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INTRODUCTION
1. BASICS
2. SCALAR PRODUCT, NORMS AND ANGLES
3. PROJECTION ON A LINE
4. ORTHOGONALIZATION: THE GRAM-SCHMIDT PROCEDURE
5. HYPERPLANES AND HALF-SPACES
6. LINEAR FUNCTIONS
7. APPLICATION: DATA VISUALIZATION BY PROJECTION ON A LINE
8. EXERCISES
9. BASICS
10. MATRIX-VECTOR AND MATRIX-MATRIX MULTIPLICATION, SCALAR PRODUCT
11. SPECIAL CLASSES OF MATRICES
12. QR DECOMPOSITION OF A MATRIX
13. MATRIX INVERSES
14. LINEAR MAPS
15. MATRIX NORMS
16. APPLICATIONS
17. EXERCISES
18. MOTIVATING EXAMPLE
19. EXISTENCE AND UNICITY OF SOLUTIONS
20. SOLVING LINEAR EQUATIONS VIA QR DECOMPOSITION
21. APPLICATIONS
22. EXERCISES
23. ORDINARY LEAST-SQUARES
24. VARIANTS OF THE LEAST-SQUARES PROBLEM
25. KERNELS FOR LEAST-SQUARES
26. APPLICATIONS
27. EXERCISES
28. QUADRATIC FUNCTIONS AND SYMMETRIC MATRICES
29. SPECTRAL THEOREM
30. POSITIVE SEMI-DEFINITE MATRICES
31. PRINCIPAL COMPONENT ANALYSIS
32. APPLICATIONS: PCA OF SENATE VOTING DATA
33. EXERCISES
34. THE SVD THEOREM
35. MATRIX PROPERTIES VIA SVD
36. SOLVING LINEAR SYSTEMS VIA SVD
37. LEAST-SQUARES AND SVD
38. LOW-RANK APPROXIMATIONS
39. APPLICATIONS
40. EXERCISES
Dimension of an affine subspace
Sample and weighted average
Sample average of vectors
Euclidean projection on a set
Orthogonal complement of a subspace
Power laws
Power law model fitting
Definition: Vector norm
An infeasible linear system
Sample variance and standard deviation
Functions and maps
Dual norm
Incidence matrix of a network
Nullspace of a transpose incidence matrix
Rank properties of the arc-node incidence matrix
Permutation matrices
QR decomposition: Examples
Backwards substitution for solving triangular linear systems.
Solving triangular systems of equations: Backwards substitution example
Linear regression via least squares
Nomenclature
Standard forms
A two-dimensional toy optimization problem
Global vs. local minima
Gradient of a function
Set of solutions to the least-squares problem via QR decomposition
Sample covariance matrix
Optimal set of least-squares via SVD
Pseudo-inverse of a matrix
SVD: A 4x4 example
Singular value decomposition of a 4 x 5 matrix
Representation of a two-variable quadratic function
Edge weight matrix of a graph
Network flow
Laplacian matrix of a graph
Hessian of a function
Hessian of a quadratic function
Gram matrix
Quadratic functions in two variables
Quadratic approximation of the log-sum-exp function
Determinant of a square matrix
A squared linear function
Eigenvalue decomposition of a symmetric matrix
Rayleigh quotients
Largest singular value norm of a matrix
Nullspace of a 4x5 matrix via its SVD
Range of a 4x5 matrix via its SVD
Low-rank approximation of a 4x5 matrix via its SVD
Pseudo-inverse of a 4x5 matrix via its SVD
Image compression via least-squares
Senate voting data matrix.
Senate voting analysis and visualization.
Beer-Lambert law in absorption spectrometry
Absorption spectrometry: Using measurements at different light frequencies.
Similarity of two documents
Image compression
Temperatures at different airports
Navigation by range measurement
Bag-of-words representation of text
Bag-of-words representation of text: Measure of document similarity
Rate of return of a financial portfolio
Single factor model of financial price data
The problem of Gauss
Control of a unit mass
Portfolio optimization via linearly constrained least-squares.
Cauchy-Schwarz inequality proof
Dimension of hyperplanes
Spectral theorem: Eigenvalue decomposition for symmetric matrices
Singular value decomposition (SVD) theorem
Rank-one matrices
Rank-one matrices: A representation theorem
Full rank matrices
Rank-nullity theorem
A theorem on positive semidefinite forms and eigenvalues
Fundamental theorem of linear algebra
The sample average of given vectors is defined as the vector , with
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