Introduction to Matrices and Vectors

Introduction to Matrices and Vectors

by Jacob T. Schwartz

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Realizing that matrices can be a confusing topic for the beginner, the author of this undergraduate text has made things as clear as possible by focusing on problem solving, rather than elaborate proofs. He begins with the basics, offering students a solid foundation for the later chapters on using special matrices to solve problems.The first three chapters present the basics of matrices, including addition, multiplication, and division, and give solid practice in the areas of matrix manipulation where the laws of algebra do not apply. In later chapters the author introduces vectors and shows how to use vectors and matrices to solve systems of linear equations. He also covers special matrices — including complex numbers, quaternion matrices, and matrices with complex entries — and transpose matrices; the trace of a matrix; the cross product of matrices; eigenvalues and eigenvectors; and infinite series of matrices. Exercises at the end of each section give students further practice in problem solving.
Prerequisites include a background in algebra, and in the later chapters, a knowledge of solid geometry. The book was designed as an introductory text for college freshmen and sophomores, but selected chapters can also be used to supplement advanced high school classes. Professionals who need a better understanding or review of the subject will also benefit from this concise guide.

Product Details

ISBN-13: 9780486143705
Publisher: Dover Publications
Publication date: 05/23/2012
Series: Dover Books on Mathematics
Sold by: Barnes & Noble
Format: NOOK Book
Pages: 192
File size: 7 MB

Table of Contents

Chapter 1. Definition, Equality, and Addition of Matrices
1. Introduction
2. What Matrices Are
3. Equality of Matrices. Specification of Matrices. The Zero Matrix
4. Addition of Matrices
5. Addition of Matrices Continued
6. Addition of Matrices Concluded
7. Numerical Multiples of Matrices
Chapter 2. Multiplication of Matrices
1. A Problem Arising in Business Management
2. Formal Definition of Multiplication
3. A Surprising Property of Matrix Multiplication
4. Matrix Multiplication Continued
5. The Unit Matrix
6. The Laws of Matrix Multiplication
7. Powers of Matrices. Laws of Exponents
8. Multiplication of Matrices by Matrices and Multiplication of Matrices by Numbers
9. Polynomials in a Matrix
Chapter 3. Division of Matrices
1. Introduction
2. Using an Equation Satisfied by a Matrix
3. The Least Equation Satisfied by a Matrix
4. Using the Least Equation Satisfied by a Matrix to Solve the Problem of Reciprocals
5. Proof of the Uniqueness of the Least Equation Satisfied by a Matrix
6. Two Theorems about Reciprocals
Chapter 4. Vectors and Linear Equations
1. Definition of Vectors. Notation and Properties
2. Vectors and Directed Segments in the Plane
3. Vectors and Directed Segments in Three-dimensional Space
4. Geometric Applications of the Algebra of Vectors
5. Distances, Cosines, and Vectors
6. More about the Lengths of Vectors
7. Expressing Systems of Linear Equations in Terms of Vectors and Matrices. Solving by Using Reciprocal Matrices
8. Solving Systems of Linear Equations by the Method of Elimination
Chapter 5. Special Matrices of Particular Interest
1. The Complex Numbers as Real Matrices
2. The Quaternion Matrices
3. Matrices with Complex Entries
Chapter 6. More Algebra of Matrices and Vectors
1. The Transpose Matrix
2. The Trace of a Matrix
3. The Cross Product of Matrices
4. 3 x 3 Skew Matrices and Vectors of Size 3
5. Geometry of the Cap Product
Chapter 7. Eigenvalues and Eigenvectors
1. Matrix Reciprocals and Vectors
2. Eigenvalues
Chapter 8. Infinite Series of Matrices
1. The Geometric Series
2. The Size of the Entries in the Powers of A
3. The Exponential Series

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