Linear Functions and Matrix Theory

Courses that study vectors and elementary matrix theory and introduce linear transformations have proliferated greatly in recent years. Most of these courses are taught at the undergraduate level as part of, or adjacent to, the second-year calculus sequence. Although many students will ultimately find the material in these courses more valuable than calculus, they often experience a class that consists mostly of learning to implement a series of computational algorithms. The objective of this text is to bring a different vision to this course, including many of the key elements called for in current mathematics-teaching reform efforts. Three of the main components of this current effort are the following: 1. Mathematical ideas should be introduced in meaningful contexts, with after a clear understanding formal definitions and procedures developed of practical situations has been achieved. 2. Every topic should be treated from different perspectives, including the numerical, geometric, and symbolic viewpoints. 3. The important ideas need to be visited repeatedly throughout the term, with students' understan9ing deepening each time. This text was written with these three objectives in mind. The first two chapters deal with situations requiring linear functions (at times, locally linear functions) or linear ideas in geometry for their understanding. These situations provide the context in which the formal mathematics is developed, and they are returned to with increasing sophistication throughout the text.

Inhalt

1 linear Functions.- 1.1 Linear Functions.- 1.2 Local Linearity.- 1.3 Matrices.- 1.4 More Linearity.- 2 Linear Geometry.- 2.1 Linear Geometry in the Plane.- 2.2 Vectors and Lines in the Plane.- 2.3 Linear Geometry in Space.- 2.4 An Introduction to Linear Perspective.- 3 Systems of Linear Equations.- 3.1 Systems of Linear Equations.- 3.2 Gaussian Elimination.- 3.3 Gauss-Jordan Elimination.- 3.4 Matrix Rank and Systems of Linear Equations.- 3.5 The Simplex Algorithm.- 4 Basic Matrix Algebra.- 4.1 The Matrix Product: A Closer Look.- 4.2 Fibonacci Numbers and Difference Equations.- 4.3 The Determinant.- 4.4 Properties and Applications of the Determinant.- 4.5 The LU-Decomposition.- 5 Key Concepts of Linear Algebra in Rn.- 5.1 Linear Combinations and Subspaces.- 5.2 Linear Independence.- 5.3 Basis and Dimension.- 6 More Vector Geometry.- 6.1 The Dot Product.- 6.2 Angles and Projections.- 6.3 The Cross Product.- 7 Eigenvalues and Eigenvectors of Matrices.- 7.1 Eigenvalues and Eigenvectors.- 7.2 Eigenspaces and Diagonalizability.- 7.3 Symmetric Matrices and Probability Matrices.- 8 Matrices as Linear Transformations.- 8.1 Linear Transformations.- 8.2 Using Linear Transformations.- 8.3 Change of Basis.- 9 Orthogonality and Least-Squares Problems.- 9.1 Orthogonality and the Gram-Schmidt Process.- 9.2 Orthogonal Projections.- 9.3 Least-Squares Approximations.- Answers to Odd-Numbered Problems.