Numerical Liner Algebra with Applications

TOPICS Chapters 1-6 provide coverage of applied linear algebra sufficient for reading the remainder of the book. Chapter 1: Matrices The chapter introduces matrix arithmetic and the very important topic of linear transformations. Rotation matrices provide an interesting and useful example of linear transformations. After discussing matrix powers, the concept of the matrix inverse and transpose concludes the chapter. Chapter 2: Linear Equations This chapter introduces Gaussian elimination for the solution of linear systems Ax = b and for the computation of the matrix inverse. The chapter also introduces the relationship between the matrix inverse and the solution to a linear homogeneous equation. Two applications involving a truss and an electrical circuit conclude the chapter. Chapter 3: Subspaces This chapter is, by its very nature, somewhat abstract. It introduces the concepts of subspaces, linear independence, basis, matrix rank, range, and null space. Although the chapter may challenge some readers, the concepts are essential for understanding many topics in the book, and it should be covered thoroughly. Chapter 4: Determinants Although the determinant is rarely computed in practice, it is often used in proofs of important results. The chapter introduces the determinant and its computation using expansion byminors and by rowelimination. The chapter ends with an interesting application of the determinant to text encryption. Chapter 5: Eigenvalues and Eigenvectors This is a very important chapter, and its results are used throughout the book. After defining the eigenvalue and an associated eigenvector, the chapter develops some of their most important properties, including their use in matrix diagonalization. The chapter concludes with an application to the solution of systems of ordinary differential equations and the problem of ranking items using eigenvectors. Chapter 6: Orthogonal Vectors and Matrices This chapter introduces the inner product and its associati