Linear Algebra for Everyone
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From the reviews:
"The author's intention was to write an introductory linear algebra text for students studying disciplines that use this subject matter as a tool. . All topics introduced are given a concrete basis in reality by use of well-chosen examples. . Summing Up: Recommended. Upper-division undergraduates." (D. S. Larson, Choice, Vol. 49 (2), October, 2011)
"This is a conventional introduction to matrices that the author has attempted to make more accessible by garnishing it with palindromes, jokes, and aphorisms. It is a translation of the 2007 Italian-language work Algebra Lineare per tutti . . The translation is smooth and easy to read . . The book takes a very concrete approach. . the work is in fact aimed at linear algebra students . ." (Allen Stenger, The Mathematical Association of America, June, 2011)
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Content
- Intro
- Title Page
- Copyright Page
- Foreword
- Introduction
- Table of Contents
- Numerical and Symbolic Computations
- The equation ax = b. Let's try to solve it
- The equation ax = b. Be careful of mistakes
- The equation ax = b. Let's manipulate the symbols
- Exercises
- Part I
- 1 Systems of Linear Equations and Matrices
- 1.1 Examples of Systems of Linear Equations
- 1.2 Vectors and Matrices
- 1.3 Generic Systems of Linear Equations and Associated Matrices
- 1.4 The Formalism of Ax = b
- Exercises
- 2 Operations with Matrices
- 2.1 Sum and the product by a number
- 2.2 Row by column product
- 2.3 How much does it cost to multiply two matrices?
- 2.4 Some properties of the product of matrices
- 2.5 Inverse of a matrix
- Exercises
- 3 Solutions of Systems of Linear Equations
- 3.1 Elementary Matrices
- 3.2 Square Linear Systems, Gaussian Elimination
- 3.3 Effective Calculation of Matrix Inverses
- 3.4 How much does Gaussian Elimination cost?
- 3.5 The LU Decomposition
- 3.6 Gaussian Elimination for General Systems of Linear Equations
- 3.7 Determinants
- Exercises
- 4 Coordinate Systems
- 4.1 Scalars and Vectors
- 4.2 Cartesian Coordinates
- 4.3 The Parallelogram Rule
- 4.4 Orthogonal Systems, Areas, Determinants
- 4.5 Angles, Moduli, Scalar Products
- 4.6 Scalar Products and Determinants in General
- 4.7 Change of Coordinates
- 4.8 Vector Spaces and Bases
- Exercises
- Part II
- 5 Quadratic Forms
- 5.1 Equations of the Second Degree
- 5.2 Elementary Operations on Symmetric Matrices
- 5.3 Quadratic Forms, Functions, Positivity
- 5.4 Cholesky Decomposition
- Exercises
- 6 Orthogonality and Orthonormality
- 6.1 Orthonormal Tuples and Orthonormal Matrices
- 6.2 Rotations
- 6.3 Subspaces, Linear Independence, Rank, Dimension
- 6.4 Orthonormal Bases and the Gram-Schmidt Procedure
- 6.5 The QR Decomposition
- Exercises
- 7 Projections, Pseudoinverses and Least Squares
- 7.1 Matrices and Linear Transformations
- 7.2 Projections
- 7.3 Least Squares and Pseudoinverses
- Exercises
- 8 Endomorphisms and Diagonalization
- 8.1 An Example of a Plane Linear Transformation
- 8.2 Eigenvalues, Eigenvectors, Eigenspaces and Similarity
- 8.3 Powers of Matrices
- 8.4 The Rabbits of Fibonac
- 8.5 Differential Systems
- 8.6 Diagonalizability of Real Symmetric Matrices
- Exercises
- Part III
- Appendix
- Problems with the computer
- Conclusion?
- References
- Index
