Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares

Contents
Preface
I Vectors
	1 Vectors
		1.1 Vectors
		1.2 Vector addition
		1.3 Scalar-vector multiplication
		1.4 Inner product
		1.5 Complexity of vector computations
		Exercises
	2 Linear functions
		2.1 Linear functions
		2.2 Taylor approximation
		2.3 Regression model
		Exercises
	3 Norm and distance
		3.1 Norm
		3.2 Distance
		3.3 Standard deviation
		3.4 Angle
		3.5 Complexity
		Exercises
	4 Clustering
		4.1 Clustering
		4.2 A clustering objective
		4.3 The k-means algorithm
		4.4 Examples
		4.5 Applications
		Exercises
	5 Linear independence
		5.1 Linear dependence
		5.2 Basis
		5.3 Orthonormal vectors
		5.4 Gram–Schmidt algorithm
		Exercises
II Matrices
	6 Matrices
		6.1 Matrices
		6.2 Zero and identity matrices
		6.3 Transpose, addition, and norm
		6.4 Matrix-vector multiplication
		6.5 Complexity
		Exercises
	7 Matrix examples
		7.1 Geometric transformations
		7.2 Selectors
		7.3 Incidence matrix
		7.4 Convolution
		Exercises
	8 Linear equations
		8.1 Linear and affine functions
		8.2 Linear function models
		8.3 Systems of linear equations
		Exercises
	9 Linear dynamical systems
		9.1 Linear dynamical systems
		9.2 Population dynamics
		9.3 Epidemic dynamics
		9.4 Motion of a mass
		9.5 Supply chain dynamics
		Exercises
	10 Matrix multiplication
		10.1 Matrix-matrix multiplication
		10.2 Composition of linear functions
		10.3 Matrix power
		10.4 QR factorization
		Exercises
	11 Matrix inverses
		11.1 Left and right inverses
		11.2 Inverse
		11.3 Solving linear equations
		11.4 Examples
		11.5 Pseudo-inverse
		Exercises
III Least squares
	12 Least squares
		12.1 Least squares problem
		12.2 Solution
		12.3 Solving least squares problems
		12.4 Examples
		Exercises
	13 Least squares data fitting
		13.1 Least squares data fitting
		13.2 Validation
		13.3 Feature engineering
		Exercises
	14 Least squares classification
		14.1 Classification
		14.2 Least squares classifier
		14.3 Multi-class classifiers
		Exercises
	15 Multi-objective least squares
		15.1 Multi-objective least squares
		15.2 Control
		15.3 Estimation and inversion
		15.4 Regularized data fitting
		15.5 Complexity
		Exercises
	16 Constrained least squares
		16.1 Constrained least squares problem
		16.2 Solution
		16.3 Solving constrained least squares problems
		Exercises
	17 Constrained least squares applications
		17.1 Portfolio optimization
		17.2 Linear quadratic control
		17.3 Linear quadratic state estimation
		Exercises
	18 Nonlinear least squares
		18.1 Nonlinear equations and least squares
		18.2 Gauss–Newton algorithm
		18.3 Levenberg–Marquardt algorithm
		18.4 Nonlinear model fitting
		18.5 Nonlinear least squares classification
		Exercises
	19 Constrained nonlinear least squares
		19.1 Constrained nonlinear least squares
		19.2 Penalty algorithm
		19.3 Augmented Lagrangian algorithm
		19.4 Nonlinear control
		Exercises
Appendices
	A Notation
	B Complexity
	C Derivatives and optimization
		C.1 Derivatives
		C.2 Optimization
		C.3 Lagrange multipliers
	D Further study
Index