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# Alignment: Introduction to Linear Algebra β’ Gilbert Strang β’ Fifth Edition

Use Nagwa in conjunction with your preferred textbook. The recommended lessons from Nagwa for each section of this textbook are provided below. This alignment is not affiliated with, sponsored by, or endorsed by the publisher of the referenced textbook. Nagwa is a registered trademark of Nagwa Limited. All other trademarks and registered trademarks are the property of their respective owners.

### Table of Contents

• 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: π΄ = πΏπ
• 2.7 Transposes and Permutations
• 3 Vector Spaces and Subspaces
• 3.1 Spaces of Vectors
• 3.2 The Nullspace of π΄: Solving π΄π₯ = 0 and ππ₯ = 0
• 3.3 The Complete Solution to π΄π₯ = π
• 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 Cramerβ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 / Linear Algebra in a Nutshell