Gilbert Strang-Introduction to Linear Algebra, Fifth Edition (2016)
English | Size: 267.61 MB
Category: Tutorial
This video standard describes a system for encoding and decoding (a “Codec”) that engineers have defined for applications like High Definition TV. It is not expected that you will know the meaning of every word — your book author does not know either. The point is to see an important example of a “standard” that is created by an industry after years of development— so all companies will know what coding system their products must be consistent with.
The words “motion compensation” refer to a way to estimate each video image from the previous one. The simplest would be to guess that successive video images are the same. Then we would only need the changes between frames — hopefully small. But if the camera is following the action, the whole scene will shift slightly and need correction. A better idea is to see which way the scene is moving and build that change into the next scene. This is MOTION COMPENSATION. In fact the motion is allowed to be different on different parts of the screen.
It is ideas like this — easy to talk about but taking years of effort to perfect — that make video technology and other technologies possible and successful. Engineers do their job. I hope these links give an idea of the detail needed.
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 andWeighted Least Squares
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