicities

* Criteria for diagonalizability