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Offering the most geometric presentation available, Linear Algebra with Applications, Fifth Edition emphasizes linear transformations as a unifying theme. This elegant textbook combines a user-friendly presentation with straightforward, lucid language to clarify and organize the techniques and applications of linear algebra. Exercises and examples make up the heart of the text, with abstract exposition kept to a minimum. Exercise sets are broad and varied and reflect the author’s creativity and passion for this course. This revision reflects careful review and appropriate edits throughout, while preserving the order of topics of the previous edition.

1. Linear Equations

1.1 Introduction to Linear Systems

1.2 Matrices, Vectors, and Gauss-Jordan Elimination

1.3 On the Solutions of Linear Systems; Matrix Algebra

2. Linear Transformations

2.1 Introduction to Linear Transformations and Their Inverses

2.2 Linear Transformations in Geometry

2.3 Matrix Products

2.4 The Inverse of a Linear Transformation

3. Subspaces of Rⁿ and Their Dimensions

3.1 Image and Kernel of a Linear Transformation

3.2 Subspace of Rⁿ ; Bases and Linear Independence

3.3 The Dimension of a Subspace of Rⁿ

3.4 Coordinates

4. Linear Spaces

4.1 Introduction to Linear Spaces

4.2 Linear Transformations and Isomorphisms

4.3 The Matrix of a Linear Transformation

5. Orthogonality and Least Squares

5.1 Orthogonal Projections and Orthonormal Bases

5.2 Gram-Schmidt Process and QR Factorization

5.3 Orthogonal Transformations and Orthogonal Matrices

5.4 Least Squares and Data Fitting

5.5 Inner Product Spaces

6. Determinants

6.1 Introduction to Determinants

6.2 Properties of the Determinant

6.3 Geometrical Interpretations of the Determinant; Cramer's Rule

7. Eigenvalues and Eigenvectors

7.1 Diagonalization

7.2 Finding the Eigenvalues of a Matrix

7.3 Finding the Eigenvectors of a Matrix

7.4 More on Dynamical Systems

7.5 Complex Eigenvalues

7.6 Stability

8. Symmetric Matrices and Quadratic Forms

8.1 Symmetric Matrices

8.2 Quadratic Forms

8.3 Singular Values

9. Linear Differential Equations

9.1 An Introduction to Continuous Dynamical Systems

9.2 The Complex Case: Euler's Formula

9.3 Linear Differential Operators and Linear Differential Equations

Appendix A. Vectors

Appendix B: Techniques of Proof

Answers to Odd-numbered Exercises

Subject Index

Name Index