# MATRIX ANALYSIS AND APPLIED LINEAR ALGEBRA PDF

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matical rigors, the challenge in teaching applied linear algebra is to expose some . book along with the solutions manual in PDF format. algebra and matrix theory and fails to learn at least the elementary aspects of what. This book is about matrix and linear algebra, and their applications. For many To provide a balanced blend of applications, theory and computation which em-. Carl Meyer's Matrix Analysis and Applied Linear Algebra is an introduc-. tion to the theory and practice of linear algebra for university students of. mathematics, engineering, and science. and a few other goodies, all. in PDF format. There is a.

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solutions. This means that you can start the back substitution with any value.. The only possible SolutionsMaster APPLIED LINEAR ALGEBRA AND MATRIX . This book is about matrix and linear algebra, and their applications. For many and applied mathematics, which means that matrix analysis plays a central role. Request PDF on ResearchGate | On Sep 1, , James Gentle and others published Matrix Analysis and Applied Linear Algebra by Carl D. Meyer; Numerical.

Short Calculus: Undergraduate Algebra. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

For many students the tools of matrix and linear algebra will be as fundamental in their professional work as the tools of calculus; thus it is important to ensure that students appreciate the utility and beauty of these subjects as well as the mechanics.

To this end, applied mathematics and mathematical modeling ought to have an important role in an introductory treatment of linear algebra.

In this way students see that concepts of matrix and linear algebra make concrete problems workable. I hope that instructors will not omit this material; that would be a missed opportunity for linear algebra! The text has a strong orientation toward numerical computation and applied mathematics, which means that matrix analysis plays a central role. All three of the basic components of linear algebra — theory, computation, and applications — receive their due.

The proper balance of these components gives students the tools they need as well as the motivation to acquire these tools. Another feature of this text is an emphasis on linear algebra as an experimental science; this emphasis is found in certain examples, computer exercises, and projects. Applications and ideas should take center stage, not software. This book is designed for an introductory course in matrix and linear algebra.

Here are some of its main goals: Each chapter has computer exercises sprinkled throughout and an optional section on computational notes. To help students to express their thoughts clearly. Requiring written reports is one vehicle for teaching good expression of mathematical ideas. To encourage cooperative learning.

Mathematics educators are becoming increasingly appreciative of this powerful mode of learning. Team projects and reports are excellent vehicles for cooperative learning. To promote individual learning by providing a complete and readable text. An outline of the book is as follows: Chapter 1 contains a thorough development of Gaussian elimination. It would be nice to assume that the student is familiar with complex numbers, but experience has shown that this material is frequently long forgotten by many.

Complex numbers and the basic language of sets are reviewed early on in Chapter 1. Basic properties of matrix and determinant algebra are developed in Chapter 2. Special types of matrices, such as elementary and symmetric, are also introduced. About determinants: These provide motivation for the more sophisticated ideas of abstract vector space, subspace, and basis, which are introduced largely in the context of the standard spaces.

Chapter 4 introduces geometrical aspects of standard vector spaces such as norm, dot product, and angle. Chapter 5 introduces eigenvalues and eigenvectors. General norm and inner product concepts for abstract vector spaces are examined in Chapter 6. Each section concludes with a set of exercises and problems. Optional sections cover tensor products, linear operators, operator norms, the Schur triangularization theorem, and the singular value decomposition.

In addition, each chapter has an optional section of computational notes and projects. I employ the convention of marking sections and subsections that I consider optional with an asterisk. There is more than enough material in this book for a one-semester course. One could increase emphasis on any one of the theoretical, applied, or computational aspects of linear algebra by the appropriate selection of syllabus topics.

The text is well suited to a course with a three-hour lecture and lab component, but computer-related material is not mandatory.

Instructors may mix and match any of the Preface ix optional sections according to their own interests, since these sections are largely independent of each other. While it would be very time-consuming to cover them all, every instructor ought to use some part of this material. The unstarred sections form the core of the book; most of this material should be covered.

There are 27 unstarred sections and 10 optional sections. The material of most of the unstarred sections can be covered at a rate of about one and one-half class periods per section. Thus, the core material could be covered in about 40 class periods. This leaves time for extra sections and in-class exams. In a two-semester course or a course of more than three hours, one could expect to cover most, if not all, of the text.

This approach reduces the number of unstarred sections to be covered from 27 to I employ the following taxonomy for the reader tasks presented in this text.

Exercises constitute the usual learning activities for basic skills; these come in pairs, and solutions to the odd-numbered exercises are given in an appendix. More advanced conceptual or computational exercises that ask for explanations or examples are termed problems, and solutions for problems are not given, but hints are supplied for those problems marked with an asterisk. Some of these exercises and problems are computer-related. As with pencil-and-paper exercises, these are learning activities for basic skills.

At the next level are projects. These assignments involve ideas that extend the standard text material, possibly some numerical experimentation and some written exposition in the form of brief project papers.

These are analogous to lab projects in the physical sciences. Finally, at the top level are reports. These require a more detailed exposition of ideas, considerable experimentation — possibly open ended in scope — and a carefully written report document. I have included some of my favorite examples of all of these activities in this textbook.

In my own classes I expect projects to be prepared with text processing software to which my students have access in a mathematics computer lab. Instructors should provide background materials to help the students through local systemdependent issues. When I assign a project, I usually make available a Maple, Matlab, or Mathematica notebook that amounts to a brief background lecture on the subject of the project and contains some of the key commands students will need to carry out the project.

This helps students focus more on the mathematics of the project rather than computer issues. Most of the computational computer tools that would be helpful in this course fall into three categories and are available for many operating systems: These software products are fairly rich in linear algebra capabilities. This is not to be construed as an endorsement or requirement of any particular software or computer.

Each system has its own strengths. In various semesters I have obtained excellent results with all these platforms. Students are open to all sorts of technology in mathematics. This openness, together with the availability of inexpensive high-technology tools, has changed how and what we teach in linear algebra.

I would like to thank my colleagues whose encouragement has helped me complete this project, particularly David Logan. I would also like to thank my wife, Muriel Shores, for her valuable help in proofreading and editing the text, and Dr. David Taylor, whose careful reading resulted in many helpful comments and corrections. I continue to develop a linear algebra home page of material such as project notebooks, supplementary exercises, errata sheet, etc.

This site can be reached at http: Basic Ideas. General Procedure. The latter problem will not be encountered until Chapter 4; it requires some background development and even the motivation for this problem is fairly sophisticated.

By contrast, the former problem is easy to understand and motivate. As a matter of fact, simple cases of this problem are a part of most high-school algebra backgrounds. We will address the problem of when a linear system has a solution and how to solve such a system for all of its solutions. Example 1. Thus, each equation above represents a line. Normally, we expect the solution to be unique, which it is in this example. We also learned how to solve such an equation algebraically: Graphical solution to Example 1.

The geometrical approach becomes impractical as a means of obtaining an explicit solution to our problem: The solution to this problem can be discerned roughly in Figure 1. Nonetheless, the geometrical approach gives us a qualitative idea of what to expect without actually solving the system of equations.

Now we know from analytical geometry that the graph of this equation is a plane in three dimensions.

In general, two planes will intersect in a line, though there are exceptional cases of the two planes represented being identical or distinct and parallel. Similarly, three planes will intersect in a plane, line, point, or nothing. Hence, we know that the above system of three equations has a solution set that is either a plane, line, point, or the empty set. Which outcome occurs with our system of equations?

Figure 1. The matter of dealing with three equations and three unknowns is a bit trickier than the problem of two equations and unknowns. Just as with two equations and unknowns, the key idea is still to use one equation to solve for one unknown. Some Key Notation Here is a formal statement of the kind of equation that we want to study in this chapter. This formulation gives us the notation for dealing with the general problem later on. A linear equation in the variables x1 , x2 ,.

But our focus is on systems that consist solely of linear equations. A linear system of m equations in the n unknowns x1 , x2 ,. This systematic way of describing the system will come in handy later, when we introduce the matrix concept. About indices: In those rare situations where confusion is possible, e. So why develop a whole subject? We shall consider a few examples whose solutions are not so apparent as those of the previous two examples. The point of this chapter, as well as that of Chapters 2 and 3, is to develop algebraic and geometrical methodologies that are powerful enough to handle problems like these.

A basic physical observation is that heat is directly proportional to temperature. In a wide range of problems this hypothesis is true, and we shall assume that we are modeling such a problem.

How can we model a steady state? Assume that the nodes are a distance h apart. Approximate y x in between nodes by connecting adjacent points xi , yi with a line segment. See Figure 1. A derivation of these equations is given in Section 1.

Suppose further that the rod is laterally insulated, but has a known internal heat source and that both the left and right ends of the rod are held at 0 degrees Fahrenheit. What are the steadystate equations approximately for this problem? Follow the notation of the discussion preceding this example. Thus we have from equation 1. It is reasonable to expect that the smaller h is, the more accurately yi will approximate y xi.

This is indeed the case. This problem is already about as large as we might want to work by hand, if not larger. The basic ideas of solving systems like this are the same as in Examples 1.

Suppose the three sectors are E nergy, M aterials, and S ervices and suppose that the demands of a sector are proportional to its output.

This is reasonable; if, for example, the materials sector doubled its output, one would expect its needs for energy, material, and services to likewise double.

We require that the system be in equilibrium in the sense that total output of the sector E should equal the amounts consumed by all sectors and consumers. Let x, y, z be the total outputs of the sectors E, M, and S respectively. Consider how we balance the total supply and demand for energy. The total output supply is x units. The demands from the three sectors E, M, and S are, according to the table data, 0.

Further, consumers demand 2 units of energy. The questions that interest economists are whether this system has solutions, and if so, what they are. An administrative unit has four divisions serving the internal needs of the unit, labeled A ccounting, M aintenance, S upplies, and T raining. The input—output table of demand rates is given by the following table.

Express the equilibrium of this system as a system of equations. Let x, y, z, w be the total outputs of the sectors A, M, S, and T, respectively.

There is an obvious, but useless, solution to this system. One hopes for nontrivial solutions that are meaningful in the sense that each variable takes on a nonnegative value. This textbook does not depend on any particular system, but certain exercises require a computational device. Solve the following systems algebraically. Determine whether each of the following systems of equations is linear. If so, put it in standard format.

Write out the linear system that results from Example 1. Exercise 8. Exercise 9. Suppose that in the input—output model of Example 1. Derive equations for prices that achieve equilibrium, that is, equations that say that the price received for a unit item equals the cost of producing it.

Derive equilibrium equations for these prices. Problem Use a symbolic calculator or CAS to solve the systems of Examples 1. Comment on your solutions. Are they sensible? Express these three conditions as a linear system of three equations in the unknowns a0 , a1 , a2.

It provides a convenient shorthand for expressing mathematical statements. We use some shorthand to indicate certain relationships between sets and elements. Usually, sets will be designated by uppercase letters such as A, B, etc. As usual, set A is a subset of set B if every element of A is an element of B, and a proper subset if it is a subset but not equal to B. Two sets A and B are said to be equal if they have exactly the same elements. Some shorthand: Let A and B be sets.

About Numbers Natural Numbers One could spend a whole course fully developing the properties of number systems. At the start of it all is the kind of numbers that everyone knows something about: One could view most subsequent expansions of the concept of number as a matter of rising to the challenge of solving new equations.

For example, we cannot solve the equation 1. Rational Numbers Rational-number arithmetic has some characteristics that distinguish it from integer arithmetic. The associative, commutative, identity, and inverse laws must hold for each of addition and multiplication; and the distributive law must hold for multiplication over addition. Story has it that this is lethal knowledge, in that followers of a Pythagorean cult claim that the gods threw overboard from a ship one of their followers who was unfortunate enough to discover that fact.

Filling in these holes leads us to the set R of real numbers, which are in one-to-one correspondence with the points on a number line.

There is one more problem to overcome. We need to extend our number system one more time, and this leads to the set C of complex numbers. Standard and polar coordinates in the complex plane. Notice that the imaginary part of z is a real number: Two complex numbers are equal precisely when they have the same real part and the same imaginary part.

We will not do so, but the fact that complex numbers behave like ordered pairs of real numbers leads to an important geometrical insight: This results in the so-called complex plane. In addition, there are several more useful ideas about complex numbers that we will need. Thus we have the following laws of complex arithmetic: Complex multiplication does not admit such a simple interpretation. Verify that the product of conjugates is the conjugate of the product.

This is just the last fact in the preceding list.

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The big surprise is that once we have the complex numbers in hand, we have a number system so complete that we can solve any polynomial equation in it. Fundamental Theorem of Algebra Theorem 1. As a matter of fact, there are no general formulas for the roots of a polynomial of degree greater than four, which means that we have to resort to numerical approximations in most practical cases.

In vector space theory the numbers in use are sometimes called scalars, and we will use this term. However, we shall see later, when we study eigensystems, that even if we are interested only in real scalars, complex numbers have a way of turning up quite naturally.

Also compute the complex conjugate and absolute value of the solution. Proceed as follows: This material will not be needed until Chapter 4. Recall from basic algebra the so-called roots theorem: If we team this fact up with the Fundamental Theorem of Algebra, we see an interesting fact about factoring polynomials over C: The numbers a that occur are exactly the roots of f z.

For now, we will settle on a few ad hoc methods for solving some important special cases. There is one detail to attend to: Quadratic equations are also simple enough: What we are really asking is this: In a few cases, such an equation is quite easy to solve.

In reference to Figure 1. It follows that a complex number may have more than one polar form. As the notation suggests, polar forms obey the laws of exponents. The solution goes as follows: We have to be a bit more careful with i. We also have the standard half angle formulas from trigonometry to help us: Put the following complex numbers into polar form and sketch them in the complex plane: Calculate the following: Solve the following systems for the unknown z: Solve the equations for the unknown z: Find the polar and standard form of the complex numbers: Find all solutions to the following equations: Find the solutions to the following equations: Exercise Verify the factorization by expanding it.

Verify that for any two complex numbers, the sum of the conjugates is the conjugate of the sum. Use the notation of Example 1. How many roots counting multiplicities should this equation have? Basic Ideas 21 1.

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The principal methodology is the method of Gaussian elimination and its variants, which we introduce by way of a few simple examples. Then it will be simple to solve for each variable one at a time, starting with the last equation, which will involve only the last variable.

In a nutshell, this is Gaussian elimination. Such will be the case for most of the problems in this chapter. An Example and Some Shorthand Example 1. We can do this easily if we take care to combine like terms as we go. Use the last equation to solve for the last variable, then work backward, solving for the remaining variables in reverse order. We could easily scratch out the solution in much less space.

But what if the system is larger, say 4 equations in 4 unknowns, or more? How do we proceed then? It pays to have a systematic strategy and notation. We also had an ulterior motive in the way we solved this system. All of the operations we will ever need to solve a linear system were illustrated in the preceding example: Take a closer look at the system of equations 1.

As long as we write numbers down systematically, there is no need to write out all the equal signs or plus signs. So we could embellish the table with a few reminders in an extra top row: Rectangular tables of numbers are very useful in representing a system of equations.

Such a table is one of the basic objects studied in this text. A matrix is a rectangular array of numbers. Basic Ideas 23 square matrix of order n. Finally, the number that occurs in the ith row and jth column is called the i, j th entry of the matrix. Therefore, we will follow a standard typographical convention: In a few cases these conventions are not followed, but the meaning of the symbols should be clear from context.

We shall need to refer to parts of a matrix. As indicated above, the location of each entry of a matrix is determined by the index of the row and column it occupies. Generally, the size of A will be clear from context. In case the type of the vector row or column is not clear from context, the default is a column vector.

In this case we say that n is the order of the matrix. Another term that we will use frequently is the following. If all entries are zero, the vector has no leading entry.

The equations of 1. First is the full matrix that describes the system, which we call the augmented matrix of the system. This notation is related to the operations that we performed on the preceding example. Now that we have the matrix notation, we could just as well perform these operations on each row of the augmented matrix, since a row corresponds to an equation in the original system.

Three types of operations were used.

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We shall catalog these and give them names, so that we can document our work in solving a system of equations in a concise way. Here are the three elementary operations we shall use, described in terms of their action on rows of a matrix; an entirely equivalent description applies to the equations of the linear system whose augmented matrix is the matrix below.

This is shorthand for the elementary operation of switching the ith and jth rows of the matrix. For instance, in Example 1. This is shorthand for the elementary operation of multiplying the ith row by the nonzero constant c. For instance, we moved from equation 1.

This is shorthand for the elementary operation of adding d times the jth row to the ith row. Read the symbols from right to left to get the right order. The whole forward-solving phase of Example 1. There is still the job of back solving, which is the second phase of Gaussian elimination. Here are the details using our shorthand for elementary operations: This is, of course, the answer we found earlier: The method of combining forward and back solving into elementary operations on the augmented matrix has a name: Solve the following system by Gauss—Jordan elimination: Otherwise, we cannot use the second equation to solve for the second variable, y.

Next back solve: Here is some handy terminology.

An entry of a matrix used to zero out entries above or below it by means of elementary row operations is called a pivot. The entries that we use in Gaussian or Gauss—Jordan elimination for pivots are always leading entries in the row that they occupy. For the sake of emphasis, in the next few examples we will put a circle around the pivot entries as they occur.

Basic Ideas 27 Solution. Neither the second nor the third row corresponds to equations that involve the variable y. Here is the point of view that we adopt in applying Gaussian elimination to this system: Perform the following operations to do Gauss—Jordan elimination on the system: What about y? The point is that there is not enough information in the system to solve for the variable y, even though we started with three distinct equations.

Somehow, they contained redundant information. Therefore, we take the point of view that y is not to be solved for; it is a free variable in the sense that we can assign it any value whatsoever and obtain a legitimate solution to the system. On the other hand, the variables x and z are bound in the sense that they will be solved for in terms of constants and free variables.

Up to this point, the linear systems we have considered had unique solutions, so every variable was solved for, and hence bound. The default is that y is allowed to take on any real value from R. But if, for some reason, we choose to work with the complex numbers as our scalars, then y would be allowed to take on any complex value from C.

To summarize, once we have completed Gauss—Jordan elimination on an augmented matrix, we can immediately spot the free and bound variables of the system: Another example will illustrate the point.

Actually, there is only one more possibility, which is illustrated by the following example. We extract the augmented matrix and proceed with Gauss—Jordan elimination. It is understood that they are done in order, starting with the top one. But something strange is going on here.

This is impossible. What this matrix is telling us is that the original system has no solution, i. Thus, one need proceed no further. The system has no solutions. A system of equations is consistent if it has at least one Consistent solution.

Otherwise it is called inconsistent. Our last example is one involving complex numbers explicitly. Solve the following system of equations: Of course, the arithmetic is a bit harder. For each of the following matrices identify the size and the i, j th entry for all relevant indices i and j: Exhibit the augmented matrix of each system and give its size.

Basic Ideas 31 Exercise 5. Show the elementary operations you use. Each of the following matrices results from applying Gauss—Jordan elimination to the augmented matrix of a linear system. Here, b1 , b2 are constants and x1 , x2 are the unknowns. Find the general solution for the system of equations in Exercise 9 of Section 1.

Is there a solution to this problem with nonnegative entries? Find the general solution for the system of equations in Exercise 10 of Section 1. Is there meaningful solution to this problem? You must also account for the possibilities that one of x, y, z is zero.

Suppose that the input—output table of Example 1. Show that the only solution to the problem with nonnegative values is the solution with all variables equal to zero. As in Exercise 12 of Section 1. Solve the network system of Problem 13 of Section 1 and exhibit all physically meaningful solutions.

General Procedure The preceding section introduced Gaussian elimination and Gauss—Jordan elimination at a practical level. In this section we will see why these methods work and what they really mean in matrix terms. A key idea that comes out of this section is the notion of the rank of a matrix. To see that linear systems are special, consider the following nonlinear system of equations. The answer lies in examining the kinds of operations we perform with these methods.

First, we need some terminology. Up to this point we have always described a solution to a linear system in terms of a list of equations. For general problems this is a bit of a nuisance. A solution vector for the general linear system given by equation 1. The set of all such solutions is called the solution set of the linear system, and two linear systems are said to be equivalent if they have the same solution set. Tuple Convention We will want to make frequent reference to vectors without having to display them in the text.

To save space in referring to column vectors, we shall adopt the convention that a column vector will also be denoted by a tuple with the same entries. The n-tuple x1 , x2 ,. Describe the solution sets of all the examples worked out in the previous section.

Here is the solution set to Example 1. General Procedure 35 The solution set for Example 1. For Example 1. Here it is: Finally, the solution set for Example 1. After all, Gaussian and Gauss—Jordan elimination amount to a sequence of elementary row operations applied to the augmented matrix of a given linear system.

Theorem 1. Then these two linear systems are B equivalent, i. Thus, every solution to the old system is also a solution to the new system resulting from performing an elementary operation. This will show that every solution to the new system is also a solution to the old system.

Let us examine each elementary operation in turn. The elementary operation of switching the ith and jth rows of the matrix. This switches the rows back. The elementary operation of multiplying the ith row by the nonzero constant c. The elementary operation of adding d times the jth row to the ith row. In Chapter 2 this notation will take on an entirely new and richer meaning. Our next objective is to describe the end result of these methods in a precise way.

So a row of zeros has no leading entry. A matrix R is said to be in reduced row form if: Consider the following matrices whose leading entries are enclosed in a circle. Which are in reduced row form? But c fails, since a zero row precedes the nonzero ones; matrix e fails to be in reduced row form because the column numbers of the leading entries do not form an increasing sequence. Matrices 1. Hence, it is the only matrix in the list in reduced row echelon form.

On the other hand, the goal of Gauss—Jordan elimination is to use elementary operations to reduce the augmented matrix of a linear system to reduced row echelon form. Is it always possible to reduce a matrix to a reduced row form or row echelon form? If so, to how many such forms? These are important questions. If we take the matrix in question to be the augmented matrix of a linear system, what we are really asking becomes, does Gaussian elimination always work on a linear system?

If so, does it lead us to answers that have the same form? Notice how the last question was phrased. Therefore, the solution sets we obtain will always be the same with either method, as sets. The last matrix is also in reduced row echelon form. Yet all three of these matrices can be obtained from each other by elementary row operations. As a matter of fact, any matrix can be reduced by elementary row operations to one and only one reduced row echelon form, which we can call the reduced row echelon form of the given matrix.

Every matrix can be reduced by a sequence of elementary row Uniqueness of operations to one and only one reduced row echelon form. Here is the algorithm we have been using: This must occur at some point since both r and s increase with each step, and when it occurs, the resulting matrix is in reduced row echelon form.

Next, we prove uniqueness. Suppose that some matrix could be reduced to two distinct reduced row echelon forms. We show this is impossible. There are two possibilities to consider. Case 1: The last column bi of either Ri has a leading entry in it. Case 2: Each bi has no leading entry in it. Hence we obtain the same solution with either augmented matrix by setting the free variables of the system equal to 0. When we do so, the bound variables are uniquely determined: Similarly, the second says that the second bound variable equals the second entry in the right-hand-side vector, and so forth.

Whether we use R1 or R2 to solve the system, we obtain the same result, since we can manipulate one such solution into the other by elementary row operations. Hence, there can be no counterexample to the theorem, which completes the proof. General Procedure 39 Corollary 1.

Let the matrix B be obtained from the matrix A by performing a sequence of elementary row operations on A. Then B and A have the same reduced row echelon form. We can obtain the reduced row echelon form of B in the following manner: First perform the elementary operations on B that undo the ones originally performed on A to get B.

The matrix A results from these operations. Now perform whatever elementary row operations are needed to reduce A to its reduced row echelon form. Since B can be reduced to one and only one reduced row echelon form, the reduced row echelon forms of A and B coincide, which is what we wanted to show.

The rank of a matrix A is the number of nonzero rows of the reduced row echelon form of A. This number is written as rank A. Rank of Matrix There are other ways to describe the rank of a matrix. One has to check that any two reduced row forms have the same number of nonzero rows.

We can count up the other columns as well. The nullity of a matrix A is the number of columns of the Nullity reduced row echelon form of A that do not contain a leading entry. This number is written as null A. One has to be a little careful about this idea of rank.

Consider the following example. Remember that the rank of A is the number of nonzero rows in one of its reduced row forms, and not the number of nonzero rows of A itself. The rank of a matrix is a nonnegative number, but it could be 0!

This happens if the matrix has only zero entries, so that it has no nonzero rows.

In this case, the nullity of the matrix is as large as possible, namely the number of columns of the matrix. Here are some simple limits on the size of rank A and null A. Given a list of real numbers a1 , a2 ,.

Also, each leading entry of a matrix in reduced row echelon form is the unique nonzero entry in its column. So there can be no more leading entries than columns n.

The number of pivot columns is rank A and the number of nonpivot columns is null A. The sum of these numbers is n. If the rank of a matrix equals its column number we say that the matrix has full column rank. Similarly, a matrix has full row rank if its rank equals the row number of the matrix.

For example, matrix A of Example 1. Since this rank is smaller than 3, A does not have full column or row rank. Here is an application of the rank concept to systems. The general linear system 1.

We can reduce A ementary operations that reduce the A part of the matrix to reduced row echelon form, then attending to the last column. On the other hand, we have already seen in the proof of Theorem 1.

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Otherwise, the system is said to be inhomogeneous. The nice feature of homogeneous systems is that they are always consistent! For obvious reasons this solution is called the trivial solution to the system. Notice that the right-hand side of zeros is never changed by an elementary row operation.

So why bother writing out the augmented matrix of such a system? In the end, the right-hand side is still a column of zeros.

Circle leading entries and determine which of the following matrices can be put into reduced row echelon form with at most one elementary operation. The rank of the following matrices can be determined by inspection. Give the rank and nullity of each matrix.

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Compute a reduced row form that can be reached in a minimum number of steps and the reduced row echelon forms of the following matrices. Given that the matrices are augmented matrices for a linear system, write out the general solution to the system. Are these systems equivalent? Solve this system by reducing the augmented matrix to reduced row echelon form. This system will have solutions for any right-hand side.

Justify this fact in terms of rank. General Procedure 45 Exercise Give a rank condition for a homogeneous system that is equivalent to the system having a unique solution. Justify your answer. Fill in the blanks: Show that a system of linear equations has a unique solution if and only if every column, except the last one, of the reduced row echelon form of the augmented matrix has a pivot entry in it.

Prove or disprove by example: Use Theorem 1. There are several reasons for this unfortunate fact. For instance, verify this simple arithmetic fact on a calculator or computational software such as Matlab but excluding computer algebra systems such as Derive, Maple, and Mathematica — since symbolic calculation is the default on these systems, they will give the correct answer: The problem is that if, for example, a calculator uses 6-digit accuracy, then 23 is calculated as 0.

Starting with erroneous data and doing an exact calculation can be as bad as starting with exact data and doing an inexact calculation. In fact, in a certain sense they are equivalent to each other. A thorough analysis can be found in the Golub and Van Loan text [9] of the bibliography, a text that is considered a standard reference work.

The text [13] is an excellent introductory treatment of this subject. We will consider this question: Is it possible that with all the arithmetic performed in Gaussian elimination the errors pile up and become large? The answer is yes.

With the advent of computers came a heightened interest in these questions. Now solve the linear system 1. This answer is spectacularly bad! The idea is fairly simple: Do not choose the next available column entry for a pivot. Rather, search down the column in question for the largest entry in absolute value. Then switch rows, if necessary, and use this entry as a pivot. But partial pivoting is not a panacea for numerical problems. In fact, it can be easily defeated.

Here the problem is a matter of scale. It can be cured by dividing each row by the largest entry of the row before beginning the Gaussian elimination process. This procedure is known as row scaling. The combination of row scaling and partial pivoting overcomes many of the numerical problems of Gaussian and Gauss—Jordan elimination but not all!

There is a more drastic procedure, known as complete pivoting. In this procedure one searches all the unused rows excluding the right-hand sides for the largest entry, then uses it as a pivot for Gaussian elimination.

The columns used in this procedure do not move in that left-to-right fashion we are used to seeing in system solving. Yet in most cases they do reasonably well. Since this combination involves much less calculation than complete pivoting, it is the method of choice for many problems.

There are deeper reasons for numerical problems in solving some systems than the one the preceding example illustrates. A classical example of this type of problem, the Hilbert matrix, is discussed in one of the projects below. The following example is extremely useful. A little experimentation with an example or two shows that the answer should be 2n. We have n entries to worry about. Consider a typical one, say the ith one. This ensures that we will have a pivot in every row of the matrix.

Recall the identities for sums of consecutive integers and their squares: For large n we have that n3 is much larger than n or n2 e.

There remains the matter of back solving. We leave as an exercise to show that the total work of back solving is quadratic in n. Hence we have the following estimate of the complexity of Gaussian elimination. Exact answer: These projects provide an opportunity to explore a subject in a greater depth than exercises permit. Also, the computing platform used for the projects will vary.

We cannot discuss every platform in this text, so we will give a few examples of implementation notes that an instructor might supply. Here are a few suggestions. Usually, you may assume that your report will be read by your supervisors, who are technical people such as yourself.

Therefore, you should write a brief statement of the problem and discussion of methodology. Have in mind a target length for your paper. Generally, a discourse should have three parts: Roughly, a beginning should consist of introductory material. In the middle you develop the ideas described or theses proposed in the introduction, and in the end you summarize your work and tie up loose ends. Use a vocabulary with which you are comfortable.

Use a spell-checker if one is available. Read and follow these instructions carefully. Use every available resource, of course. In particular, we all know that the worldwide web is a gold mine of information and disinformation! Utilize it and other resources fully, but give appropriate references and credits, just as you would with a textbook source.

Of course, rules about paper writing are not set in concrete. Also, a part can be quite short; for example, an introduction might only be a paragraph or two. Here is a sample skeleton for a report perhaps rather more elaborate than you need: Introduction title page, summary, and conclusions ; 2.

Main sections problem statement, assumptions, methodology, results, conclusions ; 3. Appendices such as mathematical analysis, graphs, possible extensions, etc. Heat Flow I Problem Description: You have been assigned the analysis of a component that is similar to a laterally insulated rod. For courses containing advanced undergraduate or graduate students, the focus can be on material in the latter sections of Chapters 4, 5, 7, and Chapter 8 Perron—Frobenius Theory of Nonnegative Matrices.

A rich two-semester course can be taught by using the text in its entirety. The overwhelming response was that the primary use of linear algebra in applied industrial and laboratory work involves the development, analysis, and implementation of numerical algorithms along with some discrete and statistical modeling.

Computing Projects Computing projects help solidify concepts, and I include many exercises that can be incorporated into a laboratory setting. It also tends to dehumanize mathematics, which is the epitome of human endeavor. But, as I came to realize, this is a perilous task because writing history is frequently an interpretation of facts rather than a statement of facts.

The solutions manual contains the solutions for each exercise given in the book. The solutions are constructed to be an integral part of the learning process. Rather than just providing answers, the solutions often contain details and discussions that are intended to stimulate thought and motivate material in the following sections.

This electronic version of the text is completely searchable and linked. In addition, the CD contains material that extends historical remarks in the book and brings them to life with a large selection of xii Preface portraits, pictures, attractive graphics, and additional anecdotes.

The supporting Internet site at MatrixAnalysis. I thank the SIAM organization and the people who constitute it the infrastructure as well as the general membership for allowing me the honor of publishing my book under their name.

I am dedicated to the goals, philosophy, and ideals of SIAM, and there is no other company or organization in the world that I would rather have publish this book. I am particularly indebted to Michele Benzi for conversations and suggestions that led to several improvements. Painter and Franklin A. Finally, neither this book nor anything else I have done in my career would have been possible without the love, help, and unwavering support from Bethany, my friend, partner, and wife.

Her multiple readings of the manuscript and suggestions were invaluable. I dedicate this book to Bethany and our children, Martin and Holly, to our granddaughter, Margaret, and to the memory of my parents, Carl and Louise Meyer. Carl D. This link seems to have been made at the outset.

The earliest recorded analysis of simultaneous equations is found in the ancient Chinese book Chiu-chang Suan-shu Nine Chapters on Arithmetic , estimated to have been written some time around B. Three sheafs of a good crop, two sheafs of a mediocre crop, and one sheaf of a bad crop are sold for 39 dou. Two sheafs of good, three mediocre, and one bad are sold for 34 dou; and one good, two mediocre, and three bad are sold for 26 dou.We will have more to say about this number in the next section.

I hope that instructors will not omit this material; that would be a missed opportunity for linear algebra! Recall from Corollary 2. The last matrix is also in reduced row echelon form. There is enough material so that professors can pick and choose topics depending on the level of a particular class, and enough depth so that it can be tailored to many different courses. Otherwise it is called inconsistent. Fundamentals of Contemporary Set Theory.

Now that we have the matrix notation, we could just as well perform these operations on each row of the augmented matrix, since a row corresponds to an equation in the original system.