Matrices and Linear Algebra

Hans Schneider, George Phillip Barker

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486660141
• eBook ISBN: 9780486139302
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

Linear algebra is one of the central disciplines in mathematics. A student of pure mathematics must know linear algebra if he is to continue with modern algebra or functional analysis. Much of the mathematics now taught to engineers and physicists requires it.
This well-known and highly regarded text makes the subject accessible to undergraduates with little mathematical experience. Written mainly for students in physics, engineering, economics, and other fields outside mathematics, the book gives the theory of matrices and applications to systems of linear equations, as well as many related topics such as determinants, eigenvalues, and differential equations.
Table of Contents:
l. The Algebra of Matrices
2. Linear Equations
3. Vector Spaces
4. Determinants
5. Linear Transformations
6. Eigenvalues and Eigenvectors
7. Inner Product Spaces
8. Applications to Differential Equations
For the second edition, the authors added several exercises in each chapter and a brand new section in Chapter 7. The exercises, which are both true-false and multiple-choice, will enable the student to test his grasp of the definitions and theorems in the chapter. The new section in Chapter 7 illustrates the geometric content of Sylvester's Theorem by means of conic sections and quadric surfaces. 6 line drawings. lndex. Two prefaces. Answer section.

chapter 1

The Algebra
of Matrices

1. MATRICES: DEFINITIONS

This book is entitled Matrices and Linear Algebra, and "linear" will be the most common mathematical term used here. This word has many related meanings, and now we shall explain what a linear equation is. An example of a linear equation is 3x₁ + 2x₂ = 5, where x₁ and x₂ are unknowns. In general an equation is called linear if it is of the form

where x₁, …, x_n are unknowns, and a₁, …, a_n and b are numbers. Observe that in a linear equation, products such as x₁x₂ or x₃⁴ and more general functions such as sin x₁ do not occur.

In elementary books a pair of equations such as

is called a pair of simultaneous equations. We shall call such a pair a system of linear equations. Of course we may have more than three unknowns and more than two equations. Thus the most general system of m equations in n unknowns is

The a_ij are numbers, and the subscript (i, j) denotes that a_ij is the coefficient of x_j in the ith equation.

So far we have not explained what the coefficients of the unknowns are, but we have taken for granted that they are real numbers such as 2,

, or π. The coefficients could just as well be complex numbers. This case would arise if we considered the equations

Note that a real number is also a complex number (with imaginary part zero), but sometimes it is important to consider either all real numbers or all complex numbers. We shall denote the real numbers by

and the ...

Table of contents

• Cover
• Title Page
• Copyright Page
• Preface to the Second Edition
• Preface to the First Edition
• Contents
• 1 The Algebra Of Matrices
• 2 Linear Equations
• 3 Vector Spaces
• 4 Determinants
• 5 Linear Transformations
• 6 Eigenvalues And Eigenvectors
• 7 Inner Product Spaces
• 8 Applications To Differential Equations
• Answers
• Symbols
• Index