Introduction to Linear Algebra

Cover
Table of Contents
Preface
1 Introduction to Vectors
	1.1 Vectors and Linear Combinations
	1.2 Lengths and Dot Products
	1.3 Matrices
2 Solving Linear Equations
	2.1 Vectors and Linear Equations
	2.2 The Idea of Elimination
	2.3 Elimination Using Matrices
	2.4 Rules for Matrix Operations
	2.5 Inverse Matrices
	2.6 Elimination = Factorization: A = LU
	2.7 Transposes and Permutations
3 Vector Spaces and Subspaces
	3.1 Spaces of Vectors
	3.2 The Nullspace of A : Solving Ax= 0 and Rx = 0
	3.3 The Complete Solution to Ax = b
	3.4 Independence, Basis and Dimension
	3.5 Dimensions of the Four Subspaces
4 Orthogonality
	4.1 Orthogonality of the Four Subspaces
	4.2 Projections
	4.3 Least Squares Approximations
	4.4 Orthonormal Bases and Gram-Schmidt
5 Determinants
	5.1 The Properties of Determinants
	5.2 Permutations and Cofactors
	5.3 Cramer\'s Rule, Inverses, and Volumes
6 Eigenvalues and Eigenvectors
	6.1 Introduction to Eigenvalues
	6.2 Diagonalizing a Matrix
	6.3 Systems of Differential Equations
	6.4 Symmetric Matrices
	6.5 Positive Definite Matrices
7 The Singular Value Decomposition (SVD)
	7.1 Image Processing by Linear Algebra
	7.2 Bases and Matrices in the SVD
	7.3 Principal Component Analysis (PCA by the SVD)
	7.4 The Geometry of the SVD
8 Linear Transformations
	8.1 The Idea of a Linear Transformation
	8.2 The Matrix of a Linear Transformation
	8.3 The Search for a Good Basis
9 Complex Vectors and Matrices
	9.1 Complex Numbers
	9.2 Hermitian and Unitary Matrices
	9.3 The Fast Fourier Transform
10 Applications
	10.1 Graphs and Networks
	10.2 Matrices in Engineering
	10.3 Markov Matrices, Population, and Economics
	10.4 Linear Programming
	10.5 Fourier Series: Linear Algebra for Functions
	10.6 Computer Graphics
	10.7 Linear Algebra for Cryptography
11 Numerical Linear Algebra
	11.1 Gaussian Elimination in Practice
	11.2 Norms and Condition Numbers
	11.3 Iterative Methods and Preconditioners
12 Linear Algebra in Probability & Statistics
	12.1 Mean, Variance, and Probability
	12.2 Covariance Matrices and Joint Probabilities
	12.3 Multivariate Gaussian and Weighted Least Squares
Matrix Factorizations
Index
Six Great Theorems of Linear Algebra/Linear Algebra in a Nutshell