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Complex numbers
Introduction to Complex numbers
THEORY
T
1.
Imaginary numbers
PRACTICE
P
2.
Imaginary numbers
7
THEORY
T
3.
The notion of complex numbers
PRACTICE
P
4.
The notion of complex numbers
12
THEORY
T
5.
Polar coordinates
PRACTICE
P
6.
Polar coordinates
7
THEORY
T
7.
Real and imaginary part
PRACTICE
P
8.
Real and imaginary part
5
Calculating with complex numbers
THEORY
T
1.
Calculating with polar coordinates
PRACTICE
P
2.
Calculating with polar coordinates
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THEORY
T
3.
The quotient
PRACTICE
P
4.
The quotient
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THEORY
T
5.
Complex conjugate
PRACTICE
P
6.
Complex conjugate
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THEORY
T
7.
Geometric interpretation
PRACTICE
P
8.
Geometric interpretation
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Complex functions
THEORY
T
1.
Complex exponents
PRACTICE
P
2.
Complex exponents
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THEORY
T
3.
Rules of calculation for complex powers
PRACTICE
P
4.
Rules of calculation for complex powers
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THEORY
T
5.
Complex sine and cosine
PRACTICE
P
6.
Complex sine and cosine
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THEORY
T
7.
Complex logarithm
PRACTICE
P
8.
Complex logarithm
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Complex polynomials
THEORY
T
1.
The notion of a complex polynomial
PRACTICE
P
2.
The notion of a complex polynomial
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THEORY
T
3.
Factorization of complex polynomials
PRACTICE
P
4.
Factorization of complex polynomials
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THEORY
T
5.
Zeros of complex polynomials
PRACTICE
P
6.
Zeros of complex polynomials
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THEORY
T
7.
Fundamental theorem of algebra
PRACTICE
P
8.
Fundamental theorem of algebra
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THEORY
T
9.
Real polynomials
PRACTICE
P
10.
Real polynomials
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End of complex numbers
THEORY
T
1.
Conclusion of Complex Numbers
Vector calculus in plane and space
Vectors in Planes and Space
THEORY
T
1.
The notion of vector
PRACTICE
P
2.
The notion of vector
9
THEORY
T
3.
Scalar multiplication
PRACTICE
P
4.
Scalar multiplication
6
THEORY
T
5.
Addition of vectors
PRACTICE
P
6.
Addition of vectors
6
THEORY
T
7.
Linear combinations of vectors
PRACTICE
P
8.
Linear combinations
8
Straight Lines and Planes
THEORY
T
1.
Straight lines and planes
PRACTICE
P
2.
Straight lines and planes
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THEORY
T
3.
Parametrization of a plane
PRACTICE
P
4.
Parametrization of a plane
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Bases, Coordinates and Equations
THEORY
T
1.
The notion of basis
PRACTICE
P
2.
The notion of base
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THEORY
T
3.
Coordinate space
PRACTICE
P
4.
Coordinate space
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THEORY
T
5.
Straight lines in the plane in coordinates
PRACTICE
P
6.
Straight lines in the plane in coordinates
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THEORY
T
7.
Planes in coordinate space
PRACTICE
P
8.
Planes in coordinate space
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THEORY
T
9.
Lines in the coordinate space
PRACTICE
P
10.
Lines in the coordinate space
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Distances, Angles and Inner Product
THEORY
T
1.
Distances, angles, and dot products
PRACTICE
P
2.
Distances, angles, and dot products
9
THEORY
T
3.
Dot product
PRACTICE
P
4.
Dot product
5
THEORY
T
5.
Properties of the dot product
PRACTICE
P
6.
Properties of the dot product
5
THEORY
T
7.
The standard dot product
PRACTICE
P
8.
The standard dot product
6
THEORY
T
9.
Normal vectors
PRACTICE
P
10.
Normal vectors
9
The Cross Product
THEORY
T
1.
Cross product in 3 dimensions
PRACTICE
P
2.
Cross product in 3 dimensions
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THEORY
T
3.
The concept of volume in space
PRACTICE
P
4.
The concept of volume in space
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THEORY
T
5.
The volume of a parallelepiped
PRACTICE
P
6.
The volume of a parallelepiped
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THEORY
T
7.
Properties of the cross product
PRACTICE
P
8.
Properties of the cross product
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THEORY
T
9.
The standard cross product
PRACTICE
P
10.
The standard cross product
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Conclusion
THEORY
T
1.
Conclusion of Vector calculus in plane and space
Systems of linear equations and matrices
Linear equations
THEORY
T
1.
The notion of linear equation
PRACTICE
P
2.
The notion of linear equation
5
THEORY
T
3.
Reduction to a base form
PRACTICE
P
4.
Reduction to base form
2
THEORY
T
5.
Solving a linear equation with a single unknown
PRACTICE
P
6.
Solving a linear equation with a single unknown
5
THEORY
T
7.
Solving a linear equation with several unknowns
PRACTICE
P
8.
Solving a linear equation with several unkowns
5
Systems of Linear Equations
THEORY
T
1.
The notion of a system of linear equations
PRACTICE
P
2.
The notion of a system of linear equations
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THEORY
T
3.
Homogeneous and inhomogeneous systems
PRACTICE
P
4.
Homogeneous and inhomogeneous systems
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THEORY
T
5.
Lines in the plane
PRACTICE
P
6.
Lines in the plane
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THEORY
T
7.
Planes in space
PRACTICE
P
8.
Planes in space
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THEORY
T
9.
Elementary operations on systems of linear equations
PRACTICE
P
10.
Several linear equations with several unknowns
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Systems and matrices
THEORY
T
1.
From systems to matrices
PRACTICE
P
2.
From systems to matrices
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THEORY
T
3.
Equations and matrices
PRACTICE
P
4.
Equations and matrices
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THEORY
T
5.
Echelon form and reduced echelon form
PRACTICE
P
6.
Echelon form and reduced echelon form
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THEORY
T
7.
Row reduction of a matrix
PRACTICE
P
8.
Row reduction of a matrix
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THEORY
T
9.
Solving linear equations by Gaussian elimination
PRACTICE
P
10.
Solving linear equations by Gaussian elimination
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THEORY
T
11.
Solvability of systems of linear equations
PRACTICE
P
12.
Solvability of systems of linear equations
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THEORY
T
13.
Systems with a parameter
PRACTICE
P
14.
Systems with a parameter
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Matrices
THEORY
T
1.
The notion of matrix
PRACTICE
P
2.
The notion of matrix
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THEORY
T
3.
Simple matrix operations
PRACTICE
P
4.
Simple matrix operations
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THEORY
T
5.
Multiplication of matrices
PRACTICE
P
6.
Multiplication of matrices
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THEORY
T
7.
Matrix equations
PRACTICE
P
8.
Matrix equations
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THEORY
T
9.
The inverse of a matrix
PRACTICE
P
10.
The inverse of a matrix
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End of Systems of linear equations and matrices
THEORY
T
1.
Applications of systems of linear equations
THEORY
T
2.
Notes
Vector spaces
Vector spaces and linear subspaces
THEORY
T
1.
The notion of vector space
PRACTICE
P
2.
The notion of vector space
6
THEORY
T
3.
The notion of linear subspace
PRACTICE
P
4.
The notion of linear subspace
8
THEORY
T
5.
Lines and planes
PRACTICE
P
6.
Lines and planes
7
THEORY
T
7.
Affine subspaces
PRACTICE
P
8.
Affine subspaces
10
Spans
THEORY
T
1.
Spanning sets
PRACTICE
P
2.
Spanning sets
8
THEORY
T
3.
Operations with spanning vectors
PRACTICE
P
4.
Operations with spanning vectors
6
THEORY
T
5.
Independence
PRACTICE
P
6.
Independence
10
THEORY
T
7.
Basis and dimension
PRACTICE
P
8.
Basis and dimension
7
THEORY
T
9.
Finding bases
PRACTICE
P
10.
Finding bases
7
More about subspaces
THEORY
T
1.
Intersection and sum of linear subspaces
PRACTICE
P
2.
Intersection and sum of linear subspaces
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THEORY
T
3.
Direct sum of two linear subspaces
PRACTICE
P
4.
Direct sum of two linear subspaces
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Coordinates
THEORY
T
1.
The notion of coordinates
PRACTICE
P
2.
The notion of coordinates
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THEORY
T
3.
Coordinates of sums of scalar multiples
PRACTICE
P
4.
Coordinates of sum and scalar multiples
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THEORY
T
5.
Basis and echelon form
PRACTICE
P
6.
Basis and echelon form
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End of Vector spaces
THEORY
T
1.
Conclusion of Vector spaces
Inner Product Spaces
Inner product, length, and angle
THEORY
T
1.
The notion of inner product
PRACTICE
P
2.
The notion of inner product
9
THEORY
T
3.
Angle
PRACTICE
P
4.
Angle
6
THEORY
T
5.
Perpendicularity
PRACTICE
P
6.
Perpendicularity
5
Orthogonal systems
THEORY
T
1.
The notion of orthonormal system
PRACTICE
P
2.
The notion of orthonormal system
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THEORY
T
3.
Properties of orthonormal systems
PRACTICE
P
4.
Properties of orthonormal systems
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THEORY
T
5.
Constructing orthonormal bases
PRACTICE
P
6.
Constructing orthonormal bases
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Orthogonal projections
THEORY
T
1.
Orthogonal projection
PRACTICE
P
2.
Orthogonal projection
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THEORY
T
3.
Orthogonal complement
PRACTICE
P
4.
Orthogonal complement
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THEORY
T
5.
Gram-Schmidt in matrix form
PRACTICE
P
6.
Gram-Schmidt in matrix form
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Complex inner product spaces
THEORY
T
1.
Inner product on complex vector spaces
PRACTICE
P
2.
Inner product on complex vector spaces
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THEORY
T
3.
Orthonormal systems in complex vector spaces
PRACTICE
P
4.
Orthonormal systems in complex vector spaces
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THEORY
T
5.
Orthogonal complements in complex inner product spaces
PRACTICE
P
6.
Complex orthogonal complements
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THEORY
T
7.
Gram-Schmidt in complex inner product spaces
PRACTICE
P
8.
Gram-Schmidt in complex inner product spaces
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End of Inner product spaces
THEORY
T
1.
Bessel's Inequality
THEORY
T
2.
End of Inner product spaces
Linear maps
Linear maps
THEORY
T
1.
The notion of linear map
PRACTICE
P
2.
The notion of linear map
10
THEORY
T
3.
Linear maps determined by matrices
PRACTICE
P
4.
Linear maps determined by matrices
4
THEORY
T
5.
Composition of linear maps
PRACTICE
P
6.
Composition of linear maps
7
THEORY
T
7.
Sums and multiples of linear maps
PRACTICE
P
8.
Sums and multiples of linear maps
5
THEORY
T
9.
The inverse of a linear map
PRACTICE
P
10.
The inverse of a linear map
10
THEORY
T
11.
Kernel and image of a linear transformation
PRACTICE
P
12.
Kernel and image of a linear transformation
7
THEORY
T
13.
Recording linear maps
PRACTICE
P
14.
Recording linear maps
10
THEORY
T
15.
Rank–nullity theorem for linear maps
PRACTICE
P
16.
Rank-nullity theorem for linear maps
10
THEORY
T
17.
Invertibility criteria for linear maps
PRACTICE
P
18.
Invertibility criteria for linear maps
4
Matrices of Linear Maps
THEORY
T
1.
The matrix of a linear map in coordinate space
PRACTICE
P
2.
The matrix of a linear map in coordinate space
11
THEORY
T
3.
Determining the matrix in coordinate space
PRACTICE
P
4.
Determining the matrix of a linear map
4
THEORY
T
5.
Coordinates
PRACTICE
P
6.
Coordinates
5
THEORY
T
7.
Basis transition
PRACTICE
P
8.
Basis transition
3
THEORY
T
9.
The matrix of a linear map
PRACTICE
P
10.
The matrix of a linear map
6
THEORY
T
11.
Coordinate transformations
PRACTICE
P
12.
Coordinate transformations
3
THEORY
T
13.
Relationship to systems of linear equations
PRACTICE
P
14.
Relationship to systems of linear equations
15
Dual vector spaces
THEORY
T
1.
The notion of dual space
PRACTICE
P
2.
The notion of dual space
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THEORY
T
3.
Dual basis
PRACTICE
P
4.
Dual basis
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THEORY
T
5.
Dual map
PRACTICE
P
6.
Dual map
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End of Linear maps
THEORY
T
1.
Overview of the correspondence between matrix and linear mapping
THEORY
T
2.
Conclusion of Linear maps
Matrix calculus
Rank and inverse of a matrix
THEORY
T
1.
Rank and column space of a matrix
PRACTICE
P
2.
Rank and column space of a matrix
14
THEORY
T
3.
Invertibility and rank
PRACTICE
P
4.
Invertibility and rank
2
Determinants
THEORY
T
1.
2-dimensional determinants
PRACTICE
P
2.
2-dimensional determinants
4
THEORY
T
3.
Permutations
PRACTICE
P
4.
Permutations
11
THEORY
T
5.
Higher-dimensional determinants
PRACTICE
P
6.
Higher-dimensional determinants
5
THEORY
T
7.
More properties of determinants
PRACTICE
P
8.
More properties of determinants
6
THEORY
T
9.
Row and column expansion
PRACTICE
P
10.
Row and column expansion
6
THEORY
T
11.
Row and column reduction
PRACTICE
P
12.
Row and column reduction
5
THEORY
T
13.
Cramer's rule
PRACTICE
P
14.
Cramer's rule
4
Matrices and coordinate transformations
THEORY
T
1.
Characteristic polynomial of a matrix
PRACTICE
P
2.
Characteristic polynomial of a matrix
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THEORY
T
3.
Conjugate matrices
PRACTICE
P
4.
Conjugate matrices
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THEORY
T
5.
Characteristic polynomial of a linear map
PRACTICE
P
6.
Characteristic polynomial of a linear map
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THEORY
T
7.
Matrix equivalence
PRACTICE
P
8.
Matrix equivalence
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Minimal polynomial
THEORY
T
1.
Cayley-Hamilton
PRACTICE
P
2.
Cayley Hamilton
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THEORY
T
3.
Division with remainder for polynomials
PRACTICE
P
4.
Division with remainder for polynomials
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THEORY
T
5.
Minimal polynomial
PRACTICE
P
6.
Minimal polynomial
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End of Matrix Calculus
THEORY
T
1.
Conclusion of Matrix Calculus
Invariant subspaces of linear maps
Eigenvalues and eigenvectors
THEORY
T
1.
Diagonal form
PRACTICE
P
2.
Diagonal form
8
THEORY
T
3.
Eigenspace
PRACTICE
P
4.
Eigenspace
12
THEORY
T
5.
Determining eigenvalues and eigenvectors
PRACTICE
P
6.
Determining eigenvalues and eigenvectors
17
Diagonalizability
THEORY
T
1.
The notion of diagonalizability
PRACTICE
P
2.
The notion of diagonalizability
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THEORY
T
3.
Diagonalizability and minimal polynomial
PRACTICE
P
4.
Diagonalizability and minimal polynomial
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THEORY
T
5.
The greatest common divisor of two polynomials
PRACTICE
P
6.
The greatest common divisor of two polynomials
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THEORY
T
7.
The Euclidean algorithm
PRACTICE
P
8.
The Euclidean algorithm
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Invariant subspaces
THEORY
T
1.
The notion of invariant subspace
PRACTICE
P
2.
The notion of invariant subspace
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THEORY
T
3.
The extended Euclidean algorithm
PRACTICE
P
4.
The extended Euclidean algorithm
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THEORY
T
5.
Direct sum decomposition into invariant subspaces
PRACTICE
P
6.
Direct sum decomposition into invariant subspaces
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THEORY
T
7.
Generalized eigenspace
PRACTICE
P
8.
Generalized eigenspace
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THEORY
T
9.
Jordan normal form
PRACTICE
P
10.
Jordan normal form
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THEORY
T
11.
From real to complex vector spaces and back
PRACTICE
P
12.
From real to complex vector spaces and back
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THEORY
T
13.
Real Jordan normal form for non-real eigenvalues
PRACTICE
P
14.
Real Jordan normal form for non-real eigenvalues
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Orthogonal and symmetric maps
Orthogonal maps
THEORY
T
1.
The notion of orthogonal map
PRACTICE
P
2.
The notion of orthogonal map
5
THEORY
T
3.
Some properties of orthogonal maps
PRACTICE
P
4.
Some properties of orthogonal maps
6
THEORY
T
5.
Orthogonal maps and orthonormal bases
PRACTICE
P
6.
Orthogonal maps and orthogonal bases
6
THEORY
T
7.
Orthogonal matrices
PRACTICE
P
8.
Orthogonal matrices
5
THEORY
T
9.
Orthogonal transformation matrices
PRACTICE
P
10.
Orthogonal transformation matrices
5
Classification of orthogonal maps
THEORY
T
1.
Lowdimensional orthogonal maps
PRACTICE
P
2.
Lowdimensional orthogonal maps
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THEORY
T
3.
Three-dimensional orthogonal maps
PRACTICE
P
4.
Three-dimensional orthogonal maps
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THEORY
T
5.
Jordan normal form for orthogonal maps
PRACTICE
P
6.
Jordan normal form for orthogonal maps
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THEORY
T
7.
Classification of orthogonal maps
PRACTICE
P
8.
Classification of orthogonal maps
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Isometries
THEORY
T
1.
The notion of isometry
PRACTICE
P
2.
The notion of isometry
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THEORY
T
3.
Isometries and orthonormal systems
PRACTICE
P
4.
Isometries and orthonormal systems
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THEORY
T
5.
Equivalence of isometries
PRACTICE
P
6.
Equivalence of isometries
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THEORY
T
7.
Characterisation of isometries
PRACTICE
P
8.
Characterisation of isometries
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Unitary maps
THEORY
T
1.
The notion of unitary map
PRACTICE
P
2.
The notion of unitary map
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THEORY
T
3.
Diagonal form for unitary maps
PRACTICE
P
4.
Diagonal form for unitary maps
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Symmetric maps
THEORY
T
1.
The notion of symmetric map
PRACTICE
P
2.
The notion of symmetric map
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THEORY
T
3.
Connection with symmetric matrices
PRACTICE
P
4.
Connection with symmetric matrices
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THEORY
T
5.
Properties of symmetric maps
PRACTICE
P
6.
Properties of symmetric maps
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THEORY
T
7.
Orthonormal bases and symmetric maps
PRACTICE
P
8.
Orthonormal bases and symmetric maps
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Applications of symmetric maps
THEORY
T
1.
Quadratic forms
PRACTICE
P
2.
Quadratic forms
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THEORY
T
3.
Quadrics
PRACTICE
P
4.
Quadrics
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THEORY
T
5.
Least square solutions of linear equations
PRACTICE
P
6.
Least square solutions of linear equations
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THEORY
T
7.
Singular value decomposition
PRACTICE
P
8.
Singular value decomposition
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