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Linear Algebra for college & university students. Contains vector calculus / spaces, matrices and matrix calculus, inner product spaces, and more.

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- 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

- Vectors in planes and space
- The notion of vector
- Scalar multipliction
- 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 base
- Coordinate space
- Straight lines in the plane coordinates
- Planes in coordinate space
- Lines in the coordinate space

- Distances, angles and dot product
- Distances, 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 cross product
- The standard cross product

- Linear equations
- The notion of linear equation
- Reduction to a base form
- Solving a linear equation with a single unkown
- Solving a linaer 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 space
- Elementary operations on systems 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 elemination
- 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

- Vector spaces and linear subspaces
- The notion of vector space
- 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 mulitples
- Basis and echelon form

- Inner product, length, and angle
- The notion of inner product
- Angle
- Perpendicularity

- Orthogonal systems
- The notion of orthonormal system
- Properties of orthonormal systems
- Constucting orthonormal bases

- Orthogonal projections
- Orthogonal projection
- Orthogonal complement
- Gram-Schmidt in matrix form

- Complex inner product spaces
- Inner product on complex vector spaces
- Orthonormal systems in complex vector spaces
- Orthogonal complements in complex inner product spaces
- Complex orthogonal complements
- Gram-Schmidt in complex inner product spaces

- Linear maps
- The notion of linear map
- Linear maps determined by matrices
- Composition of linear maps
- Sums and multiples of linear maps
- The inverse of a linear map
- Kernel and image of a linear transformation
- Recording linear maps
- Rank-nullity theorem for linear maps
- 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
- Basis transition
- The 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

- 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 transforms
- Characteristic polynomial of a matrix
- Conjugate matrices
- Characteristic polynomial of a linear map
- Matrix equivalence

- Minimal polynomial
- Cayley-Hamilton
- Division with remainder for polynomials
- Minimal polynomial

- Eigenvalues and eigenvectors
- Diagonal form
- Eigenspace
- Determining eigenvalues and eigenvectors

- Diagonalizability
- The notion of diagonalizability
- Diagonalizability and minimal polynomial
- The greatest common divisor of two polynomials
- The Euclidean algorithm

- Invariant subspaces
- The notion of an invariant subspace
- The extended Euclidean algorithm
- Direct sum decomposition into invariant subspaces
- Generalized eigenspace
- Jordan normal form
- From real to complex vector spaces and back
- Real Jordan normal form for non-real eigenvalues

- Orthogonal maps
- The notion of orthogonal map
- Properties of orthogonal maps
- More properties of orthogonal maps
- Orthogonal matrices
- Orthogonal transformation matrices

- Classification of orthogonal maps
- Low-dimensional orthogonal maps
- Jordan normal form for orthogonal maps
- Classification of orthogonal maps

- Unitary maps
- The notion of unitary map
- Diagonal form for unitary maps

- Isometries
- The notion of isometry
- Equivalence of isometries
- Characterisation of isometries

- Symmetric maps
- The notion of symmetric map
- Connection with symmetric matrices
- Properties of symmetric maps
- Orthonormal bases and symmetric maps

- Application of symmetric maps
- Quadratic forms
- Quadrics
- Least square solutions of linear equations
- Singular value decomposition

- Differential equations and Laplace transform
- The Laplace transform
- The inverse Laplace transform
- Laplace transforms of differential equations
- Convolution
- Laplace transforms of Heaviside functions
- Laplace transforms of periodic functions
- Riemann-Stieltjes intergration
- Laplace transforms for delta function
- Transfer and response functions