Elementary Linear Algebra 1st Edition

James R. Kirkwood
Visit to download the full and correct content document:
https://textbookfull.com/product/elementary-linear-algebra-1st-edition-james-r-kirkwoo
d/
More products digital (pdf, epub, mobi) instant
download maybe you interests ...

Elementary Linear Algebra Ron Larson

https://textbookfull.com/product/elementary-linear-algebra-ron-
larson/

Elementary Linear Algebra 12th Edition Howard Anton

https://textbookfull.com/product/elementary-linear-algebra-12th-
edition-howard-anton/

Elementary linear algebra : Metric version Eighth

Edition Ron Larson

https://textbookfull.com/product/elementary-linear-algebra-
metric-version-eighth-edition-ron-larson/

Instructor's Solutions Manual for Elementary Linear

Algebra with Applications, 9th Edition Bernard Kolman

https://textbookfull.com/product/instructors-solutions-manual-
for-elementary-linear-algebra-with-applications-9th-edition-
bernard-kolman/
Linear Algebra Seymour Lipschutz

https://textbookfull.com/product/linear-algebra-seymour-
lipschutz/

Basic Linear Algebra Jürgen Müller

https://textbookfull.com/product/basic-linear-algebra-jurgen-
muller/

Linear Algebra M. Thamban Nair

https://textbookfull.com/product/linear-algebra-m-thamban-nair/

Linear Algebra G. Shanker Rao

https://textbookfull.com/product/linear-algebra-g-shanker-rao/

Extending The Linear Model With R Second Edition

Julian James Faraway

https://textbookfull.com/product/extending-the-linear-model-with-
r-second-edition-julian-james-faraway/
Elementary Linear Algebra
 TEXTBOOKS in MATHEMATICS

Series Editors: Al Boggess and Ken Rosen

PUBLISHED TITLES
ABSTRACT ALGEBRA: A GENTLE INTRODUCTION
Gary L. Mullen and James A. Sellers
ABSTRACT ALGEBRA: AN INTERACTIVE APPROACH, SECOND EDITION
William Paulsen
ABSTRACT ALGEBRA: AN INQUIRY-BASED APPROACH
Jonathan K. Hodge, Steven Schlicker, and Ted Sundstrom
ADVANCED LINEAR ALGEBRA
Hugo Woerdeman
ADVANCED LINEAR ALGEBRA
Nicholas Loehr
ADVANCED LINEAR ALGEBRA, SECOND EDITION
Bruce Cooperstein
APPLIED ABSTRACT ALGEBRA WITH MAPLE™ AND MATLAB®, THIRD EDITION
Richard Klima, Neil Sigmon, and Ernest Stitzinger
APPLIED DIFFERENTIAL EQUATIONS: THE PRIMARY COURSE
Vladimir Dobrushkin
APPLIED DIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS
Vladimir Dobrushkin
APPLIED FUNCTIONAL ANALYSIS, THIRD EDITION
J. Tinsley Oden and Leszek Demkowicz
A BRIDGE TO HIGHER MATHEMATICS
Valentin Deaconu and Donald C. Pfaff
COMPUTATIONAL MATHEMATICS: MODELS, METHODS, AND ANALYSIS WITH
MATLAB® AND MPI, SECOND EDITION
Robert E. White
A CONCRETE INTRODUCTION TO REAL ANALYSIS, SECOND EDITION
Robert Carlson
A COURSE IN DIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS,
SECOND EDITION
Stephen A. Wirkus, Randall J. Swift, and Ryan Szypowski
A COURSE IN ORDINARY DIFFERENTIAL EQUATIONS, SECOND EDITION
Stephen A. Wirkus and Randall J. Swift
PUBLISHED TITLES CONTINUED
DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE, SECOND
EDITION
Steven G. Krantz

DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE WITH

BOUNDARY VALUE PROBLEMS
Steven G. Krantz

DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES,

THIRD EDITION
George F. Simmons

DIFFERENTIAL EQUATIONS WITH MATLAB®: EXPLORATION, APPLICATIONS,

AND THEORY
Mark A. McKibben and Micah D. Webster

DISCOVERING GROUP THEORY: A TRANSITION TO ADVANCED MATHEMATICS

Tony Barnard and Hugh Neill
DISCRETE MATHEMATICS, SECOND EDITION
Kevin Ferland
ELEMENTARY DIFFERENTIAL EQUATIONS
Kenneth Kuttler
ELEMENTARY NUMBER THEORY
James S. Kraft and Lawrence C. Washington
THE ELEMENTS OF ADVANCED MATHEMATICS: FOURTH EDITION
Steven G. Krantz
ESSENTIALS OF MATHEMATICAL THINKING
Steven G. Krantz
EXPLORING CALCULUS: LABS AND PROJECTS WITH MATHEMATICA®
Crista Arangala and Karen A. Yokley
EXPLORING GEOMETRY, SECOND EDITION
Michael Hvidsten
EXPLORING LINEAR ALGEBRA: LABS AND PROJECTS WITH MATHEMATICA®
Crista Arangala
EXPLORING THE INFINITE: AN INTRODUCTION TO PROOF AND ANALYSIS
Jennifer Brooks
GRAPHS & DIGRAPHS, SIXTH EDITION
Gary Chartrand, Linda Lesniak, and Ping Zhang
INTRODUCTION TO ABSTRACT ALGEBRA, SECOND EDITION
Jonathan D. H. Smith

INTRODUCTION TO ANALYSIS
Corey M. Dunn
PUBLISHED TITLES CONTINUED
INTRODUCTION TO MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED
MATHEMATICS, SECOND EDITION
Charles E. Roberts, Jr.
INTRODUCTION TO NUMBER THEORY, SECOND EDITION
Marty Erickson, Anthony Vazzana, and David Garth
INVITATION TO LINEAR ALGEBRA
David C. Mello
LINEAR ALGEBRA, GEOMETRY AND TRANSFORMATION
Bruce Solomon
MATHEMATICAL MODELING: BRANCHING BEYOND CALCULUS
Crista Arangala, Nicholas S. Luke, and Karen A. Yokley
MATHEMATICAL MODELING FOR BUSINESS ANALYTICS
William P. Fox
MATHEMATICAL MODELLING WITH CASE STUDIES: USING MAPLE™ AND
MATLAB®, THIRD EDITION
B. Barnes and G. R. Fulford
MATHEMATICS IN GAMES, SPORTS, AND GAMBLING–THE GAMES PEOPLE PLAY,
SECOND EDITION
Ronald J. Gould
THE MATHEMATICS OF GAMES: AN INTRODUCTION TO PROBABILITY
David G. Taylor
A MATLAB® COMPANION TO COMPLEX VARIABLES
A. David Wunsch
MEASURE AND INTEGRAL: AN INTRODUCTION TO REAL ANALYSIS, SECOND
EDITION
Richard L. Wheeden
MEASURE THEORY AND FINE PROPERTIES OF FUNCTIONS, REVISED EDITION
Lawrence C. Evans and Ronald F. Gariepy
NUMERICAL ANALYSIS FOR ENGINEERS: METHODS AND APPLICATIONS, SECOND
EDITION
Bilal Ayyub and Richard H. McCuen
ORDINARY DIFFERENTIAL EQUATIONS: AN INTRODUCTION TO THE
FUNDAMENTALS
Kenneth B. Howell
PRINCIPLES OF FOURIER ANALYSIS, SECOND EDITION
Kenneth B. Howell
REAL ANALYSIS AND FOUNDATIONS, FOURTH EDITION
Steven G. Krantz
PUBLISHED TITLES CONTINUED
RISK ANALYSIS IN ENGINEERING AND ECONOMICS, SECOND EDITION
Bilal M. Ayyub
SPORTS MATH: AN INTRODUCTORY COURSE IN THE MATHEMATICS OF SPORTS
SCIENCE AND SPORTS ANALYTICS
Roland B. Minton
A TOUR THROUGH GRAPH THEORY
Karin R. Saoub
TRANSITION TO ANALYSIS WITH PROOF
Steven G. Krantz
TRANSFORMATIONAL PLANE GEOMETRY
Ronald N. Umble and Zhigang Han
UNDERSTANDING REAL ANALYSIS, SECOND EDITION
Paul Zorn
Elementary Linear Algebra

James R. Kirkwood and Bessie H. Kirkwood

MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the
accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products
does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular
use of the MATLAB® software.

CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742

© 2018 by Taylor & Francis Group, LLC

CRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S. Government works

Printed on acid-free paper

International Standard Book Number-13: 978-1-4987-7846-6 (Hardback)

This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been
made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity
of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright
holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this
form has not been obtained. If any copyright material has not been acknowledged, please write and let us know so we
may rectify in any future reprint.

Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized
in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying,
microfilming, and recording, or in any information storage or retrieval system, without written permission from the
publishers.

For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://
www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923,
978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For
organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for
identification and explanation without intent to infringe.

Visit the Taylor & Francis Web site at

http://www.taylorandfrancis.com

and the CRC Press Web site at

http://www.crcpress.com
This book is dedicated to Katie and Elizabeth, who have brought indescribable joy, meaning, and

happiness into our lives

Contents

Preface..............................................................................................................................................xv

1. Matrices......................................................................................................................................1
1.1 Matrix Arithmetic...........................................................................................................1
1.1.1 Matrix Arithmetic.............................................................................................3
1.1.1.1 Matrix Addition.................................................................................3
1.1.1.2 Scalar Multiplication.........................................................................3
1.1.1.3 Matrix Multiplication....................................................................... 3
Exercises.....................................................................................................................................7
1.2 The Algebra of Matrices................................................................................................9
1.2.1 Properties of Matrix Addition, Scalar Multiplication,
and Matrix Multiplication...............................................................................9
1.2.2 The Identity Matrix.........................................................................................12
1.2.3 The Inverse of a Square Matrix.....................................................................12
1.2.4 Determinants...................................................................................................14
1.2.5 Elementary Matrices.......................................................................................17
1.2.6 Matrices That Interchange Two Rows of a Matrix.....................................18
1.2.7 Multiplying a Row of a Matrix by a Constant............................................18
1.2.8 Adding a Multiple of One Row to Another Row.......................................19
1.2.9 Computing the Inverse of a Matrix..............................................................19
1.2.10 The Transpose of a Matrix.............................................................................22
Exercises...................................................................................................................................23
1.3 The LU Decomposition of a Square Matrix (Optional)...........................................29
Exercises...................................................................................................................................32

2. Systems of Linear Equations...............................................................................................33

2.1 Basic Definitions...........................................................................................................33
Exercises...................................................................................................................................34
2.2 Solving Systems of Linear Equations (Gaussian Elimination)..............................34
2.2.1 Solving Systems of Linear Equations...........................................................36
2.2.2 Using Technology to Accomplish Gaussian Elimination..........................40
Exercises...................................................................................................................................40
2.3 Equivalent Systems of Linear Equations..................................................................41
2.3.1 Row Reduced Form of a Matrix....................................................................43
Exercises...................................................................................................................................44
2.4 Expressing the Solution of a System of Linear Equations......................................44
2.4.1 Systems of Linear Equations That Have No Solutions............................. 44
2.4.2 Systems of Linear Equations That Have Exactly One Solution...............45
2.4.3 Systems of Linear Equations That Have Infinitely Many Solutions.......45
2.4.4 Application of Linear Systems to Curve Fitting.........................................47
Exercises...................................................................................................................................51
2.5 Expressing Systems of Linear Equations in Other Forms......................................53
2.5.1 Representing a System of Linear Equations as a Vector Equation..........53
2.5.2 Equivalence of a System of Linear Equations and a Matrix Equation....55
Exercises...................................................................................................................................58
xi
xii Contents

2.6 Applications..................................................................................................................60
2.6.1 Flow Problems.................................................................................................60
2.6.2 Example: Kirchoff’s Laws..............................................................................61
2.6.3 Balancing Chemical Equations Using Linear Algebra..............................63
Exercises...................................................................................................................................64
2.6.4 Markov Chains................................................................................................65
Exercises...................................................................................................................................71

3. Vector Spaces..........................................................................................................................73
3.1 Vector Spaces in ℝn.......................................................................................................73
Exercises...................................................................................................................................80
3.2 Axioms and Examples of Vector Spaces...................................................................81
3.2.1 Some Examples of Sets That Are Not Vector Spaces.................................83
3.2.2 Additional Properties of Vector Spaces.......................................................84
Exercises...................................................................................................................................86
3.3 Subspaces of a Vector Space.......................................................................................87
Exercises...................................................................................................................................91
3.4 Spanning Sets, Linearly Independent Sets and Bases.............................................92
Exercises.................................................................................................................................101
3.5 Converting a Set of Vectors to a Basis.....................................................................103
3.6 A Synopsis of Sections 3.3.4 and 3.3.5.....................................................................108
Exercises.................................................................................................................................109
3.7 Change of Bases.......................................................................................................... 110
Exercises................................................................................................................................. 117
3.8 Null Space, Row Space, and Column Space of a Matrix...................................... 118
Exercises.................................................................................................................................128
3.9 Sums and Direct Sums of Vector Spaces (Optional)..............................................130
Exercises.................................................................................................................................134

4. Linear Transformations......................................................................................................137
4.1 Properties of a Linear Transformation....................................................................137
4.1.1 Null Space and Range (Image) of a Linear Transformation...................143
Exercises.................................................................................................................................144
4.2 Representing a Linear Transformation...................................................................146
4.2.1 Representation of a Linear Transformation in the Usual Basis..............147
Exercises.................................................................................................................................152
4.3 Finding the Representation of a Linear Operator
with respect to Different Bases.................................................................................154
Exercises.................................................................................................................................156
4.4 Composition (Multiplication) of Linear Transformations....................................158
Exercises.................................................................................................................................160

5. Eigenvalues and Eigenvectors...........................................................................................163

5.1 Determining Eigenvalues and Eigenvectors..........................................................163
5.1.1 Finding the Eigenvectors after the Eigenvalues Have Been Found......165
Exercises.................................................................................................................................170
Contents xiii

5.2 Diagonalizing a Matrix..............................................................................................172

5.2.1 Algebraic and Geometric Multiplicities of an Eigenvalue......................173
5.2.2 Diagonalizing a Matrix................................................................................173
Exercises.................................................................................................................................176
5.3 Similar Matrices..........................................................................................................177
Exercises.................................................................................................................................182
5.4 Eigenvalues and Eigenvectors in Systems of Differential Equations.................183
Exercises.................................................................................................................................188

6. Inner Product Spaces...........................................................................................................191

6.1 Some Facts about Complex Numbers.....................................................................191
Exercises.................................................................................................................................192
6.2 Inner Product Spaces.................................................................................................194
Exercises.................................................................................................................................196
6.3 Orthogonality.............................................................................................................197
Exercises.................................................................................................................................200
6.4 The Gram–Schmidt Process......................................................................................201
6.4.1 Algorithm for the Gram–Schmidt Process................................................201
Exercises.................................................................................................................................205
6.5 Representation of a Linear Transformation on Inner Product Spaces (Optional)....207
Exercises.................................................................................................................................209
6.6 Orthogonal Complement..........................................................................................210
Exercises.................................................................................................................................212
6.7 Four Subspaces Associated with a Matrix (Optional)..........................................213
6.8 Projections...................................................................................................................214
Exercises.................................................................................................................................221
6.9 Least-Squares Estimates in Statistics (Optional)....................................................222
Exercise...................................................................................................................................225
6.10 Weighted Inner Products (Optional).......................................................................225
Reference������������������������������������������������������������������������������������������������������������������������������� 226

7. Linear Functionals, Dual Spaces, and Adjoint Operators...........................................227

7.1 Linear Functionals.....................................................................................................227
7.1.1 The Second Dual of a Vector Space (Optional)............................................231
Exercises.................................................................................................................................232
7.2 The Adjoint of a Linear Operator............................................................................233
7.2.1 The Adjoint Operator...................................................................................233
7.2.2 Adjoint on Weighted Inner Product Spaces (Optional)...........................239
Exercises.................................................................................................................................240
7.3 The Spectral Theorem................................................................................................242
Exercises.................................................................................................................................248

Appendix A: A Brief Guide to MATLAB®.............................................................................249

Appendix B: An Introduction to R...........................................................................................263
Appendix C: Downloading R to Your Computer..................................................................279
Answers to Selected Exercises..................................................................................................281
Index���������������������������������������������������������������������������������������������������������������������������������������������301
Preface

In beginning to write a linear algebra text, a question that surfaces even before the first
keystroke takes place is who is the audience and what do we want to accomplish. The
answer to this is more complex than in several other areas of mathematics because of the
breadth of potential users. The book was written with the idea that a typical student would
be one who has completed two semesters of calculus but who has not taken courses that
emphasize abstract mathematics. The goals of the text are to present the topics that are
traditionally covered in a first-level linear algebra course so that the computational meth-
ods become deeply ingrained and to make the intuition of the theory as transparent as
possible.
Many disciplines, including statistics, economics, environmental science, engineering,
and computer science, use linear algebra extensively. The sophistication of the applications
of linear algebra in these areas can vary greatly. Students intending to study mathematics
at the graduate level, and many others, would benefit from having a second course in lin-
ear algebra at the undergraduate level.
Some of the computations that we feel are especially important are matrix computations,
solving systems of linear equations, representing a linear transformation in standard bases,
finding eigenvectors, and diagonalizing matrices. Of less emphasis are topics such as con-
verting the representation of vectors and linear transformations between nonstandard
bases and converting a set of vectors to a basis by expanding or contracting the set.
In some cases, the intuition of a proof is more transparent if an example is presented
before a theorem is articulated or if the proof of a theorem is given using concrete cases
rather than an abstract argument. For example, by using three vectors instead of
n vectors.
There are places in Chapters 4 through 7 where there are results that are important
because of their applications, but the theory behind the result is time consuming and is
more advanced than a typical student in a first exposure would be expected to digest. In
such cases, the reader is alerted that the result is given later in the section and omitting the
derivation will not compromise the usefulness of the results. Two specific examples of this
are the projection matrix and the Gram–Schmidt process.
The exercises were designed to span a range from simple computations to fairly direct
abstract exercises.
We expect that most users will want to make extensive use of a computer algebra system
for computations. While there are several systems available, MATLAB® is the choice of
many, and we have included a tutorial for MATLAB in the appendix. Because of the exten-
sive use of the program R by statisticians, a tutorial for that program is also included.

A Note about Mathematical Proofs

As a text for a first course in linear algebra, this book has a major focus on demonstrating
facts and techniques of linear systems that will be invaluable in higher mathematics
and fields that use higher mathematics. This entails developing many computational tools.

xv
xvi Preface

A first course in linear algebra also serves as a bridge to mathematics courses that are pri-
marily theoretical in nature and, as such, necessitates understanding and, in some cases,
developing mathematical proofs. This is a learning process that can be frustrating.
We offer some suggestions that may ease the transition.
First, the most important requirement for abstract mathematics is knowing the impor-
tant definitions well. These are analogous to the formulas of computational mathematics.
A casual perusal will not be sufficient.
You will often encounter the question “Is it true that …?” Here you must show that a
result holds in all cases allowed by the hypotheses or find one example where the claim
does not hold. Such an example is called a counter example. If you are going to prove that
a result always holds, it is helpful (perhaps necessary) to believe that it holds. One way to
establish an opinion for its validity is to work through some examples. This may highlight
the essential properties that are crucial to a proof. If the result is not valid, this will often
yield a counter example. Simple examples are usually the best. In linear algebra, this usu-
ally means working with small dimensional objects, such as 2 × 2 matrices.
In the text, we will sometimes give the ideas of a proof rather than a complete proof
when we feel that “what is really going on” is obscured by the abstraction, for example, if
the idea of a proof seems clear by examining 2 × 2 matrices but would be difficult to follow
for n × n matrices.
In constructing a mathematical proof, there are two processes involved. One is to develop
the ideas of the proof, and the second is to present those ideas in appropriate language.
There are different methods of proof that can be broadly divided into direct methods and
indirect methods. An example of an indirect method is proof by contradiction. The contra-
positive of “if statement A is true, then statement B is true” is “if statement B is false, then
statement A is false.” Proving the contrapositive of an “if-then” statement is logically
equivalent to proving the original statement. The contrapositive of

“If x = 4, then x is an integer”

is

“If x is not an integer, then x ¹ 4.”

The converse of an “if-then” statement reverses the hypothesis and the conclusion. The
converse of “if statement A is true, then statement B is true” is “ if statement B is true, then
statement A is true.”
Unlike the contrapositive, the converse is not logically equivalent to the original state-
ment. The converse of

“If x = 4, then x is an integer”

is

“If x is an integer , then x = 4.”

Here the original statement is true, but the converse is false. You will sometimes have a
problem of the type “show ‘A if and only if B.’” In this case, you must prove

“If A, then B”
Preface xvii

and

“If B, then A”

are both true.

In constructing a proof, it is necessary to completely understand what is given and what
is to be shown. Sometimes, what is given can be phrased in different ways, and one way
may be more helpful than others. Often in trying to construct a proof, it is worthwhile writ-
ing explicitly “this is what I know” and “this is what I want to show” to get started.
In the proofs presented in the text, we make a concerted effort to present the intuition
behind the proof and why something is “the reasonable thing to do.”

MATLAB® is a registered trademark of The MathWorks, Inc. For product information,

please contact:

The MathWorks, Inc.

3 Apple Hill Drive
Natick, MA, 01760-2098 USA
Tel: 508-647-7000
Fax: 508-647-7001
E-mail: info@mathworks.com
Web: www.mathworks.com
1
Matrices

Linear algebra is the branch of mathematics that deals with vector spaces and linear
transformations.
For someone just beginning their study of linear algebra, that is probably a meaningless
statement. It does, however, convey the idea that there are two concepts in our study that
will be of utmost importance, namely, vector spaces and linear transformations. A primary
tool in our study of these topics will be matrices. In this chapter, we give the rules that
govern matrix algebra.

1.1 Matrix Arithmetic
The term “matrix” was first used in a mathematical sense by James Sylvester in 1850 to
denote a rectangular array of numbers from which determinants could be formed. The
ideas of using arrays to solve systems of linear equations go as far back as the Chinese from
300 BC to 200 AD.
A matrix is a rectangular array of numbers. Matrices are the most common way of
r­epresenting vectors and linear transformations and play a central role in nearly every
computation in linear algebra. The size of a matrix is described by its dimensions, that is,
the number of rows and columns in the matrix, with the number of rows given first. A 3 × 2
(three by two) matrix has three rows and two columns. Matrices are usually represented by
uppercase letters.
The matrix A below is an example of a 3 × 2 matrix.

æ0 -5 ö
ç ÷
A=ç 6 3 ÷.
ç -1 3 ÷ø
è

We will often want to represent a matrix in an abstract form. The example below is
typical.

æ a11 a12  a1n ö

ç ÷
a21 a22  a2 n ÷
A=ç
ç     ÷
çç ÷
è am1 am 2  amn ÷ø

1
2 Elementary Linear Algebra

It is customary to denote the numbers that make up the matrix, called the entries of the
matrix, as the lowercase version of the letter that names the matrix, with two subscripts.
The subscripts denote the position of the entry. The entry aij occupies the ith row and jth
column of the matrix A. The notation (A)ij is also common, depending on the setting.
Two matrices A and B are equal if they have the same dimensions and aij = bij for every
i and j.

Definition

A square matrix is a matrix that has the same number of rows as columns, that is, an
n × n matrix. If A is a square matrix, the entries a11, a22, …, ann make up the main diagonal of
A. The trace of a square matrix is the sum of the entries on the main diagonal.
A square matrix is a diagonal matrix if the only nonzero entries of A are on the main
diagonal. A square matrix is upper (lower) triangular if the only nonzero entries are above
(below) or on the main diagonal. A matrix that consists of either a single row or a single
column is called a vector. The matrix

æ 1ö
ç ÷
ç 5÷
ç6÷
è ø

is an example of a column vector, and

( 4, -1, 0, 8 )

is an example of a row vector.

In Figure 1.1, matrix (a) is a diagonal matrix, but the others are not.

2 0 0 0 0 0 0 0
0 0 0 0 0 0 5 0
0 0 –1 0 0 3 0 0
0 0 0 5 2 0 0 0
(a) (b)

0 1 0 0
0 3 0 0 1 0 0
0 0 7 0 0 2 0
0 0 0 2
(c) (d)

FIGURE 1.1
Matrix (a) is diagonal, the others are not.
Matrices 3

1.1.1 Matrix Arithmetic
We now define three arithmetic operations on matrices: matrix addition, scalar multiplica-
tion, and matrix multiplication.

1.1.1.1 Matrix Addition
In order to add two matrices, the matrices must have the same dimensions, and then one
simply adds the corresponding entries. For example

æ7 -1 0 ö æ -2 -3 4ö æ 7 - 2 -1 - 3 0 + 4ö æ5 -4 4ö
ç ÷+ç ÷=ç ÷=ç ÷
è -2 6 6ø è 3 5 0 ø è -2 + 3 6+5 6 + 0ø è1 11 6ø

but

æ1 0ö
æ7 -1 0ö ç ÷
ç ÷+ 2 -1 ÷
è -2 6 6 ø çç
è0 5 ÷ø

is not defined.
To be more formal—and to begin to get used to the abstract notation—we could express
this idea as

( A + B )ij = Aij + Bij .

1.1.1.2 Scalar Multiplication
A scalar is a number. Scalar multiplication means multiplying a matrix by a number
and is accomplished by multiplying every entry in the matrix by the scalar. For
example

æ1 -4 ö æ 3 -12 ö
3ç ÷=ç ÷.
è2 0 ø è6 0 ø

1.1.1.3 Matrix Multiplication
Matrix multiplication is not as intuitive as the two prior operations. A fact that we will
demonstrate later is that every linear transformation can be expressed as multiplication by
a matrix. The definition of matrix multiplication is based in part on the idea that composi-
tion of two linear transformations should be expressed as the product of the matrices that
represent the linear transformations.
We describe matrix multiplication by stages.
4 Elementary Linear Algebra

Case 1

Multiplying a matrix consisting of one row by a matrix consisting of one column, where
the row matrix is on the left and the column matrix is on the right. The process is

æ b1 ö
ç ÷
( a1, a2 ,… , an ) ç b2 ÷ = a1b1 + a2b2 +  + anbn .
ç ÷
çç ÷÷
è bn ø

This can be accomplished only if there are the same number of entries in the row matrix as
in the column matrix. Notice that the product of a (1 × n) matrix and a (n × 1) matrix is a
number that, in this case, we will describe as a (1 × 1) matrix.

Case 2

Multiplying a matrix consisting of one row by a matrix consisting of more than one
­column, where the row matrix is on the left and the column matrix is on the right. Two
examples are

æ b1 c1 ö
( a1, ¼ , an ) çç  ÷
 ÷ = ( a1b1 +  + anbn , a1c1 +  + ancn )
ç bn cn ÷ø
è

æ b1 c1 d1 ö
( a1, ¼ , an ) çç  
÷
 ÷ = ( a1b1 +  + anbn , a1c1 +  + ancn , a1d1 +  + an dn ) .
ç bn cn dn ÷ø
è

Thus, the product of a (1 × n) matrix and a (n × k) matrix is a (1 × k) matrix.

It may help to visualize this as applying Case 1 multiple times.

Case 3

Multiplying a matrix consisting of more than one row by a matrix consisting of one
c­ olumn, where the row matrix is on the left and the column matrix is on the right. Two
examples of this are

æ c1 ö
æ a1 , ¼ , an ö ç ÷ æ a1c1 +  + ancn ö
( 2 ´ n ) ´ ( n ´ 1) ç ÷ç  ÷ = ç ÷ 2´1
è b1 , ¼ , bn ø ç c ÷ è b1c1 +  + bncn ø
è nø

æ a1 , ¼ , an ö æ d1 ö æ a1d1 +  + an dn ö
( 3 ´ n ) ´ ( n ´ 1) çç b1 , ¼ , bn ÷÷ çç  ÷÷ = çç b1d1 +  + bndn ÷÷ 3 ´ 1.
ç c1 , ¼ , cn ÷ ç dn ÷ ç c1d1 +  + cn dn ÷
è øè ø è ø
Matrices 5

Case 4

Multiplying any two “compatible” matrices. In order to be compatible for multiplication,

the matrix on the left must have the same number of entries in a row as there are entries in
a column of the matrix on the right. One example is

æ c1 c2 ö
æ a1 a2 a3 ö ç ÷ æ a1c1 + a2d1 + a3 e1 a1c2 + a2d2 + a3 e2 ö
( 2 ´ 3) ´ ( 3 ´ 2) ç ÷ d1 d2 ÷ = ç ÷,
è b1 b2 b3 ø çç b1c1 + b2d1 + b3 e1 b1c2 + b2d2 + b3 e2 ø
è e1 e2 ÷ø è

which is a (2 × 2) matrix.
We describe a formula for computing the product of two compatible matrices.
Suppose that A is an m × k matrix and B is a k × n matrix, say

æ a11 a12  a1k ö æ b11 b12  b1n ö

ç ÷ ç ÷
a21 a22  a2 k ÷ b21 b22  b2 n ÷
A=ç B=ç .
ç     ÷ ç     ÷
çç ÷ çç ÷÷
è am1 am 2  amk ÷ø è bk 1 bk 2  bkn ø

The product matrix AB is m × n and the formula for the i, j entry of the matrix AB is
k

( AB )ij = åailblj .
l =1

One can think of this as multiplying the ith row of A by the jth column of B.

1.1.1.3.1 A Summary of Matrix Multiplication

If A is an m × k matrix and B is an s × n matrix, then AB is defined if and only if k = s. If k = s,
then AB is an m × n matrix whose i, j entry is
k

( AB )ij = åais bsj .

s=1

Another way to visualize matrix multiplication that is useful is to consider the product AB
in the following way: Let b̂i be the vector that is the ith column of the matrix B, so that if

æ b11 b12 b13 ö

ç ÷
B = ç b21 b22 b23 ÷
ç b31 b32 b33 ÷ø
è

then

æ b11 ö æ b12 ö æ b13 ö

ç ÷ ç ÷ ç ÷
bˆ1 = ç b21 ÷ , bˆ2 = ç b22 ÷ , bˆ3 = ç b23 ÷ .
ç b31 ÷ ç b32 ÷ ç b33 ÷
è ø è ø è ø
6 Elementary Linear Algebra

We can then think of the matrix B as a “vector of vectors”

B = ébˆ1 bˆ2 bˆ3 ù

ë û

and we have

AB = é Abˆ1 Abˆ2 Abˆ3 ù .

ë û

This will often be used in the text.

We verify this in the case

æ a11 a12 a13 ö

A=ç ÷.
è a21 a22 a23 ø

We have

æ b11 ö
æ a11 a12 a13 ö ç ÷ æ a11b11 + a12 b21 + a13 b31 ö
Abˆ1 = ç ÷ b21 = ç ÷
è a21 a22 a23 ø çç ÷÷ è a21b11 + a22 b21 + a23 b31 ø
b
è ø31

æ b12 ö
æ a11 a12 a13 ö ç ÷ æ a11b12 + a12 b22 + a13 b32 ö
Abˆ2 = ç ÷ b22 = ç ÷
è a21 a22 a23 ø çç ÷÷ è a21b12 + a22 b22 + a23 b32 ø
è b32 ø

æ b13 ö
æ a11 a12 a13 ö ç ÷ æ a11b13 + a12 b23 + a13 b33 ö
Abˆ3 = ç ÷ b23 = ç ÷
è a21 a22 a23 ø çç ÷÷ è a21b13 + a22 b23 + a23 b33 ø
b
è ø33

so

a b +a b +a b a11b12 + a12 b22 + a13 b32 a11b13 + a12 b23 + a13 b33 ö
é Abˆ1 Abˆ2 Abˆ3 ù = æç 11 11 12 21 13 31 ÷.
ë û è a21b11 + a22 b21 + a23 b31 a21b12 + a22 b22 + a23 b32 a21b13 + a22 b23 + a23 b33 ø

Also,

æ b11 b12 b13 ö

æ a11 a12 a13 ö ç ÷
AB = ç ÷ b21 b22 b23 ÷
è a21 a22 a23 ø çç
è b31 b32 b33 ÷ø

æ a11b11 + a12b21 + a13b31 a11b12 + a12b22 + a13b32 a11b13 + a12b23 + a13b33 ö

=ç ÷.
è a21b11 + a22b21 + a23b31 a21b12 + a22b22 + a23b32 a21b13 + a22b23 + a23b33 ø

As a final cautionary remark, we note that if A is an m × k matrix and B is a k × n matrix,

then AB is defined and is an m × n matrix, but BA is not defined unless m = n.
Matrices 7

Exercises
1. Suppose

æ -1 2ö æ4 -2 2ö
æ2 0ö ç ÷ æ 1 5 2ö ç ÷ æ1 7 ö
A=ç ÷ , B = ç -3 2÷, C = ç ÷, D = ç0 1 5 ÷, E = ç ÷.
è3 1ø ç 1 4 ÷ø è -3 0 6ø ç2 0 -3 ÷ø è4 -3 ø
è è

Where the computations are possible, find

(a) 3A − 2E
(b) BC
(c) CB + 4D
(d) C − 4B
(e) BA
(f) EC
2. Compute Abˆ1 , Abˆ2 , é Abˆ1 , Abˆ2 ù , and AB for
ë û

æ2 -1 ö æ1 0 -1 ö
A=ç ÷, B = ç ÷.
è1 3ø è2 1 4ø

3. Find a 3 × 3 matrix A for which

æ x ö æ 4x - 2y ö
ç ÷ ç ÷
(a) A ç y ÷ = ç 3 z ÷
çz÷ ç 0 ÷
è ø è ø
æxö æ zö
ç ÷ ç ÷
(b) A ç y ÷ = ç y ÷
ç z÷ çx÷
è ø è ø
æxö æ x + y + z ö
ç ÷ ç ÷
(c) A ç y ÷ = ç 2x + 2 y + 2z ÷
ç z ÷ ç -x - y - z ÷
è ø è ø
4. Show that
æ a1 b1 ö æ x ö æ a1 ö æ b1 ö
(a) ç ÷ç ÷ = xç ÷ + y ç ÷
a
è 2 b2 øè øy a
è 2ø è b2 ø
æ a1 b1 c1 ö æ x ö æ a1 ö æ b1 ö æ c1 ö
ç ÷ç ÷ ç ÷ ç ÷ ç ÷
(b) ç a2 b2 c2 ÷ ç y ÷ = x ç a2 ÷ + y ç b2 ÷ + z ç c2 ÷
ç a3 b3 c3 ÷ø çè z ÷ø ç a3 ÷ ç b3 ÷ ç c3 ÷
è è ø è ø è ø

5. If AB is a 5 × 7 matrix, how many columns does B have?

8 Elementary Linear Algebra

6. (a) If A and B are 2 × 2 matrices, show that

( AB )11 + ( AB )22 = ( BA )11 + ( BA )22.

(b) If A and B are n × n matrices, compare the trace of AB and the trace of BA.
7. If A and B are matrices so that AB is defined, then

jth column of AB = A éë jth column of B ùû

ith row of AB = éë ith row of A ùû B.

Demonstrate this principle by finding the second row and third column of AB for

æ3 -1 0ö æ2 5 -3 ö
ç ÷ ç ÷
A = ç1 5 -2 ÷ B = ç -1 0 4 ÷.
ç4 3 1 ÷ø ç4 2 -5 ÷ø
è è

8. Construct a 5 × 5 matrix, not all of whose entries are zero, that has the following
properties:
(a) Aij = 0 if i < j.
(b) Aij = 0 if i ≥ j.
(c) Aij = 0 if |i − j| = 1.
9. Compute

æe fö æe fö
( a, b ) ç g ÷
hø
and ( c, d ) ç g ÷
hø
è è

and compare with

æa b öæ e fö
ç ÷ç ÷.
èc døè g hø

Can you make any conjectures based on this example?

10. If AB is a 3 × 3 matrix, show that BA is defined.
11. If

æ3 8 0ö æ2 1 1ö
A=ç ÷, B = ç ÷
è4 -1 -2 ø è3 -1 7ø

find the matrix C for which

æ0 1 2ö
3 A - 5B + 2C = ç ÷.
è2 -3 4ø
Matrices 9

FIGURE 1.2
A simple graph that is not directed.

12. A graph is a set of vertexes and a set of edges between some of the vertices. If the
graph is simple, then there is no edge from a vertex to itself. If the graph is not
directed, then an edge from vertex i to vertex j is also an edge from j to vertex i.
Figure 1.2 shows a simple graph that is not directed.
Associated with each graph is a matrix A defined by

ì1 if there is an edge from vertex i to vertex j

ï
Aij = í .
ï0 otherwise
î

(a) If there are n vertices, what are the dimensions of the associated matrix A?
What else can you conclude about A?
We say there is a path of length k from vertex i to vertex j if it is possible to
traverse from vertex i to vertex j by traversing across exactly k edges.
(Traversing the same edge more than once is allowed.)
(b) How could the matrix A be used to determine whether there is a path from i to
vertex j in exactly 2 steps? In exactly k steps?
(c) How could you tell how many paths there are from i to vertex j in exactly
k steps?
(i) For the graph in Figure 1.2
(ii) Construct the associated graph.
(d) Use the associated matrix to determine whether it is possible to go from ver-
tex 1 to vertex 4 by traversing across exactly 3 edges using the matrix you
constructed.

1.2 The Algebra of Matrices

1.2.1 Properties of Matrix Addition, Scalar Multiplication, and Matrix Multiplication
Matrix addition, scalar multiplication, and matrix multiplication obey the rules of the next
theorem.
10 Elementary Linear Algebra

Theorem 1

If the sizes of the matrices A, B, and C are so that the operations can be performed, and
α and β are scalars, then

(a) A+B=B+A
(b) (A + B) + C = A + (B + C)
(c) (AB)C = A(BC)
(d) A(B + C) = AB + AC
(e) (B + C)A = BA + CA
(f) α(A + B) = αA + αB
(g) (α + β)A = αA + βA
(h) α(βA) = (αβ)A
(i) α(AB) = (αA)B = A(αB)

This theorem is simply saying that these rules of combining matrices obey the usual laws
of arithmetic. When we more closely examine rules that govern matrix multiplication, we
will see that some of these rules are different from what we might hope.
The proofs of these results are not deep but can be tedious. We demonstrate a proof of
part (c).
Suppose that A is a k × l matrix, B is an l × m matrix, and C is an m × n matrix. Then AB
is a k × m matrix, (AB)C is a k × n matrix, BC is an l × n matrix, and A(BC) is a k × n matrix.
Thus, (AB)C and A(BC) have the same dimensions. To complete the proof, we must show
that the entries of the two matrices are the same. That is, we must show

éë( AB ) C ùû = éë A ( BC ) ùû for r = 1, ¼ , k ; s = 1, ¼ , n.
rs rs

We have

m m
é l
ù m
é l
ù m
é l
ù
éë( AB ) C ùû =
rs å ( AB)rj Cjs = å êåAriBij ú Cjs = å êå ( AriBij ) Cjs ú = å êåAri ( BijCjs )ú
j =1 j =1 êë i =1 úû j =1 êë i =1 úû j =1 êë i =1 úû

l é m ù l é m ù l

= åå ê
ê
Ari ( BijC js ) ú =
ú å Ari ê
ê å ( BijCjs )úú = åAri ( BC )is = éë A ( BC )ùûrs .
i =1 ë j =1 û i =1 ë j =1 û i =1

We have defined the arithmetic operations on matrices. Matrices together with these oper-
ations form a mathematical structure. To expand this structure, we need to introduce some
additional matrices.
In arithmetic, the numbers 0 and 1 have special significance. The number 0 is important
because it is the additive identity, that is,

0+a= a

for every number a. The number 1 is important because it is the multiplicative identity, that is,

1´ a = a

for every number a.

Another random document with
no related content on Scribd:
supply. But people who are not used to Croton water, or Schuylkill, or
Cochituate water do not seem to care much for this. They are glad
enough to get water at all.
In other parts of Europe, and in this continent too, the water is
carried about by men and women.
In the opposite picture you may see how some of these water-
carriers supply their customers.
In Malaga a jaunty Spaniard with a cigar in his mouth, and two jars
of water hanging from his shoulders and arms, walks up and down
the streets selling the precious fluid at so much a quart or a pint.
In Pongo the water is conveyed in a great leathern jar on the back
of a stout, bare-legged fellow who carries a long funnel, so that he
can pour the water into the pitcher and pails without taking his jar
from his shoulders.
In parts of Mexico the jars are fastened to broad straps which pass
around the water-carrier’s head, while in Guaymas, the carrier has
no load at all himself, but puts two great skins of water on the back
of a little donkey.
The French water-carrier has a stick on his shoulders with a pail of
water on each end; and when one shoulder is tired he can shift his
load to the other, which is, perhaps, the next best thing to having a
donkey.
But the water-carriers of Arabia and Egypt, who very often are
women, are the most graceful and in some respects the most
sensible of all. They carry their jar of water on their heads.
As this makes it necessary for them to keep themselves very
erect, it gives them fine, straight figures, and a graceful walk. The
disadvantage of their plan is that they cannot carry very much water
at a time.
Carrying water on the head reminds me of a little negro girl I once
saw in the South. This girl had been to a spring to get a pail of water.
The pail was so large and the girl was so small that she had a hard
time of it as she staggered along, holding the handle of the pail with
both hands, and with
the greatest difficulty
keeping it from
touching the ground.
I pitied the poor
little creature, for her
load was a great deal
too heavy for her.
But at length she
reached a stump of a
tree, and by great
efforts she got the
pail on the top of this.
Then she stooped
down and managed
to slide the pail from
the stump to the
crown of her head.
Then she stood up.
She was all right! She
seemed to forget that
she had a load, and
skipped away as if
she had nothing
heavier on her head
than a spring bonnet.
She did not go
directly to the house
where she was to WATER CARRIERS.
carry the water, but 1. Water Carrier of 4. Water Carrier of
trotted over to where Malaga. Guaymas.
some children were 2. Pongo. 5. French Water Carrier.
playing, and began
3. Water Carrier of 6. Arabian Woman at the
running around in a
perfectly easy and Mexico. Fountain.
unconcerned way, VARIOUS METHODS OF CARRYING WATER.
not appearing to think
at all of her pail. But
she did not spill a drop of the water.
The Southern negroes are very dexterous in this matter of carrying
things on their heads.
On some of the water-melon plantations there may sometimes be
seen long lines of men walking from the fields to the boats which are
to be loaded with these melons, and each man carries a water-
melon under each arm and one on his head.
Sometimes one of these men will drop a water-melon from under
his arm, but no one ever drops one from his head.
Such a thing would be considered a disgrace.
I think it is likely that very few of us would ever have a pail of water
or a water-melon, if we were obliged to carry either of them very far
on our heads.
 THE LAND OF THE WHITE ELEPHANT.

This is the kingdom of Siam in Southern Asia. It has this name

because the white elephant is the national emblem, and is
represented on the Siamese flag, as the eagle is on the American
flag.
Siam is a very pleasant country to live in, and a good many
Europeans have from time to time had their homes there, so that the
Siamese, who seem to be a teachable people, have learned a great
deal from them, and have copied some of their ways. The
missionaries, too, have done very much to improve the manners and
customs of the Siamese. But still they retain many of their old
customs, and the result is a queer sort of mixture—in some things
the people show themselves to be manly and intelligent, and, in
others, they appear very ignorant, and degraded.
For instance, a fine-looking Siamese gentleman will be standing
by your side, conversing with you. He is a nobleman of the country,
dressed in silk and diamonds. He will talk with you about foreign
countries, perhaps about books, and you will be astonished at his
information, and will regard him as a very superior man; as, indeed,
he is. But suddenly, he will go down on his hands and knees, right in
the dust. What has happened? The King has appeared upon the
scene! If he wishes to speak to the king this nobleman will crawl up
to him on all fours, and, as long as he is in the king’s presence, if it is
for two or three hours, he remains in this degrading position. The
king may be a very well-informed, and a kind-hearted man, but it has
never occurred to him that this old custom of his country is ridiculous
and disgraceful.
The Siamese are very fond of ornament. On the next page is a
portrait of one of the little princes of the royal family.
His silken suit is covered with gold and silver embroidery, and with
rows of precious stones. I hope that high pointed affair he has upon
his head is not very heavy. It would be a very inconvenient thing to
take off when he
wished to make a
bow to a lady. But
then he never does
wish to make a bow
to a lady. That is a
piece of good
manners that no
Siamese boy is ever
taught.
When this young
prince has his meals
his attendants crouch
before him on their
hands and knees.
When he wishes
anything they crawl
towards him with the
articles. To stand
erect in his presence
would be an unheard
of impropriety.
When he goes out
for an airing he rides
upon an elephant.
Perhaps you think
that would not be a
very agreeable way
of traveling, but there
you are mistaken.
The motion of an
elephant is very easy,
A SIAMESE PRINCE.
and pleasant to the
rider; and it is a much
more intelligent animal than the horse, and quite as gentle, and
docile. A little child can lead a well-trained, tame elephant. The
disadvantage of this kind of steed is that when it does take it into its
great head to behave badly, it is sometimes very difficult to control,
for it is exceedingly strong, and capable of doing a great deal of
mischief. But these trained elephants of the Siamese seldom get into
tantrums.
When the young prince takes his ride he has, at least, one
attendant to walk by the elephant, and keep things all straight. The
prince sits in a little ornamental tower on the beast’s back. From this
lofty seat he gets a good view of the surrounding country.
Near Ayuthia, in Siam, there is a large stockade, into which the
king’s elephants are driven once a year, and the finest ones are
selected for use during the ensuing year. This stockade is made of
posts of teak wood, driven firmly into the ground, a few feet apart. In
the middle of the enclosure, thus made, is a small tower-like house,
built on poles, and surrounded by strong stakes. In this are the men
who are to secure the animals after they are chosen.
The king and his nobles are on a raised platform near the
stockade; and they select those of the animals that have been driven
into the enclosure, that they consider the most desirable.
The fine points in an elephant are these: a color approaching to
white or red, black nails on the toes, and tails that have not been
injured. Elephants are so fond of fighting each other that it is a rare
thing to find one in a herd that has not lost some portion of its tail in a
battle.
 ELEPHANT HERD.
It occasionally happens, when a hunting party is out, that a white
elephant is captured. This is considered a very fortunate
circumstance, as the possession of a white elephant by the king is
supposed to bring prosperity upon the whole kingdom. The fortunate
finder of this precious animal is received with great honor upon his
return to court, and is magnificently rewarded.
The elephant is placed in a large enclosure, and treated with great
distinction. It is caparisoned with cloth of gold; and is fed with all the
dainties that elephants like. Rings of gold are placed on its tusks,
and a diadem on its head. When it is sick the court physician attends
it, the priests pray for it; and when it dies the whole kingdom mourns.
Of late years the people of Siam have grown less superstitious,
and do not pay as many honors to white elephants as they did while
in an entirely uncivilized state. But they still retain the white elephant
on their flag as the emblem of their country.
 CURIOSITIES OF VEGETABLE LIFE.

It is not necessary to travel in order to find a great many curious

things in vegetable growth. They lie around us everywhere.
We will find a great deal to surprise us, if we study the habits of
the trees and plants about us. Some have very peculiar methods of
growth; some go to sleep, and wake up at regular hours; some set
little traps for catching insects; some often change the colors of their
flowers; and many other curious ways they have.
But men, who travel in various countries, and study the vegetable
growth of all climates, meet with very marvelous things indeed. Let
us follow them about the world for awhile. But we will have to travel
very swiftly, and to skip from one country to another, and back again,
perhaps, with great haste.
We will first look at some trees that surprise us by their size.
On Mount Etna, in Sicily, there is a famous chestnut tree. It stands
on one of the lower slopes of the mountain, so that it is often visited.
There are quite a number of huge chestnut trees in that
neighborhood, each of which has a distinctive name. But the
“Chestnut of a Hundred Horses” is much larger than any of the
others. It is a very, very old tree, and the people who now live near it,
are not sure how it first got its name. Some say it was named many
years ago by a Spanish queen because its thick wide branches once
sheltered her and her party of a hundred horsemen, from the rain.
Others say it is so called because a hundred horses can be
sheltered within, and around it. It is now the home of a shepherd,
who has built a hut for himself, and a fold for his sheep, within the
hollow of the tree.
The trunk measures 190 feet around. It looks as if there were
several trees growing together, but it is known to be all one tree.
 THE GREAT CHESTNUT TREE OF MOUNT ETNA.
In the centre of a graveyard, in the village of Allouville, in
Normandy, there stands an oak that is over nine-hundred years old.
Near the ground its trunk measures thirty feet in circumference. Two
hundred years ago it was fitted up as a little chapel, and is used for
that purpose to this day.
The tree is hollow, as are all these very old trees. The lower part of
this hollow is lined with wood, carefully plastered and wainscoted.
This is the chapel. Above it, is a second story, and in this room lives
a solitary man—a hermit. Above, in the branches, is a belfry,
ornamented with a cross.
In another part of France, there is an oak that is known to be
fifteen hundred years old. It also is hollow, but every year, like the
Hundred Horse Chestnut, and the Allouville Oak, it covers itself with
thick and luxuriant foliage. The circumference of this oak is over 80
feet, and its branches spread over a circumference of 380 feet.
 The inside of this tree is used for a dining hall by pleasure parties.
A circular bench has been cut out of the wood, and a dozen persons
can sit comfortably around the table. The room has a glass door, and
a window. Beautiful ferns and mosses spring out of the sides of the
tree, and decorate this hall.
Not far from Smyrna, in Asia Minor, there is a very old, and a very
large Plane tree. It has three stems from one root. These join into
one trunk at the height of about twenty feet from the ground, thus
forming a gateway. The main road runs right through this gateway,
and cavalcades of horsemen, and camel riders, and vehicles pass
under this singular arch.
There is another Plane tree in the island of Cos, which is almost
as well known as that of Smyrna. Its spreading branches cover the
whole of a large square of a city, and are so heavy that the old trunk
is not able to bear their weight. The inhabitants of the city, proud of
their tree, and anxious to keep it, have built columns of marble under
the branches to support them.
 CHAPEL OAK OF ALLOUVILLE.
We have been visiting single trees of different kinds. There is a
family of trees, every member of which attains a great size. The
Baobabs ought to be large, for they require 800 years to attain their
full growth. They then measure, usually, from 70 to 77 feet in girth.
Enormous branches spread out from the central stem, each one of
which is a respectable size for a tree. So these trees give a splendid
shade, covering a space of ground 300 feet in circumference. They
do not grow very tall. These trees are found chiefly in Africa.
The Baobabs remind us of another marvel of the vegetable
kingdom—the great age of some trees. We have mentioned one 900
years old, and another 1,500. The ages of these are known because
the trees have been traced back historically for that length of time.
But these are babies in age by the side of the Baobabs. Botanists
calculate the ages of the largest Baobabs to be over 5,000 years!
We must remember this is calculation, not certainty. It is positively
known, however, that some of them are several hundred years old;
and there are olive trees known to have lived over a thousand years.
This is a very good old age when we consider that man seldom lives
to a hundred.

THE GIANT CANDLE.

The height to which some tropical plants that are not trees grow is
surprising. You have, no doubt, all seen the queer fleshy-leaved
cactus that is cultivated in green-houses. This plant has no woody
stem, and yet one species of it grows to the height of twenty and
thirty feet.
It is called the giant candle, and it certainly looks like one.
The great height of this plant is the more surprising because it
grows right out of crevices of rocks where no soil can be seen, and
pushes its straight, fleshy stem up into the air without anything to
shelter it from the furious winds that often sweep over the country.
But it braves the winds, and grows, and grows; and every year puts
forth its large white flowers, and bears upon its queer stalk a very
savory fruit.
The largest palms rise to a height of 45 feet, and more or
sometimes as high as 70 feet—before putting out a single branch.
Then they spread out a great plume of feathery leaves. The wax
palm of the Andes is said to grow to the height of 200 feet.
In New Zealand, the ferns, that are here so fragile and delicate,
grow so high that they look, at a little distance like small palms.
 The tallest trees in
the world are the
giant trees of
California. These are
from 300 to 350 feet
high; and one that
was cut down was
450 feet high.
We cannot very
well speak of the
height of vines, but
there are species of
these that grow to a
very great length.
Sometimes one stalk
will stretch itself out
to the length of 150
yards. Some of the
sea-weeds, thrown
upon the shore,
measure 500 yards in
a single strip.
We must again go
to the tropics if we
want to find large
leaves. You have, no
doubt, heard of the
great water-lily—the
Victoria Regia, but I
think you would open
TREE FERNS. your eyes if you could
get a sight of a river
filled with these
floating mammoths of leaves. They are from four to eight feet in
diameter. They are, in shape, almost circular, and are turned up a
little around the edges.
 The strength of these leaves is almost as surprising as their size.
The fibres are large, and are so woven together on the under side
that they form a solid framework to support the upper part of the leaf,
which is of a beautiful green, and thick, and velvety. The water-fowl
choose these leaves often for sleeping places in the hot nights, and
find it very pleasant, doubtless, to be thus rocked on the cool water,
in a velvet bed, that will not sink.
Palms have ridges running lengthwise of their leaves, as you may
see by examining a palm leaf fan. One ridged leaf of the talipot palm,
when well grown, it is said, will shelter forty persons. This sounds
like a traveler’s story, but single leaves of this tree have been
brought to this country, and one of them will completely cover the
ceiling of a good-sized room.
 THE GUTTA PERCHA TREE.
The leaves of the cocoanut palm are several feet long.
The juices of many vegetables possess very singular properties.
The cow-tree, is so called because the sap that flows from it closely
resembles milk, and is used as such by the natives.
 A substance, with which you are very familiar, India-rubber, is
nothing but the sap of a tree. A very useful sap it is; and, when
hardened, and properly prepared, is impervious to water; and shoes,
coats, coverlets, &c., are made of it. Put through another hardening
process it takes a fine polish, and is made up into beautiful
ornaments and useful articles.
India-rubber trees are found in South America, the East Indies,
and in some parts of Africa.
In the same countries there grows a beautiful tree, which yields a
thick sap, called gutta-percha. This is similar, in substance, to India-
rubber, and is used for a great variety of purposes, from making
lifeboats to knife-handles.
Sugar and molasses are made from the juice of a cane; maple
sugar from the sap of a tree that grows plentifully in all our mountain
districts. You think it wonderful that milk can be taken from a tree. Is
it not quite as strange to find sugar there? I suppose you will reply to
this that the milk is ready prepared, while we have to make the
sugar. That is true, but we add nothing to what we take from the tree.
We simply apply heat to the sap, and behold the sugar! The chief
reason, I think, why we are not surprised at this sugar tree is that we
are familiar with it. The inhabitants of Central America do not see
anything strange in the fact of a tree bearing milk.
But there is a tree that produces sugar ready-made. This is the
manna tree of Sicily. The sap hardens on the trunk and branches
into sugary particles, which are scraped off with wooden knives. This
kind of sugar is used principally in medicine. It is insipid in taste.
A species of Laurel in India contains camphor in all its juices.
Break up twigs, stems, and leaves of this tree, and heat them, and
the liquid that comes from them will soon condense into camphor
gum.
The seeds of one kind of palm produce a fine oil; and the stalks of
another give us wax, of which candles are made. This wax forms on
the outside of the stalk, at the places where the leaves join the stem.
From another palm is extracted a juice that when exposed to the air
for a short time, becomes wine.
In arid deserts, and in unwholesome marshes plants flourish
luxuriantly, the leaves of which contain pure, sweet, freshwater,
always ready for the thirsty traveler.
So far we have only spoken of the wholesome juices of plants and
trees, but a large number are full of deadly poisons. Many of these
grow in our own woods and fields. Some of the poisons have been
utilized in medicines, for it has been found that, properly prepared,
and given in small quantities, they can frequently arrest disease.
Such is opium, which we get from the poppy, strychnine, and prussic
acid. All of these are terrible poisons, but when administered by a
physician in small doses, they relieve pain, and help to cure disease.
Other vegetable poisons seem to be only destructive. If a man
should, ignorantly take refuge under a Machineel tree from a shower,
and remain there for any length of time after the rain began to drip
upon him, he would suddenly discover that blisters were breaking
out on his skin, and that sharp pains were running through his limbs,
and he might well feel thankful if he escaped without a fit of sickness.
This, at least is the story that the savages tell who live in the regions
where the Machineel grows. It is probable that they exaggerate the
facts, but they will none of them go near a tree of this species if they
can help it; and there is no doubt that it is very poisonous.
 THE DEADLY UPAS.
The most celebrated of the poison-trees is the Upas, which grows
in several tropical countries, but chiefly in the island of Java. The
accounts given of the Machineel tree are nothing compared to the
wonderful stories told of the Upas. No plant, not even grass, will
grow under one of these trees, or anywhere near it. A drop of water
falling from a leaf on any one beneath it, will produce inflammation.
Whoever walks under one of these trees bare-headed, must expect
to lose all his hair. To stay under it for a short time will cause
sickness. To sleep under it is certain death. Birds fly over the tree
with great difficulty, and if one should chance to alight upon it, woe
be to him! Instantly he drops down dead. The wild beasts know the
fatal tree, and shun it, but, if one venture beneath its shadow, he
never comes forth again, but leaves his bones there.
These stories which the natives really believe, they told to
travelers, and, for a long time they were supposed to be true. One
thing which caused these accounts to be so readily credited was the
condition of things in a valley in Java, where these trees were found
in abundance.
The natives of Java told the Europeans of a wonderful valley that
they called the Valley of Death, because the air was so poisoned
with the noxious vapors of the Upas trees that no animal ever went
through the valley. It dropped dead before the short journey was
completed, and the ground was strewn with the bones of creatures
that had perished there. The natives were willing to conduct their
visitors to a hill that overlooked this valley at a safe distance. When
they arrived at this hill, there, sure enough, was the valley, Upas
trees, dead grass, bones, and all, just as the savages had described
it. This settled the matter for a great many years in regard to the
death dealing Upas tree. Other travelers to Java, rode up this hill,
looked at the Valley of Death, shuddered, and rode down again.
At last there arrived on the hill-top a man who made up his mind
he would ride through this valley! And so he did! Everybody said he
was going to his death, but he was resolved to solve the mystery of
the valley. He rode into it, and through it, and back again, and came
out alive and well! Nevertheless he found the skeletons of wild
animals, and of birds strewn the whole length of the valley.
It was a mystery, but it was all cleared up afterwards. The valley
was fatal to all animals except man. But the Upas tree had nothing to
do with it. The valley was filled with carbonic acid gas to the height of
a couple of feet from the ground.