Computational and Algorithmic Linear Algebra and n-Dimensional Geometry

The following sections are included:

• Contents
• Preface
• Glossary of Symbols and Abbreviations
• A Poem on Linear Algebra

The following sections are included:

• What Are Algorithms?
• Some Example Applications Involving Linear Equations
• Solving a System of Linear Equations is a Fundamental Computational Tool
• The Detached Coefficient Tableau Form
• Operations on Vectors, the Zero and the Unit Vectors, Subspaces and Coordinate Subspaces of Rⁿ
• Notation to Denote the General System of Linear Equations
• Constraint Functions, Linearity Assumptions, Linear and Affine Functions
• Solutions to a System, General Solution; Nonbasic (Independent), Basic (Dependent) Variables in Parametric Representation
• Linear Dependence Relation Among Constraints, Redundant Constraints, Linearly Independent Systems
• The Fundamental Inconsistent Equation “0 = 1”
• Elementary Row Operations on a Tableau Leading to an Equivalent System
• Memory Matrix to Keep Track of Row Operations
• What to Do With “0 = 0”, Or “0 = 1” Rows Encountered After Row Operations?
• The Elimination Approach to Solve Linear Equations
• The Pivot Step, a Convenient Tool For Carrying Out Elimination Steps
• The Gauss–Jordan (GJ) Pivotal Method for Solving Linear Equations
• A Theorem of Alternatives for Systems of Linear Equations
• Special Triangular Systems of Linear Equations
• The Gaussian Pivot Step
• The Gaussian (G) Elimination Method for Solving Linear Equations
• Comparison of the GJ and the G Elimination Methods
• Infeasibility Analysis
• Homogeneous Systems of Linear Equations
• Solving Systems of Linear Equations And/Or Inequalities
• Can We Obtain a Nonnegative Solution to a System of Linear Equations by the GJ or the G Methods Directly?
• Additional Exercises
• References

The following sections are included:

• A Matrix as a Rectangular Array of Numbers, Its Rows and Columns
• Matrix-Vector Multiplication
• Three Ways of Representing the General System of Linear Equations through Matrix Notation
• Scalar Multiplication and Addition of Matrices, Transposition
• Matrix Multiplication and Its Various Interpretations
• Some Special Matrices
• Row Operations and Pivot Steps on a Matrix
• Determinants of Square Matrices
• Additional Exercises
• References

The following sections are included:

• Viewing n-tuple Vectors as Points of n-Dimensional Space Rⁿ by Setting Up Cartesian Coordinate Axes
• Translating the Origin to the Point a, Translation of a Set of Points
• Parametric Representation of Subsets of Rⁿ
• Straight Lines in Rⁿ
• Half-Lines and Rays in Rⁿ
• Line Segments in Rⁿ
• Straight Lines through the Origin
• Geometric Interpretation of Addition of Two Vectors, Parallelogram Law
• Linear Combinations, Linear Hulls or Spans, Subspaces
• Column Space, Row Space, and Null Space of a Matrix
• Affine Combinations, Affine Hulls or Spans, Affinespaces
• Convex Combinations, Convex Hulls
• Nonnegative Combinations, Pos Cones
• Hyperplanes
• Halfspaces
• Convex Polyhedra
• Convex and Nonconvex Sets
• Euclidean Distance, and Angles
• The Determinant Interpreted as a Volume Function
• Orthogonal Projections (Nearest Points)
• Procedures for Some Geometric Problems
• Additional Exercises
• References

The following sections are included:

• Linear Dependence, Independence of Sets of Vectors
• Rank and Bases for a Set of Vectors
• Two Algorithms to Check Linear Dependence, Find Rank and a Basis for a Given Set of Vectors
• How to Get an Alternate Basis by Introducing a Nonbasic Vector into A Basis?
• The Rank of a Matrix
• Computing the Inverse of a Nonsingular Square Matrix Using GJ Pivot Steps
• Pivot Matrices, Elementary Matrices, Matrix Factorizations, and the Product Form of the Inverse
• How to Update the Inverse When a Column of the Matrix Changes
• Results on the General System of Linear Equations Ax = b
• Optimization, Another Important Distinction Between LA, LP, and IP; and the Duality Theorem for Linear Optimization Subject to Linear Equations
• Memory Matrix Versions of GJ, G Elimination Methods
• Orthogonal and Orthonormal Sets of Vectors, Gram–Schmidt Orthogonalization, QR-Factorization, Orthogonal Matrices, Orthonormal Bases
• Additional Exercises
• References

The following sections are included:

• Expressing a Quadratic Function in n Variables Using Matrix Notation
• Convex, Concave Functions; Positive (Negative) (Semi) Definiteness; Indefiniteness
• Algorithm to Check Positive (Semi) Definiteness Using G Pivot Steps
• Diagonalization of Quadratic Forms and Square Matrices

The following sections are included:

• Definitions, Some Applications
• Relationship of Eigen Values and Eigen Vectors to the Diagonalizability of a Matrix
• An Illustrative Application of Eigen Values, Eigen Vectors, and Diagonalization in Differential Equations
• How to Compute Eigen Values?
• References

Nowadays there are several commercially available and highly popular software systems for solving the problems discussed in previous chapters. Among them the best known are: MATLAB, EXCEL, MATHEMATICA, MAPLE, and MACAULAY. We will provide illustrative numerical examples of how to solve these problems using MATLAB (this name is derived from Matrix Laboratory) in this chapter. For using the other software systems, see the references cited on them.

The following sections is included:

• Index