Chapter 1: Complex Numbers

- Introduction to Complex Numbers
- - Imaginary numbers
- - The notion of complex numbers
- - Polar coordinates
- - Real and imaginary part
- Calculating with complex numbers
- - Calculating with polar coordinates
- - The quotient
- - Complex conjugate
- - Geometric interpretation
- Complex functions
- - Complex exponents
- - Rules of calculation for complex powers
- - Complex sine and cosine
- - Complex logarithm
- Complex polynomials
- - The notion of a complex polynomial
- - Factorization of complex polynomials
- - Zeros of complex polynomials
- - Fundamental theorem of algebra
- - Real polynomials

Chapter 2: Vector Calculus in Dimensions 2 and 3

- Introduction
- - The coordinate system
- Vectors in planes and space
- - The notion of vector
- - Scalar multiplication
- - Addition of vectors
- - Linear combinations of vectors
- Straight lines and planes
- - Straight lines and planes
- - Parametrization of a plane
- Bases, coordinates and equations
- - The notion of a base
- - Coordinate space
- - Straight lines in the plane in coordinates
- - Planes in coordinate space
- - Lines in the coordinate space
- Distances, angles and inner product
- - Distance, angles, and dot products
- - Dot product
- - Properties of the dot product
- - The standard dot product
- - Normal vectors
- The cross product
- - Cross product in 3 dimensions
- - The concept of volume in space
- - The volume of a parallelepiped
- - Properties of the cross product
- - The standard cross product

Chapter 3: Systems of linear equations and matrices

- Linear equations
- - The notion of linear equation
- - Reduction to a base form
- - Solving a linear equation with a single unknown
- - Solving a linear equation with several unknowns
- Systems of linear equations
- - The notion of a system of linear equations
- - Homogeneous and inhomogeneous systems
- - Lines in the plane
- - Planes in the space
- - Elementary operations on systems of linear equations
- - Several linear equations with several unknowns
- - The notion of a system of linear equations
- Systems and matrices
- - From systems to matrices
- - Equations and matrices
- - Echelon form and reduced echelon form
- - Row reduction of a matrix
- - Solving linear equations by Gaussian elimination
- - Solvability of systems of linear equations
- - Systems with a parameter
- Matrices
- - The notion of a matrix
- - Simple matrix operations
- - Multiplication of matrices
- - Matrix equations
- - The inverse of a matrix
- Applications
- - Applications of systems of linear equations and matrices

Chapter 4: Vector spaces

- Vector spaces and linear subspaces
- - The notion of vector spaces
- - The notion of linear subspace
- - Lines and planes
- - Affine subspaces
- Spans
- - Spanning sets
- - Operations with spanning vectors
- - Independence
- - Basis and dimension
- - Finding bases
- More about subspaces
- - Intersection and sum of linear subspaces
- - Direct sum of two linear subspaces
- Coordinates
- - The notion of coordinates
- - Coordinates of sums of scalar multiples
- - Basis and echelon form

Chapter 5: Linear maps

- Linear maps
- - The notion of linear maps
- - Linear maps determined by matrices
- - Composition of linear maps
- - Sum and multiples of linear maps
- - The inverse of a linear map
- - Kernel and image of a linear map
- - Recording linear map
- - Rank-nullity theorem of a linear map
- - Invertibility criteria for linear maps
- Matrices of linear maps
- - The matrix of a linear map in coordinate space
- - Determining the matrix in coordinate space
- - Coordinates
- - Basic transition
- - Matrix of a linear map
- - Coordinate transformations
- - Relationship to systems of linear equations
- Dual vector spaces
- - The notion of dual space
- - Dual basis
- - Dual map

Chapter 6: Matrix calculus

- Rank and inverse of a matrix
- - Rank and column space of a matrix
- - Invertibility and rank
- Determinants
- - 2-dimensional determinants
- - Permutations
- - Higher-dimensional determinants
- - More properties of determinants
- - Row and column expansion
- - Row and column reduction
- - Cramer's rule
- Matrices and coordinate transformations
- - Characteristic polynomial of a matrix
- - Conjugate matrices
- - Characteristic polynomial of a linear map
- Minimal polynomial
- - Cayley Hamilton
- - Minimal polynomial
- - Division with remainder for polynomials