Linear Algebra

Table of Contents

1. Frontmatter

2. Chapter 1. Matrices and Linear Equations

   Henry Helson

   Abstract

   Algebra is one of the great branches of mathematics, with analysis, geometry, number theory and foundations. Its origins are in antiquity. During the nineteenth century, like analysis, it blossomed in a remarkable way, and in this century it continues vigorous growth in both pure and applied directions.

3. Chapter 2. Vector Spaces

   Henry Helson

   Abstract

   Now we are ready to build a house of abstraction into which the furniture of the last chapter will fit.

4. Chapter 3. Linear Transformations

   Henry Helson

   Abstract

   A linear transformation from a vector space V to another (or the same) vector space W is a function F from V into W such that

   $$F\left( {X + Y} \right) = F\left( X \right) + F\left( Y \right),\quad F\left( {tX} \right) = tF\left( X \right)$$

   (1.1)

   for all X, Y in V and all scalars t. (V and W can be either real or complex vector spaces, but both of the same type.)

5. Chapter 4. Determinants

   Henry Helson

   Abstract

   A linear transformation from R2 to R is a real function F of two real variables such that

   $$F(x + x',y + y') = F(x,y) + F(x' + y'),\quad F(tx,ty) = tF(x,y)$$

   (1.1)

   for all real numbers x, y, x′, y′, t. The only such functions are

   $$F(x,y) = ax + by$$

   (1.2)

   where a, b are real constants.

6. Chapter 5. Reduction of Matrices

   Henry Helson

   Abstract

   The central problem of linear algebra is to understand the structure of linear transformations in geometrical terms. If the matrix of a given linear transformation in some basis has a special form, for example if it is diagonal, then we can say something about how the transformation acts. Equivalently, if a given matrix is similar to a matrix of a special kind, then the linear transformation it determines (in any basis) will be special in a corresponding way.

7. Chapter 6. Matrix Factorings

   Henry Helson

   Abstract

   Many matrix problems are solved by writing a given matrix as a product of factors with special properties. The possibility of a factoring is a mathematical result, but usually one also needs to find the factors explicitly. A good algorithm for doing so is one that minimizes the calculation required, and introduces as little rounding error as possible. This chapter presents four factoring theorems that are important for applications, with suggestions for carrying out the factoring. It is intended to be an introduction to a course in Numerical Analysis.

8. Backmatter