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Linear Functions and Matrix Theory

This textbook provides an introduction to linear algebra for students who have completed the first year of calculus. It presents an elementary understanding of important ideas in linear algebra, emphasizing conceptual understanding, developing applied examples, and working with realistic numerical data before introducing formal mathematical definition and operations. This text emphasizes geometric, symbolic, and numeric presentations of the subject. The first two chapters cover linear phenomena in both numeric and geometric settings. The symbolic manipulation of vectors and matrices is then introduced as a tool for the study of specific problems. Many examples, student exercises, and group project ideas are included.
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Table of Contents

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.

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