Functional Linear Algebra
Linear algebra is an extremely versatile and useful subject. It rewards those who study it with powerful computational tools, lessons about how mathematical theory is built, examples for later study in other classes, and much more.
Functional Linear Algebra is a unique text written to address the need for a one-term linear algebra course where students have taken only calculus. It does not assume students have had a proofs course.
The text offers the following approaches:
- More emphasis is placed on the idea of a linear function, which is used to motivate the study of matrices and their operations. This should seem natural to students after the central role of functions in calculus.
- Row reduction is moved further back in the semester and vector spaces are moved earlier to avoid an artificial feeling of separation between the computational and theoretical aspects of the course.
- Chapter 0 offers applications from engineering and the sciences to motivate students by revealing how linear algebra is used.
- Vector spaces are developed over R, but complex vector spaces are discussed in Appendix A.1.
- Computational techniques are discussed both by hand and using technology. A brief introduction to Mathematica is provided in Appendix A.2.
As readers work through this book, it is important to understand the basic ideas, definitions, and computational skills. Plenty of examples and problems are provided to make sure readers can practice until the material is thoroughly grasped.
Dr. Hannah Robbins is an associate professor of mathematics at Roanoke College, Salem, VA. Formerly a commutative algebraist, she now studies applications of linear algebra and assesses teaching practices in calculus. Outside the office, she enjoys hiking and playing bluegrass bass.
0. Motivation. 1. Vectors. 1.1. Vector Operations. 1.2. Span. 1.3. Linear Independence. 2. Functions of Vectors. 2.1. Linear Functions. 2.2. Matrices. 2.3. Matrix Operations. 2.4. Matrix Vector Spaces. 2.5. Kernel and Range. 2.6. Row Reduction. 2.7. Applications of Row Reduction. 2.8. Solution Sets. 2.9. Large Matrix Computations. 2.10. Invertibility. 2.11. The Invertible Matrix Theorem. 3. Vector Spaces. 3.1. Basis and Coordinates. 3.2. Polynomial Vector Spaces. 3.3. Other Vector Spaces. 4. Diagonalization. 4.1. Eigenvalues and Eigenvectors. 4.2. Determinants. 4.3. Eigenspaces. 4.4. Diagonalization. 4.5. Change of Basis Matrices. 5. Computational Vector Geometry. 5.1. Length. 5.2. Orthogonality. 5.3. Orthogonal Projection. 5.4. Orthogonal Basis. A. Appendices. A.1. Complex Numbers. A.2. Mathematica. A.3. Solutions to Odd Exercises. Bibliography. Index.