Exploring Linear Algebra: Labs and Projects with Mathematica

Features Covers the core topics of linear algebra, including matrices, invertibility, and vector spaces Discusses applications to statistics and differential equations Provides straightforward explanations of the material with integrated exercises that promote an inquiry-based learning experience Includes 81 theorems and problems throughout the labs Motivates students to make conjectures and develop proofs Offers interesting problems for undergraduate-level research projects Summary

Exploring Linear Algebra: Labs and Projects withMathematica® is a hands-on lab manual for daily use in the classroom. Each lab includes exercises, theorems, and problems that guide your students on an exploration of linear algebra.

The exercises section integrates problems, technology,Mathematica® visualization, and Mathematica CDFs, enabling students to discover the theory and applications of linear algebra in a meaningful way. The theorems and problems section presents the theoretical aspects of linear algebra. Students are encouraged to discover the truth of each theorem and problem, to move toward proving (or disproving) each statement, and to present their results to their peers.

Each chapter also contains a project set consisting of application-driven projects that emphasize the material in the chapter. Students can use these projects as the basis for further undergraduate research.

Matrix Operations 
Lab 0: An Introduction to Mathematica 
Lab 1: Matrix Basics and Operations 
Lab 2: A Matrix Representation of Linear Systems 
Lab 3: Powers, Inverses, and Special Matrices 
Lab 4: Graph Theory and Adjacency Matrices 
Lab 5: Permutations and Determinants 
Lab 6: 4 x 4 Determinants and Beyond 
Project Set 1

Invertibility 
Lab 7: Singular or Nonsingular? Why Singularity Matters 
Lab 8: Mod It Out, Matrices with Entries in Zp
Lab 9: It’s a Complex World 
Lab 10: Declaring Independence: Is It Linear? 
Project Set 2

Vector Spaces 
Lab 11: Vector Spaces and Subspaces
Lab 12: Basing It All on Just a Few Vectors 
Lab 13: Linear Transformations 
Lab 14: Eigenvalues and Eigenspaces 
Lab 15: Markov Chains, An Application of Eigenvalues 
Project Set 3

Orthogonality 
Lab 16: Inner Product Spaces 
Lab 17: The Geometry of Vector and Inner Product Spaces
Lab 18: Orthogonal Matrices, QR Decomposition, and Least Squares Regression 
Lab 19: Symmetric Matrices and Quadratic Forms 
Project Set 4

Matrix Decomposition with Applications 
Lab 20: Singular Value Decomposition (SVD) 
Lab 21: Cholesky Decomposition and Its Application to Statistics
Lab 22: Jordan Canonical Form 
Project Set 5

Applications to Differential Equations 
Lab 23: Linear Differential Equations 
Lab 24: Higher-Order Linear Differential Equations 
Lab 25: Phase Portraits, Using the Jacobian Matrix to Look Closer at Equilibria 
Project Set 6

Mathematica Demonstrations and References

Index