This introductory text emphasizes linear transformations as a unifying theme. Students are able to do both computational and abstract math in each chapter. This is the most geometric presentation now available. Half way through the text, when eigenvectors are reached, a second theme, on dynamical systems, emerges for the second half of the text. There is also a wider range of problem sets in this text than any other in this market. Free to users is an accompanying website at prenhall.com/bretscher.
Offers a careful sequencing of material with
a focus on smooth transitions and good motivations for new concepts.
Uses visualization and geometrical interpretations
extensively throughout, e.g.:
The geometrical interpretations of the determinant.
The use of phase portraits for dynamical systems.
The geometrical interpretations of the QR-factorization
and the singular value decomposition.
Avoids the wall of vector spaces that often
overwhelms and confuses students, leaving them lost and discouraged
early in the course.
Introduces the abstract concepts gradually and gently
throughout the text and brings up the notion of a vector space
(or linear space) only in the last chapter.
Provides an early introduction to linear transformation
to make the discussion of matrix manipulations more meaningful
and easier to visualize.
Includes a discussion of phase portraits. Features a large number of superb problems and exercises,
Probing, thought-provoking exercises some abstract
and some focusing on applications.
One application in dynamical systemsruns throughout
the text to unify the presentation and provide motivation for readers.
Extensive historical references are found throughout to
provide motivation and meaning.
1. Linear Equations.
2. Linear Transformations.
3. Subspaces of R^n and Their Dimension.
4. Orthogonality and Least Squares.
6. Eigenvalues and Eigenvectors.
7. Coordinate Systems.
8. Linear Systems of Differential Equations.
9. Linear Spaces.
Appendix A. Vectors.
Answers to Odd-Numbered Exercises.