Linear Algebra and Its Applications

The following sections are included:

• Dedication
• Preface
• Contents

In ancient times, humans studied the equation

The solution is x = a⁻¹b. There are two lines of development: (1) one goes to a higher degree in one variable, (2) one goes to a system of linear equations in many variables. Certainly this fuses into modern Algebraic Geometry which studies the systems of equations involving higher degree polynomials of many variables. The ancient Greeks studied curves of higher degrees. And the ancient Chinese mathematical book, Nine Chapters on Arithmetic (10th century BC–2nd century BC), presented systems of linear equations in many variables and used row operations to produce the reduced row echelon form to solve a system of linear equations. This method is usually called Gaussian row operations now. We adopt the usual name for it…

The main theorem in this chapter is the theorem on the Smith Normal Form for a matrix over a P.I.D.

The following sections are included:

• Further Discussion of the Determinant. Volume in ${\mathbb{R}}^n$
• Cauchy–Binet Formula
• Tensor Product and Exterior Product
• Dual Space. Application to Physics

The following sections are included:

• Inner Product
• Gram–Schmidt Theorem. Hooke’s Law. Least Square Approximation
• Hilbert Space
• Orthogonal Complement
• Adjoint and Self-Adjoint Operator. Uncertainty Principle
• Minimal Problems and Positive-Definite Matrices. Many Variable Functions
• Unitary Operator
• Spectrum Theorem

The following sections are included:

• Bilinear Forms. Sylvester’s Law of Inertia
• Groups Preserving Bilinear Forms. Special Relativity
• Singular Value Decompositions. Its Applications

The following sections are included:

• Numerical Model, Finite Difference Method and Finite Element Method
• Power Method. Perron–Frobenius Theorem. Google Search
• QR Method. Find the Eigenvalues
• Approximate Solutions to a System of Linear Equations
• Linear Programming

The following section is included:

• Index