Michael L. O'Leary - Linear Algebra

1 Cover

2 Title Page Linear Algebra Michael L. O’Leary

3 Copyright

4 Dedication

5 Preface

6 Acknowledgments

7 About the Companion Website

8 CHAPTER 1: Logic and Set Theory1.1 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 Matrices3.1 Linear Transformations 3.2 Matrix Algebra 3.3 Linear Operators 3.4 Injections and Surjections 3.5 Gauss‐Jordan Elimination

11 CHAPTER 4: Invertibility4.1 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 Spaces6.1 Inner Products 6.2 Orthonormal Bases

14 CHAPTER 7: Matrix Theory7.1 Eigenvectors and Eigenvalues 7.2 Minimal Polynomial 7.3 Similar Matrices 7.4 Diagonalization

15 Further Reading

16 Index

17 End User License Agreement

1 Chapter 1 Figure 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 ℝ 3.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+ wwith 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 ℝ 2through the line L .Figure 3.5 A reflection in ℝ 3through the xy ‐plane.Figure 3.6 A rotation of θ centered at O .Figure 3.7 A rotation in ℝ 2through θ .Figure 3.8 A rotation in ℝ 3about the z ‐axis through θ .Figure 3.9 A rotation in ℝ 3about the z ‐axis through θ viewed fro...Figure 3.10 A rotation in ℝ 3of e 3and e 1about 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 uand v.Figure 4.2 The parallelogram described by T uand 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 Figure 5.4 Diagram for the change of basis transformation .Figure 5.5 Change of basis with linear transformation T .Figure 5.6 Diagram for the standard transformation .

6 Chapter 6Figure 6.1 The projection of u 2onto v 1.Figure 6.2 Finding a vector orthogonal to S( v 1, v 2). Figure 6.3 The projection of u 3onto v 1.

7 Chapter 7Figure 7.1 The ellipse 7 x 2+ 4 xy + 4 y 2= 16 wi...

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2 Table of Contents

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