Course content

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
- 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
- The notion of a matrix
- Simple matrix operations
- Multiplication of matrices
- Matrix equations
- The inverse of a matrix
- 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
- 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
- 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
- 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

Linear Algebra

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