A First Course in Linear Algebra

A First Course in Linear Algebra provides an introduction to the algebra and geometry of vectors, matrices, and linear transformations.

This book is designed as a background for second-year courses in calculus of several variables and differential equations where the theory of linear differential equations parallels that of linear algebraic equations. The topics discussed include the multiplication of vectors by scalars, vectors in n-space, planes and lines, and composites of linear mappings. The symmetric matrices and mappings, quadratic forms, change of coordinates, and effect of change of basis on matrices of linear functions are also described. This text likewise considers the computation of determinants, diagonalizable transformations, computation of eigenvalues and eigenvectors, and principal axis theorem.

This publication is suitable for college students taking a course in linear algebra.

Inhalt

1. Vectors
2. Coordinate Systems
3. Vectors and Their Components
4. Length of a Vector
5. Multiplication of Vectors by Scahrs
6. Addition of Vectors; Linear Combinations
7. Dot Products
8. Cross Products
   8, Vectors in n-Space
9. Still More General Vector Spaces
   
   2. Planes and Lines
10. Planes
11. Lines
12. Vector Functions of Scalars
    
    3. Linear Functions
13. Definition
14. Matrices
15. Sums and Scalar Multiples of Linear Mappings
16. Composites of Linear Mappings and Products of Matrices
17. Inverses
18. Kernel and Image
    
    4. Solution of Equations
       1 Solution Process; Reduction to Echelon Form
       2, Closer Analysis of the Solutions and the Solution Method
    5. Dimension
19. Subspaces
20. Dimension
21. Rank
    
    6. Determinants and Transposes
22. Computation of Determinants
23. Explicit Formulas
24. Transposes
25. Symmetric Matrices and Mappings
    
    7. Eigenvalues
26. Diagonalizable Transformations
27. Computation of Eigenvalues and Eigenvectors
28. Principal Axis Theorem
    
    8. Quadratic Forms and Change of Basis
29. Quadratic Forms
30. Change of Coordinates
31. Effect of Change of Basis on Matrices of Linear Functions
    Appendix I. A Smattering of Logic
    Appendix II. Existencc of Real Eigenvalues of Symmetric
    Mappings
    Answers
    Index