Linear Algebra. Michael L. O'Leary

      

      Table of Contents

      1  Cover

      2  Title Page

      3  Copyright

      4  Dedication

      5  Preface

      6  Acknowledgments

      7  About the Companion Website

      8  CHAPTER 1: Logic and Set Theory 1.1 Statements 1.2 Sets and Quantification 1.3 Sets and Proofs 1.4 Functions

      9  CHAPTER 2: Euclidean Space 2.1 Vectors 2.2 Dot Product 2.3 Cross Product

      10  CHAPTER 3: Transformations and Matrices 3.1 Linear Transformations 3.2 Matrix Algebra 3.3 Linear Operators 3.4 Injections and Surjections 3.5 Gauss‐Jordan Elimination

      11  CHAPTER 4: Invertibility 4.1 Invertible Matrices 4.2 Determinants 4.3 Inverses and Determinants 4.4 Applications

      12  CHAPTER 5: Abstract Vectors 5.1 Vector Spaces 5.2 Subspaces 5.3 Linear Independence 5.4 Basis and Dimension 5.5 Rank and Nullity 5.6 Isomorphism

      13  CHAPTER 6: Inner Product Spaces 6.1 Inner Products 6.2 Orthonormal Bases

      14  CHAPTER 7: Matrix Theory 7.1 Eigenvectors and Eigenvalues 7.2 Minimal Polynomial 7.3 Similar Matrices 7.4 Diagonalization

      15  Further Reading

      16  Index

      17  End User License Agreement

      List of Illustrations

      1 Chapter 1Figure 1.1 A function f ⊆ A × B .Figure 1.2 The composition of f and g.Figure 1.3 g is not a one‐to‐one function.Figure 1.4 h is a one‐to‐one function.Figure 1.5 f is an onto function.Figure 1.6 g is a bijection.Figure 1.7 The image of C under f.Figure 1.8 The inverse image of D under f.

      2 Chapter 2Figure 2.1 Two interpretations of vector as an arrow.Figure 2.2 Addition of arrows.Figure 2.3 Scaling ofarrows.Figure 2.4 Finding the distance between vectors.Figure 2.5 A triangle in ℝ³ .Figure 2.6 The distance between u and v is ‖u − v‖...Figure 2.7 The line L containing u with direction vector m.Figure 2.8 The plane P containing u with direction vectors m and n.Figure 2.9 Lines L and L′ with normal n.Figure 2.10 u = r v + w with w orthogonal to v.Figure 2.11 The distance from the vector u to the plane P.Figure 2.12 The distance from the vector u to the plane P, side view.Figure 2.13 Computing the cross product.Figure 2.14 The parallelogram described by u and v.Figure 2.15 The parallelepiped described by u, v, and w.

      3 Chapter 3Figure 3.1 Linear transformations preserve addition.Figure 3.2 Linear transformations preserve scalar multiplication.Figure 3.3 A reflection through the line ℓ.Figure 3.4 A reflection in ℝ² through the line L.Figure 3.5 A reflection in ℝ³ through the xy‐plane.Figure 3.6 A rotation of θ centered at O.Figure 3.7 A rotation in ℝ² through θ.Figure 3.8 A rotation in ℝ³ about the z‐axis through θ.Figure 3.9 A rotation in ℝ³ about the z‐axis through θ viewed fro...Figure 3.10 A rotation in ℝ³ of e ₃ and e ₁ about the y‐axis through θ...Figure 3.11 A reflection is an isometry.Figure 3.12 A rotation is an isometry.Figure 3.13 A translation is anisometry.Figure 3.14 A shear along L.Figure 3.15 A horizontal shear with shear factor k.

      4 Chapter 4Figure 4.1 The parallelogram described by u and v.Figure 4.2 The parallelogram described by T u and T v.Figure 4.3 The areas of a parallelogram and its image under a linear operato...

      5 Chapter 5Figure 5.1 Illustrating the Rank‐Nullity Theorem.Figure 5.2 Writing w as a linear combination of two bases.Figure 5.3 Change of basis