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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Praise for the Third Edition
"This volume is ground-breaking in terms of mathematical texts in that it does not teach from a detached perspective, but instead, looks to show students that competent mathematicians bring an intuitive understanding to the subject rather than just a master of applications."
—Electric Review
Learn foundational and advanced topics in linear algebra with this concise and approachable resource
A comprehensive introduction, Linear Algebra: Ideas and Applications, Fifth Edition provides a discussion of the theory and applications of linear algebra that blends abstract and computational concepts. With a focus on the development of mathematical intuition, the book emphasizes the need to understand both the applications of a particular technique and the mathematical ideas underlying the technique.
The book introduces each new concept in the context of explicit numerical examples, which allows the abstract concepts to grow organically out of the necessity to solve specific problems. The intuitive discussions are consistently followed by rigorous statements of results and proofs. Linear Algebra: Ideas and Applications, Fifth Edition also features:
- A new application section on section on Google’s Page Rank Algorithm.
- A new application section on pricing long term health insurance at a Continuing Care Retirement Community (CCRC).
- Many other illuminating applications of linear algebra with self-study questions for additional study.
- End-of-chapter summaries and sections with true-false questions to aid readers with further comprehension of the presented material
- Numerous computer exercises throughout using MATLAB code
Linear Algebra: Ideas and Applications, Fifth Edition is an excellent undergraduate-level textbook for one or two semester undergraduate courses in mathematics, science, computer science, and engineering. With an emphasis on intuition development, the book is also an ideal self-study reference.
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Table of Contents
Cover
LINEAR ALGEBRA
Copyright
PREFACE
FEATURES OF THE TEXT
ACKNOWLEDGMENTS
ABOUT THE COMPANION WEBSITE
CHAPTER 1: SYSTEMS OF LINEAR EQUATIONS
1.1 THE VECTOR SPACE OF
m × n
MATRICES
1.2 SYSTEMS
1.3 GAUSSIAN ELIMINATION
1.4 COLUMN SPACE AND NULLSPACE
Notes
CHAPTER 2: LINEAR INDEPENDENCE AND DIMENSION
2.1 THE TEST FOR LINEAR INDEPENDENCE
2.2 DIMENSION
2.3 ROW SPACE AND THE RANK‐NULLITY THEOREM
Notes
CHAPTER 3: LINEAR TRANSFORMATIONS
3.1 THE LINEARITY PROPERTIES
3.2 MATRIX MULTIPLICATION (COMPOSITION)
3.3 INVERSES
3.4 The LU Factorization
3.5 THE MATRIX OF A LINEAR TRANSFORMATION
Note
CHAPTER 4: DETERMINANTS
4.1 DEFINITION OF THE DETERMINANT
4.2 REDUCTION AND DETERMINANTS
4.3 A FORMULA FOR INVERSES
CHAPTER 5: EIGENVECTORS AND EIGENVALUES
5.1 EIGENVECTORS
5.2 DIAGONALIZATION
5.3 COMPLEX EIGENVECTORS
CHAPTER 6: ORTHOGONALITY
6.1 THE SCALAR PRODUCT IN
6.2 PROJECTIONS: THE GRAM–SCHMIDT PROCESS
6.3 FOURIER SERIES: SCALAR PRODUCT SPACES
6.4 ORTHOGONAL MATRICES
6.5 LEAST SQUARES
6.6 QUADRATIC FORMS: ORTHOGONAL DIAGONALIZATION
6.7 THE SINGULAR VALUE DECOMPOSITION (SVD)
6.8 HERMITIAN SYMMETRIC AND UNITARY MATRICES
Note
CHAPTER 7: GENERALIZED EIGENVECTORS
7.1 GENERALIZED EIGENVECTORS
EXERCISES
7.2 CHAIN BASES
Note
CHAPTER 8: NUMERICAL TECHNIQUES
8.1 CONDITION NUMBER
8.2 COMPUTING EIGENVALUES
Notes
Index
ANSWERS AND HINTS
End User License Agreement
List of Tables
Chapter 1
TABLE 1.1 Profits: 2019
Chapter 3
TABLE 3.1 Demand Table
TABLE 3.2 Annual Fees: 2020–2025 (in thousands)
Chapter 5
TABLE 5.1
Gauss Auto Rental
List of Illustrations
Chapter 1
FIGURE 1.1 Coordinates in
.
FIGURE 1.2 Vector Algebra.
FIGURE 1.3 Example 1.1.
FIGURE 1.4 Coordinates in
.
FIGURE 1.5 Dependence in
.
FIGURE 1.6 Three vectors in
.
FIGURE 1.7 Route map.
FIGURE 1.8 Dominance in an extended family.
FIGURE 1.9 Exercise 1.1.
FIGURE 1.10 Exercise 1.43.
FIGURE 1.11 Only one solution.
FIGURE 1.12 Two planes.
FIGURE 1.13 Solution set is a line.
FIGURE 1.14 Three planes intersecting in a line.
FIGURE 1.15 No solution.
FIGURE 1.16 Pressure drop.
FIGURE 1.17 Voltage drop.
FIGURE 1.18 Example 1.7.
FIGURE 1.19 Hypothetical Flows
FIGURE 1.20 Exercise 1.51.a
FIGURE 1.21 Exercises 1.51.b and 1.51.c.
FIGURE 1.22 Exercise 1.52.
FIGURE 1.23 Exercise 1.53.
FIGURE 1.24 Two traffic patterns.
FIGURE 1.25 Exercise 1.79.
FIGURE 1.26 The column space.
FIGURE 1.27 Spaces in
.
FIGURE 1.28 Spaces in
.
FIGURE 1.29 The translation theorem.
Chapter 2
Figure 2.1 Co‐planar and non‐co‐planar vectors.
Figure 2.2 The standard basis in
.
Figure 2.3 Exercise 64.
Chapter 3
FIGURE 3.1 Rotating a triangle.
FIGURE 3.2 A shear.
FIGURE 3.3 Exercise 3.3.
FIGURE 3.4 Exercise 3.4.
FIGURE 3.5 Exercise 3.5.
FIGURE 3.6 A rotated ellipse.
FIGURE 3.7 A composite transformation.
FIGURE 3.8 Composite transformations.
FIGURE 3.9 My car picture.
FIGURE 3.10 Two‐step connections.
FIGURE 3.11 Links between web pages.
FIGURE 3.12 Triangle.
FIGURE 3.13 Natural and skewed coordinates in
.
FIGURE 3.14 Example 3.15.
FIGURE 3.15 The
coordinate is doubled.
FIGURE 3.16 An ellipse?
FIGURE 3.17 A hyperbola.
Chapter 4
FIGURE 4.1 The area of a parallelogram.
FIGURE 4.2 Proof of Proposition 4.2.
FIGURE 4.3 Volume of a parallelepiped.
Chapter 5
Figure 5.1 Addition of complex numbers.
Figure 5.2 Polar form.
Figure 5.3 Example 5.9.
Chapter 6
Figure 6.1 Distance in
.
Figure 6.2 Three orthogonal vectors.
Figure 6.3 Projections to line and planes.
Figure 6.4 Orthogonal complement.
Figure 6.5 The Grahm–Schmidt process.
Figure 6.6 A rasp.
Figure 6.7 A “pure” tone.
Figure 6.8 Four‐tone approximation.
Figure 6.9 Ten‐tone approximation.
Figure 6.10 “Close” functions.
Figure 6.11 Exercises 6.38 and 6.39.
Figure 6.12 Exercises 6.41 and 6.42.
Figure 6.13 Rotation and reflection.
Figure 6.14 A three‐dimensional rotation.
Figure 6.15 Orthogonal matrices in
.
Figure 6.16 Reflection in
.
Figure 6.17 Formula for Reflections.
Figure 6.18 Rotation in a plane.
Figure 6.19 Fitting a straight line.
Figure 6.20 Example 6.16.
Figure 6.21 A hyperboloid
2
Figure 1.1 1.60.a Assumed flows.
Figure 3.2 Exercise 3.1.
Figure 3.3 Exercise 3.7(b).
Guide
Cover Page
Title Page
Copyright
Table of Contents
Begin Reading
Index
WILEY END USER LICENSE AGREEMENT
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LINEAR ALGEBRA
Ideas and Applications
Fifth Edition
RICHARD C. PENNEY
Purdue University
This edition first published 2021
© 2021 John Wiley & Sons, Inc.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of Richard C. Penney to be identified as the author of this work has been asserted in accordance with law.
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Library of Congress Cataloging‐in‐Publication Data
Names: Penney, Richard C., author.
Title: Linear algebra : ideas and applications / Richard Cole Penney,
Purdue University.
Description: Fifth edition. | Hoboken : Wiley, 2021. | Includes index.
Identifiers: LCCN 2020029281 (print) | LCCN 2020029282 (ebook) | ISBN
9781119656920 (cloth) | ISBN 9781119656937 (adobe pdf) | ISBN
9781119656951 (epub)
Subjects: LCSH: Algebras, Linear–Textbooks.
Classification: LCC QA184.2 .P46 2021 (print) | LCC QA184.2 (ebook) | DDC
512/.5–dc23
LC record available at https://lccn.loc.gov/2020029281
LC ebook record available at https://lccn.loc.gov/2020029282
Cover Design: Wiley
Cover Image: © Flávio Tavares /Getty Images
PREFACE
I wrote this book because I have a deep conviction that mathematics is about ideas, not just formulas and algorithms, and not just theorems and proofs. The text covers the material usually found in a one or two semester linear algebra class. It is written, however, from the point of view that knowing why is just as important as knowing how.
To ensure that the readers see not only why a given fact is true but also why it is important, I have included a number of beautiful applications of linear algebra.
Most of my students seem to like this emphasis. For many, mathematics has always been a body of facts to be blindly accepted and used. The notion that they personally can decide mathematical truth or falsehood comes as a revelation. Promoting this level of understanding is the goal of this text.
RICHARD PENNEY
West Lafayette, Indiana
October 2020
FEATURES OF THE TEXT
Parallel Structure Most linear algebra texts begin with a long, basically computational, unit devoted to solving systems of equations and to matrix algebra and determinants. Students find this fairly easy and even somewhat familiar. But, after a third or more of the class has gone by peacefully, the boom falls. Suddenly, the students are asked to absorb abstract concept after abstract concept, one following on the heels of the other. They see little relationship between these concepts and the first part of the course or, for that matter, anything else they have ever studied. By the time the abstractions can be related to the first part of the course, many students are so lost that they neither see nor appreciate the connection.
This text is different. We have adopted a parallel mode of development in which the abstract concepts are introduced right from the beginning, along with the computational. Each abstraction is used to shed light on the computations. In this way, the students see the abstract part of the text as a natural outgrowth of the computational part. This is not the “mention it early but use it late” approach adopted by some texts. Once a concept such as linear independence or spanning is introduced, it becomes part of the vocabulary to be used frequently and repeatedly throughout the rest of the text.
The advantages of this kind of approach are immense. The parallel development allows us to introduce the abstractions at a slower pace, giving students a whole semester to absorb what was formerly compressed into two‐thirds of a semester. Students have time to fully absorb each new concept before taking on another. Since the concepts are utilized as they are introduced, the students see why each concept is necessary. The relation between theory and application is clear and immediate.
Gradual Development of Vector Spaces One special feature of this text is its treatment of the concept of vector space. Most modern texts tend to introduce this concept fairly late. We introduce it early because we need it early. Initially, however, we do not develop it in any depth. Rather, we slowly expand the reader's understanding by introducing new ideas as they are needed.
This approach has worked extremely well for us. When we used more traditional texts, we found ourselves spending endless amounts of time trying to explain what a vector space is. Students felt bewildered and confused, not seeing any point to what they were learning. With the gradual approach, on the other hand, the question of what a vector space is hardly arises. With this approach, the vector space concept seems to cause little difficulty for the students.
Treatment of Proofs It is essential that students learn to read and produce proofs. Proofs serve both to validate the results and to explain why they are true. For many students, however, linear algebra is their first proof‐based course. They come to the subject with neither the ability to read proofs nor an appreciation for their importance.
Many introductory linear algebra texts adopt a formal “definition–theorem–proof” format. In such a treatment, a student who has not yet developed the ability to read abstract mathematics can perceive both the statements of the theorems and their proofs (not to mention the definitions) as meaningless abstractions. They wind up reading only the examples in the hope of finding “patterns” that they can imitate to complete the assignments. In the end, such students wind up only mastering the computational techniques, since this is the only part of the course that has any meaning for them. In essence, we have taught them to be nothing more than slow, inaccurate computers.
Our point of view is different. This text is meant to be read by the student — all of it! We always work from the concrete to the abstract, never the opposite. We also make full use of geometric reasoning, where appropriate. We try to explain “analytically, algebraically, and geometrically.” We use carefully chosen examples to motivate both the definitions and theorems. Often, the essence of the proof is already contained in the example. Despite this, we give complete and rigorous student readable proofs of most results.
Conceptual Exercises Most texts at this level have exercises of two types: proofs and computations. We certainly do have a number of proofs and we definitely have lots of computations. The vast majority of the exercises are, however, “conceptual, but not theoretical.” That is, each exercise asks an explicit, concrete question which requires the student to think conceptually in order to provide an answer. Such questions are both more concrete and more manageable than proofs and thus are much better at demonstrating the concepts. They do not require that the student already have facility with abstractions. Rather, they act as a bridge between the abstract proofs and the explicit computations.
Applications Sections Doable as Self‐Study Applications can add depth and meaning to the study of linear algebra. Unfortunately, just covering the “essential” topics in the typical first course in linear algebra leaves little time for additional material, such as applications.
Many of our sections are followed by one or more application sections that use the material just studied. This material is designed to be read unaided by the student and thus may be assigned as outside reading. As an aid to this, we have provided two levels of exercises: self‐study questions and exercises. The self‐study questions are designed to be answerable with a minimal investment of time by anyone who has carefully read and digested the relevant material. The exercises require more thought and a greater depth of understanding. They would typically be used in parallel with classroom discussions.
We feel that, in general, there is great value in providing material that the students are responsible for learning on their own. Learning to read mathematics is the first step in learning to do mathematics. Furthermore, there is no way that we can ever teach everything the students need to know; we cannot even predict what they need to know. Ultimately, the most valuable skill we teach is the ability to teach oneself. The applications form a perfect vehicle for this in that an imperfect mastery of any given application will not impede the student's understanding of linear algebra.
Early Orthogonality Option We have designed the text so that the chapter on orthogonality, with the exception of the last three sections, may be done immediately following Chapter 3 rather than after the section on eigenvalues.
True–False Questions We have included true–false questions for most sections.
Chapter Summaries At the end of each chapter, there is a chapter summary that brings together major points from the chapter so students can get an overview of what they just learned.
Student Tested This text has been used over a period of years by numerous instructors at both Purdue and other universities nationwide. We have incorporated comments from instructors, reviewers, and (most important) students.
Technology Most sections of the text include a selection of computer exercises under the heading Computer Projects. Each exercise is specific to its section and is designed to support and extend the concepts discussed in that section.
These exercises have a special feature: They are designed to be “freestanding.” In principle, the instructor should not need to spend any class time at all discussing computing. Everything most students need to know is right there. In the text, the discussion is based on MATLAB. However, translations of the exercises into Maple are contained in the Student Resource Manual. They are also available free on the World Wide Web at http://www.math.purdue.edu/ ∼ rcp/LAText.
Meets LACSG Recommendations The Linear Algebra Curriculum Study Group (LACSG) recommended that the first class in linear algebra be a “student‐oriented” class that considers the “client disciplines” and that makes use of technology. The above comments make it clear that this text meets these recommendations. The LACSG also recommended that the first class be “matrix‐oriented.” We emphasize matrices throughout.
ACKNOWLEDGMENTS
The author would like to thank all of students, teachers, and reviewers of the text who have made comments, many of which have had significant impact on the final form of the third edition. In particular, he would like to thank Bill Dunbar and his students, Ken Rubenstein and Steven Givant for their particularly useful input.
ABOUT THE COMPANION WEBSITE
This book is accompanied by a companion website:
www.wiley.com/go/penney/linearalgebra5e
The website includes:
Instructors' Solutions
Manual Figures
CHAPTER 1SYSTEMS OF LINEAR EQUATIONS
1.1 THE VECTOR SPACE OF m × n MATRICES
It is difficult to go through life without seeing matrices. For example, the 2019 annual report of Acme Squidget might contain the Table 1.1, which shows how much profit (in millions of dollars) each branch made from the sale of each of the company's three varieties of squidgets in 2019.
If we were to enter this data into a computer, we might enter it as a rectangular array without labels. Such an array is called a matrix. The Acme profits for 2019 would be described by the following matrix. This matrix is a matrix (read “five by four”) in that it has five rows and four columns. We would also say that its “size” is . In general, a matrix has size if it has rows and columns.
Definition 1.1 The set of all matrices is denoted .
TABLE 1.1 Profits: 2019
Red
Blue
Green
Total
Kokomo
11.4
5.7
6.3
23.4
Philly
9.1
6.7
5.5
21.3
Oakland
14.3
6.2
5.0
25.5
Atlanta
10.0
7.1
5.7
22.8
Total
44.8
25.7
22.5
93.0
Each row of an matrix may be thought of as a matrix. The rows are numbered from top to bottom. Thus, the second row of the Acme profit matrix is the matrix
This matrix would be called the “profit vector” for the Philly branch. (In general, any matrix with only one row is called a row vector. For the sake of legibility, we usually separate the entries in row vectors by commas, as above.)
Similarly, a matrix with only one column is called a column vector. The columns are numbered from left to right. Thus, the third column of the Acme profit matrix is the column vector
This matrix is the “green squidget profit vector.”
If is a sequence of column vectors, then the matrix that has the as columns is denoted
Similarly, if is a sequence of row vectors, then the matrix that has the as rows is denoted
In general, if a matrix is denoted by an uppercase letter, such as , then the entry in the th row and th column may be denoted by either or , using the corresponding lowercase letter. We shall refer to as the “ entry of .” For example, for the matrix above, the entry is . Note that the row number comes first. Thus, the most general matrix is
We will also occasionally write “,” meaning that “ is the matrix whose entry is .”
At times, we want to take data from two tables, manipulate it in some manner, and display it in a third table. For example, suppose that we want to study the performance of each division of Acme Squidget over the two‐year period 2018–2019. We go back to the 2018 annual report, finding the 2018 profit matrix to be
If we want the totals for the two‐year period, we simply add the entries of this matrix to the corresponding entries from the 2018 profit matrix. Thus, for example, over the two‐year period, the Kokomo division made million dollars from selling blue squidgets. Totaling each pair of entries, we find the two‐year profit matrix to be
In matrix notation, we indicate that was obtained by summing corresponding entries of and by writing
In general, if and are matrices, then is the matrix defined by the formula
For example
Addition of matrices of different sizes is not defined.
What if, instead of totals for each division and each product, we wanted two‐year averages? We would simply multiply each entry of by . The notation for this is “.” Specifically,
In general, if is a number and is an matrix, we define
Hence,
There is also a notion of subtraction of matrices. In general, if and are matrices, then we define to be the matrix defined by the formula
Thus,
In linear algebra, the terms scalar and “number” mean essentially the same thing. Thus, multiplying a matrix by a real number is often called scalar multiplication.
The Space
We may think of a column vector as representing the point in the plane with coordinates . as in Figure 1.1. We may also think of as representing the vector from the point to —that is, as an arrow drawn from to . We will usually denote the set of matrices by when thought of as points in two‐dimensional space.
Like matrices, we can add pairs of vectors and multiply vectors by scalars. Specifically, if and are vectors with the same initial point, then is the diagonal of the parallelogram with sides and beginning at the same initial point (Figure 1.2b). For a positive scalar , is the vector with the same direction as that of , but with magnitude expanded (or contracted) by a factor of .
FIGURE 1.1 Coordinates in .
FIGURE 1.2 Vector Algebra.
Figure 1.2a shows that when two elements of are added, the corresponding vectors add as well. Similarly, multiplication of an element of by a scalar corresponds to multiplication of the corresponding vector by the same scalar. If , the direction of the vector is reversed and the vector is then expanded or contracted by a factor of (Figure 1.2b).
Example 1.1
Compute the sum of the vectors represented by and and draw a diagram illustrating your computation.
Solution. The sum is computed as follows:
The vectors (along with their sum) are plotted in Figure 1.3.
FIGURE 1.3Example 1.1.
FIGURE 1.4 Coordinates in .
Similarly, we may think of the matrix
as representing either the point in three‐dimensional space or the vector from to as in Figure 1.4. Matrix addition and scalar multiplication are describable as vector addition just as in two dimensions. We will usually denote the set of matrices by when thought of as points in three‐dimensional space.
What about matrices? Even though we cannot visualize dimensions, we still envision matrices as somehow representing points in dimensional space. The set of matrices will be denoted as when thought of in this way.
Definition 1.2 is the set of all matrices.
Linear Combinations and Linear Dependence
We can use our Acme Squidget profit matrices to demonstrate one of the most important concepts in linear algebra. Consider the last column of the 2019 profit matrix. Since this column represents the total profit for each branch, it is just the sum of the other columns in the profit matrix:
This last column does not tell us anything we did not already know in that we could have computed the sums ourselves. Thus, while it is useful to have the data explicitly displayed, it is not essential. We say that this data is “dependent on” the data in the other columns. Similarly, the last row of the profit matrix is dependent on the other rows in that it is just their sum.
For another example of dependence, consider the two profit matrices and and their average
The matrix depends on and —once we know and , we can compute .
These examples exhibit an especially simple form of dependence. In each case, the matrix we chose to consider as dependent was produced by multiplying the other matrices by scalars and adding. This leads to the following concept.
Definition 1.3 Let , be a set of elements of . An element of is linearly dependent on if there are scalars such that
We also say that “ is a linear combination of the .”
Remark. In set theory, an object that belongs to a certain set is called an element of that set. The student must be careful not to confuse the terms “element” and “entry.” The matrix below is one element of the set of matrices. Every element of the set of matrices has four entries.
The expression “” means that is an element of the set .
One particular element of is linearly dependent on every other element of . This is the matrix, which has all its entries equal to 0. We denote this matrix by . It is referred to as “the zero element of .” Thus, the zero element of is
The zero matrix depends on every other matrix because, for any matrix ,
We can also discuss linearly dependent sets of matrices:
Definition 1.4 Let be a set of elements of . Then is linearly dependent if at least one of the is a linear combination of the other elements of —that is, is a linear combination of the set of elements with . We also define the set , where is the zero element of , to be linearly dependent. is said to be linearly independent if it is not linearly dependent. Hence, is linearly independent if none of the are linear combinations of other elements of .
Thus, from formula (1.3), the set is linearly dependent. In addition, from formula (1.2), the set of columns of is linearly dependent. The set of rows of is also linearly dependent since the last row is the sum of the other rows.
Example 1.2
Is a linearly independent set where the are the following matrices?
Solution. By inspection
showing that is linearly dependent.
Remark. Note that is not a combination of the other since the entry of is nonzero, while all the other are zero in this position. This demonstrates that linear dependence does not require that each of the be a combination of the others.
Example 1.3
Let , , and be as shown. Is a linearly dependent set?
Solution. We begin by asking ourselves whether is linearly dependent on and —that is, are there scalars and such that
The answer is no since the last entries of both and are 0, while the last entry of is 1.
Similarly, we see that is not a linear combination of and (from the second entries) and is not a linear combination of and (from the third entries). Thus, the given three matrices form a linearly independent set.
Example 1.3 is an example of the following general principle that we use several times later in the text.
Proposition 1.1 Suppose that is a set of matrices such that each has a nonzero entry in a position where all the other are zero—that is, for each there is a pair of indices such that while for all . Then is linearly independent.
Proof. Suppose that is linearly dependent. Then there is a such that
Let be as described in the statement of the proposition. Equating the entries on both sides of the above equation shows that
contradicting the hypothesis that .
Linear independence also has geometric significance. Two vectors and in will be linearly independent if and only if neither is a scalar multiple of the other—that is, they are noncollinear (Figure 1.5a). We will prove in Section 2.2 that any three vectors , , and in are linearly dependent (Figure 1.5b).
FIGURE 1.5 Dependence in .
FIGURE 1.6 Three vectors in .
In , the set of linear combinations of a pair of linearly independent vectors lies in the plane they determine. Thus, three noncoplanar vectors will be linearly independent (Figure 1.6).
In general, the set of all matrices that depend on a given set of matrices is called the span of the set:
Definition 1.5 Let be a set of elements of . Then the span of is the set of all elements of the form
where the are scalars.
The span of , then, is just the set of all linear combinations of the elements of . Thus, for example, if , , and are as in Example 1.3, then
is one element of .
In and , the span of a single vector is the line through the origin determined by it. From Figure 1.6, the span of a set of two linearly independent vectors will be the plane they determine.
What Is a Vector Space?
One of the advantages of matrix notation is that it allows us to treat a matrix as if it were one single number. For example, we may solve for in formula (1.3):
The preceding calculations used a large number of properties of matrix arithmetic that we have not discussed. In greater detail, our argument was as follows:
We certainly used the associative law , the laws and , as well as several other laws. In Theorem 1.1, we list the most important algebraic properties of matrix addition and scalar multiplication.
These properties are called the vector space properties. Experience has proved that these properties are all that one needs to effectively deal with any computations such as those just done with , , and . For the sake of this list, we let for some fixed and .1 Thus, for example, might be the set of all matrices.
Theorem 1.1 (The Vector Space Properties). Let , and be elements of . Then
is a well‐defined element of
.
commutativity
.
associativity
.
There is an element denoted
in
such that
for all
. This element is referred to as the “zero element.”
For each
, there is an element
such that
.
Additionally, for all scalars and :
is a well‐defined element of
.
.
.
.
.
The proofs that the properties from this list hold for are left as exercises for the reader. However, let us prove property (c) as an example of how such a proof should be written.
Example 1.4
Prove property (c) for .
Solution. Let , , and be elements of . Then
When we introduced linear independence, we mentioned that for any matrix
This is very simple to prove:
This proof explicitly uses the fact that we are dealing with matrices. It is possible to give another proof that uses only the vector space properties. We first note from property (i) that
Next, we cancel from both sides using the vector space properties:
Both proofs are valid for matrices. We, however, prefer the second. Since it used only the vector space properties, it will be valid in any context in which these properties hold. For example, let denote the set of all real‐valued functions which are defined for all real numbers. Thus, the functions and are two elements of . We define addition and scalar multiplication for functions by the formulas
Thus, for example,
defines an element of . Since addition and scalar multiplication of functions is defined using the corresponding operations on numbers, it is easily proved that the vector space properties (a)–(j) hold if we interpret , , and as functions rather than matrices. (See Example 1.5.)
Thus, we can automatically state that , where represents any function and is the zero function. Admittedly, this is not an exciting result. (Neither, for that matter, is for matrices.) However, it demonstrates an extremely important principle: Anything we prove about matrices using only the vector space properties will be true in any context for which these properties hold.
As we progress in our study of linear algebra, it will be important to keep track of exactly which facts can be proved directly from the vector space properties and which require additional structure. We do this with the concept of “vector space.”
Definition 1.6 A set is a vector space if it has a rule of addition and a rule of scalar multiplication defined on it so that all the vector space properties – from Theorem 1.1 hold. By a rule of addition we mean a well‐defined process for taking an arbitrary pair of elements and from and producing a third element in . . By a rule of scalar multiplication we mean a well‐defined process for taking an arbitrary scalar and an arbitrary element of and producing a second element of .
The following theorem summarizes our discussion of functions. We leave most of the proof as an exercise.
Theorem 1.2 The set of real‐valued functions on is a vector space under the operations defined by formula .
Example 1.5
Prove vector space property (h) for .
Solution. Let and be real‐valued functions and let . Then
showing that , as desired.
Any concept defined for solely in terms of addition and scalar multiplication will be meaningful in any vector space . One simply replaces by , where is a general vector space. Specifically:
The concept an element
in
depending on a set
of elements of
is defined as in Definition 1.3.
The concepts of linear independence/dependence for a set
of elements of
are defined as in
Definition 1.4
.
The concept of the span of a set
of elements of
is defined as in
Definition 1.5
.
Example 1.6
Show that the set of functions is linearly dependent in .
Solution. This is clear from the formula
Theorem 1.1 states that for, each and , is a vector space. The set of all possible matrices is not a vector space, at least under the usual rules of addition and scalar multiplication. This is because we cannot add matrices unless they are the same size: For example, we cannot add a matrix to a matrix. Thus, our “rule of addition” is not valid for all matrices.
At the moment, the spaces, along with , are the only vector spaces we know. This will change in Section 1.4, where we describe the concept of “subspace of a vector space.” However, if we say that something is “true for all vector spaces,” we are implicitly stating that it can be proved solely on the basis of the vector space properties. Thus, the property that is true for all vector spaces. Another important vector space property is the following: The proof (which must use only the vector space properties or their consequences) is left as an exercise.
Proposition 1.2 Let be an element of a vector space . Then .
Before ending this section, we need to make a comment concerning notation. Writing column vectors takes considerable text space. There is a handy space‐saving notation that we shall use often. Let be an matrix. The “main diagonal” of refers to the entries of the form . (Note that all these entries lie on a diagonal line starting at the upper left‐hand corner of .) If we flip along its main diagonal, we obtain an matrix, which is denoted and called the transpose of . Mathematically, is the matrix defined by the formula
Thus, if
Notice that the columns of become rows in . Thus, is a space efficient way of writing the column vector
Remark. The reader will discover in later sections that the transpose of a matrix has importance far beyond typographical convenience.
Why Prove Anything?
There is a fundamental difference between mathematics and science. Science is founded on experimentation. If certain principles (such as Newton's laws of motion) seem to be valid every time experiments are done to verify them, they are accepted as a “law.”
They will remain a law only as long as they agree with experimental evidence. Thus, Newton's laws were eventually replaced by the theory of relativity when they were found to conflict with the experiments of Michelson and Morley. Mathematics, on the other hand, is based on proof. No matter how many times some mathematical principle is observed to hold, we will not accept it as a “theorem” until we can produce a logical argument that shows the principle can never be violated.
One reason for this insistence on proof is the wide applicability of mathematics. Linear algebra, for example, is essential to a staggering array of disciplines including (to mention just a few) engineering (all types), biology, physics, chemistry, economics, social sciences, forestry, and environmental science. We must be certain that our “laws” hold, regardless of the context in which they are applied. Beyond this, however, proofs also serve as explanations of why our laws are true. We cannot say that we truly understand some mathematical principle until we can prove it.
Mastery of linear algebra, of course, requires that the student learn a body of computational techniques. Beyond this, however, the student should read and, most important, understand the proofs. The student will also be asked to create his or her own proofs. This is because it cannot be truly said that we understand something until we can explain it to someone else.
In writing a proof, the student should always bear in mind that proofs are communication. One should envision the “audience” as another student who wants to be convinced of the validity of what is being proved. This other student will question anything that is not a simple consequence of something that he or she already understands.
True‐False Questions: Justify your answers.
1.1 A subset of a linearly independent set is linearly independent.
1.2 A subset of a linearly dependent set is linearly dependent.
1.3 A set that contains a linearly independent set is linearly independent.
1.4 A set that contains a linearly dependent set is linearly dependent.
1.5 If a set of elements of a vector space is linearly dependent, then each element of the set is a linear combination of the other elements of the set.
1.6 A set of vectors that contains the zero vector is linearly dependent.
1.7 If
is in the span of
,
, and
, then the set
is linearly independent as long as the
are independent.
1.8 If
is linearly dependent, then
is in the span of
,
, and
.
1.9 The following set of vectors is linearly independent:
1.10 The following matrices form a linearly independent set:
1.11 If
is a linearly dependent set of matrices, then
is also a linearly dependent set.
1.12 The set of functions
is a linearly independent set of elements of the vector space of all continuous functions on the interval
.
EXERCISES
1.1
In each case, explicitly write out the matrix
, where
. Also, give the third row (written as a row vector) and the second column (written as a column vector).
, where
and
, where
and
, where
and
