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Math 2500: Linear Algebra
Contents
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Contents
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Front Matter
Preface
Acknowledgements
1
Vectors, the Dot Product, and Matrices
Vectors
Getting Started with SageMath
The Dot Product
Matrices
Going Further with the Basic Objects
2
Linear Equations
Three Geometric Models
Solving Systems
Elimination using Matrices
Going Further with Elimination
Matrix Algebra
Matrix Inverses
The \(LU\) Decomposition
Permutation Matrices
Matrix Algebra: Going Further
3
Vector Spaces and Subspaces
Subspaces of \(\mathbb{R}^n\)
The Nullspace
Rank and the RREF
Solving a System
Going Further with Solving Systems
Bases
The Four Subspaces
Going Further with the Fundamental Theorem
4
Orthogonality
Orthogonality and the Four Subspaces
Projections Onto Subspaces
Going Further: Projections
Approximate Solutions: Least Squares
Orthonormal Bases and Gram-Schmidt
Going Further: Least Squares and Gram-Schmidt
5
Determinants
Properties of the Determinant
Computing Determinants
Going Further: The Determinant
6
Eigendata and the Singular Value decomposition
Eigenvectors and Eigenvalues
Diagonalizing Matrices
The Spectral Theorem
Going Further with Eigendata
Singular Value Decomposition
Going Further with the SVD
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Authored in PreTeXt
Math 2500: Linear Algebra
Preface
Acknowledgements