# LINEAR ALGEBRA KOLMAN PDF

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"Introductory Linear Algebra" by "Bernard Kolman" & "David R Hill" 8th Edition EIGHTH EDITION INTRODUCTORY LINEAR ALGEBRA AN APPLIED FIRST COURSE Bernard Kolman Drexel University David R. Hill Temple Download pdf. We have designed Elementary Linear Algebra, Sixth Edition, for the Students embarking on a linear algebra course should have a thorough knowledge of. In most mathematics programs linear algebra comes in the first or second year . algebra is about vector spaces and linear maps, is not taken to the exclusion of.

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Mar 27, G.3 What is Linear Algebra: 3 × 3 Matrix Example .. ˆ “Introductory Linear Algebra, An Applied First Course”, B. Kolman and D. Hill. instructor's solutions manual elementary linear algebra with applications ninth edition bernard kolman drexel university david hill temple university editorial. INTRODUCTORY LINEAR ALGEBRA. WITH APPLICATIONS. B. KOLMAN, D. R. HILL. Eighth Edition, Prentice Hall, Dr. Grigore C˘ALUG˘AREANU.

Most readers have had some experience with this technique in high school algebra courses. If we deal with two, three, or four unknowns, we shall often write them as x, y, z, and w. In this section we use the method of elimination as it was studied in high school. In Section 1. The rules of the trust state that both a certificate of deposit CD and a long-term bond must be used. The director determines the amount x to invest in the CD and the amount y to invest in the bond as follows: We have eliminated the unknown x.

Again, we decide to eliminate x. This means that the linear system 4 has no solution. We might have come to the same conclusion from observing that in 4 the left side of the second equation is twice the left side of the first equation, but the right side of the second equation is not twice the right side of the first equation.

The importance of the procedure lies in the fact that the linear systems 5 and 8 have exactly the same solutions. System 8 has the advantage that it can be solved quite easily, giving the foregoing values for x, y, and z. This means that the linear system 9 has infinitely many solutions. Every time we assign a value to r , we obtain another solution to 9. Since 14 and 13 have the same solutions, we con- clude that 13 has no solutions. These examples suggest that a linear system may have one solution a unique solution , no solution, or infinitely many solutions.

We have seen that the method of elimination consists of repeatedly per- forming the following operations: Interchange two equations. Multiply an equation by a nonzero constant. Add a multiple of one equation to another. It is not difficult to show Exercises T. The new linear system can then be solved quite readily. Thus we have not indicated any rules for selecting the unknowns to be eliminated.

Before providing a systematic description of the method of elimination, we introduce, in the next section, the notion of a matrix, which will greatly simplify our notation and will enable us to develop tools to solve many important problems.

Consider now a linear system of two equations in the unknowns x and y: The graph of each of these equations is a straight line, which we denote by l1 and l2 , respectively.

See Figure 1. Thus we are led geometrically to the same three possibilities mentioned previously. The system has a unique solution; that is, the lines l1 and l2 intersect at exactly one point. The system has no solution; that is, the lines l1 and l2 do not intersect.

The system has infinitely many solutions; that is, the lines l1 and l2 coin- cide. Figure 1. The graph of each of these equations is a plane, denoted by P1 , P2 , and P3 , respectively. As in the case of a linear system of two equations in two un- knowns, the linear system in 16 can have a unique solution, no solution, or infinitely many solutions. These situations are illustrated in Figure 1. For a more concrete illustration of some of the possible cases, the walls planes of a room intersect in a unique point, a corner of the room, so the linear system has a unique solution.

Next, think of the planes as pages of a book. Three pages of a book when held open intersect in a straight line, the spine. Thus, the linear system has infinitely many solutions. On the other hand, when the book is closed, three pages of a book appear to be parallel and do not intersect, so the linear system has no solution. A, B, and C. Each product must go through two processing machines: X and Y.

The products require the following times in machines X and Y: One ton of A requires 2 hours in machine X and 2 hours in machine Y. One ton of B requires 3 hours in machine X and 2 hours in machine Y. One ton of C requires 4 hours in machine X and 3 hours in machine Y.

Machine X is available 80 hours per week and machine Y is available 60 hours per week. Since management does not want to keep the expensive machines X and Y idle, it would like to know how many tons of each product to make so that the machines are fully utilized.

It is assumed that the manufacturer can sell as much of the products as is made. To solve this problem, we let x1 , x2 , and x3 denote the number of tons of products A, B, and C, respectively, to be made. This linear system has infinitely many solutions. The reader should observe that one solution is just as good as the other. There is no best solution unless additional information or restrictions are given. Key Terms Linear equation Solution to a linear system No solution Unknowns Method of elimination Infinitely many solutions Solution to a linear equation Unique solution Manipulations on a linear system Linear system 1.

Without using the method of elimination, solve the 9. Without using the method of elimination, solve the linear system If there is, find it. A plastics manufacturer makes two types of plastic: If there is, find regular and special.

Each ton of regular plastic requires it. Describe the number of points that simultaneously lie in each of the three planes shown in each part of Figure 1. A dietician is preparing a meal consisting of foods A, B, and C.

Each ounce of food A contains 2 units of protein, Describe the number of points that simultaneously lie in 3 units of fat, and 4 units of carbohydrate.

Each ounce of each of the three planes shown in each part of Figure 1. Each ounce of food C contains 3 P1 P1 units of protein, 3 units of fat, and 2 units of P3 carbohydrate. If the meal must provide exactly 25 units P2 of protein, 24 units of fat, and 21 units of carbohydrate, P2 P3 how many ounces of each type of food should be used?

A manufacturer makes 2-minute, 6-minute, and a b 9-minute film developers. Each ton of 2-minute developer requires 6 minutes in plant A and 24 minutes in plant B.

Each ton of 6-minute developer requires 12 minutes in plant A and 12 minutes in plant B. Each ton P1 P2 of 9-minute developer requires 12 minutes in plant A and 12 minutes in plant B. If plant A is available 10 hours per day and plant B is available 16 hours per day, how many tons of each type of developer can be P3 produced so that the plants are fully utilized?

An oil refinery produces low-sulfur and high-sulfur fuel. If the blending trusts, with the second trust receiving twice as much as plant is available for 3 hours and the refining plant is the first trust. How utilized? Theoretical Exercises T. Show that the linear system obtained by interchanging equation in 2 by itself plus a multiple of another two equations in 2 has exactly the same solutions as equation in 2 has exactly the same solutions as 2.

Does the linear system T. Show that the linear system obtained by replacing an always have a solution for any values of a, b, c, and d? Only the numbers in front of the unknowns x1 , x2 ,. Thus we might think of looking for a way of writing a linear system without having to carry along the unknowns.

In this section we define an ob- ject, a matrix, that enables us to do this—that is, to write linear systems in a compact form that makes it easier to automate the elimination method on a computer in order to obtain a fast and efficient procedure for finding solutions. The use of a matrix is not, however, merely that of a convenient notation. We now develop operations on matrices plural of matrix and will work with ma- trices according to the rules they obey; this will enable us to solve systems of linear equations and solve other computational problems in a fast and efficient manner.

Of course, as any good definition should do, the notion of a matrix provides not only a new way of looking at old problems, but also gives rise to a great many new questions, some of which we study in this book.

For the sake of simplicity, we restrict our attention in this book, except for Appendix A, to matrices all of whose entries are real numbers.

However, matrices with complex entries are studied and are important in applications. For convenience, we focus much of our attention in the illustrative ex- amples and exercises in Chapters 1—7 on matrices and expressions containing only real numbers.

Complex numbers will make a brief appearance in Chap- ters 8 and 9. An introduction to complex numbers, their properties, and exam- ples and exercises showing how complex numbers are used in linear algebra may be found in Appendix A. When n is understood, we refer to n-vectors merely as vectors. In Chapter 4 we discuss vectors at length. The set of all n-vectors with real entries is denoted by R n. Similarly, the set of all n-vectors with complex entries is denoted by C n.

As we have already pointed out, in the first seven chapters of this book we will work almost entirely with vectors in R n. For example, plant 2 makes units of product 3 in one week. EXAMPLE 5 The wind chill table that follows shows how a combination of air temperature and wind speed makes a body feel colder than the actual temperature.

A large measure of the effectiveness of the search engine Google c is the manner in which matrices are used to determine which sites are referenced by other sites. To build the connections proceed as follows. When it is detected that Web site j links to Web site i, set entry ai j equal to one.

Since n is quite large, about 3 billion as of December , most entries of the connectivity matrix A are zero. Such a matrix is called sparse. If row i of A contains many ones, then there are many sites linking to site i. Such sites would appear near the top of a list returned by a Google search on topics related to the information on site i. Since Google updates its connectivity matrix about every month, n increases over time and new links and sites are adjoined to the connectivity matrix.

The fundamental technique used by Google c to rank sites uses linear algebra concepts that are somewhat beyond the scope of this course. Further information can be found in the following sources. Berry, Michael W.

Siam, Moler, Cleve. Whenever a new object is introduced in mathematics, we must define when two such objects are equal. For example, in the set of all rational num- bers, the numbers 23 and 46 are called equal although they are not represented in the same manner. Accordingly, we now have the following definition. We shall now define a number of operations that will produce new matri- ces out of given matrices.

These operations are useful in the applications of matrices. That is, C is obtained by adding corresponding elements of A and B. Thus far, addition of matrices has only been defined for two matrices. Our work with matrices will call for adding more than two matrices.

Theorem 1. Additional properties of matrix addition are considered in Section 1. Each model is partially made in factory F1 in Taiwan and then finished in factory F2 in the United States. The total cost of each product consists of the manufacturing cost and the shipping cost. That is, B is obtained by multiplying each element of A by r.

Using scalar multiplication and matrix addition, we can compute C: Thus, the entries in each row of A T are the entries in the corresponding column of A. Hence computations, like linear combi- nations, are determined using matrix properties and standard arithmetic base However, the continued expansion of computer technology has brought to the forefront the use of binary base 2 representation of information.

In most computer applications like video games, FAX communications, ATM money transfers, satellite communications, DVD videos, or the generation of music CDs, the underlying mathematics is invisible and completely transparent to the viewer or user. Binary coded data is so prevalent and plays such a central role that we will briefly discuss certain features of it in appropriate sections of this book. We begin with an overview of binary addition and multiplication and then introduce a special class of binary matrices that play a prominent role in information and communication theory.

Binary representation of information uses only two symbols 0 and 1. In- formation is coded in terms of 0 and 1 in a string of bits.

## Freely available

The coefficients of the powers of 2 determine the string of bits, , which provide the binary representation of 5. Just as there is arithmetic base 10 when dealing with the real and complex numbers, there is arithmetic using base 2; that is, binary arithmetic. Table 1. We will not digress to review such topics at this time. However, our focus will be on a particular type of ma- trix and vector that contain entries that are single binary digits.

This class of matrices and vectors are important in the study of information theory and the mathematical field of error-correcting codes also called coding theory. That is, each entry is either 0 or 1. Using the definition of matrix addition 0 1 1 0 and Table 1. Thus the additive inverse of 0 is 0 as usual and the additive inverse of 1 is 1.

Hence to compute the difference of bit matrices A and B we proceed as follows: We see that the difference of bit matrices contributes nothing new to the alge- braic relationships among bit matrices. If possible, compute the indicated linear combination: If possible, compute: Justify your answer.

Compute each of the following. The elements above the main diagonal are zero. A T is lower triangular. Make a list of all possible bit 2-vectors. Make a list of all possible bit 3-vectors. Upper triangular matrix T. Make a list of all possible bit 4-vectors.

How many are The elements below the main diagonal are zero. How many bit 5-vectors are there? How many bit T. A standard light switch has two positions or states ; n-vectors are there? You are suppresses the display of the contents of matrix H. For more information on the hilb commands that appear in Sections Use bingen to solve Exercises T. Use bingen to solve Exercise T. Compare the elements of B matrix contains the same number of entries as an from part a with the current display.

Note that n 2 -vector. Reset the format to format short. Solve Exercise 11 using binadd.

Unlike matrix addition, matrix multiplication has some properties that distinguish it from multiplication of real numbers. The dot product of vectors in C n is defined in Appendix A. The dot product is an important operation that will be used here and in later sections. It is discussed in detail at the end of this section. The basic properties of matrix multiplication will be considered in the following section.

However, multiplication of matrices requires much more care than their addition, since the algebraic properties of matrix multiplication differ from those satisfied by the real numbers. Part of the problem is due to the fact that AB is defined only when the number of columns of A is the same as the number of rows of B. What about B A? Four different situations may occur: If AB and B A are both of the same size, they may be equal. If AB and B A are both of the same size, they may be unequal.

One might ask why matrix equality and matrix addition are defined in such a natural way while matrix multiplication appears to be much more com- plicated.

Example 11 provides a motivation for the definition of matrix multi- plication. How- ever, some of the pesticide is absorbed by the plant. The pesticides are ab- sorbed by herbivores when they eat the plants that have been sprayed. To determine the amount of pesticide absorbed by a herbivore, we proceed as fol- lows.

Suppose that we have three pesticides and four plants. Let ai j denote the amount of pesticide i in milligrams that has been absorbed by plant j. If we now have p carnivores such as man who eat the herbivores, we can repeat the analysis to find out how much of each pesticide has been absorbed by each carnivore.

It can be shown Exercise T. This observation will be used in Chapter 3. Now define the following matrices: The augmented matrix of 5 will be written as A b. Conversely, any matrix with more than one column can be thought of as the augmented matrix of a linear system. The coefficient and augmented matrices will play key roles in our method for solving linear systems.

Of course, the partitioning can be carried out in many different ways. We thus speak of partitioned matrices. Similarly, if A is a partitioned matrix, then the scalar multiple c A is obtained by forming the scalar multiple of each submatrix. We verify that C11 is this expression as follows: Partitioned matrices can be used to great advantage in dealing with matrices that exceed the memory capacity of a computer.

Thus, in mul- tiplying two partitioned matrices, one can keep the matrices on disk and only bring into memory the submatrices required to form the submatrix products.

The latter, of course, can be put out on disk as they are formed. The parti- tioning must be done so that the products of corresponding submatrices are defined.

In contemporary computing technology, parallel-processing comput- ers use partitioned matrices to perform matrix computations more rapidly. Partitioning of a matrix implies a subdivision of the information into blocks or units. The reverse process is to consider individual matrices as blocks and adjoin them to form a partitioned matrix. The only requirement is that after joining the blocks, all rows have the same number of entries and all columns have the same number of entries.

Similarly, results of new laboratory experiments are adjoined to existing data to update a database in a research facility. At times we shall need to solve several linear systems in which the coefficient matrix A is the same but the right sides of the systems are different, say b, c, and d.

See Section 6. The letter i is called the index of summation; it is a dummy variable that can be replaced by another letter. It is not difficult to show Exercise T. It is not difficult to show, in general Exercise T. If we add up the entries in each row of A and then add the resulting numbers, we obtain the same result as when we add up the entries in each column of A and then add the resulting numbers. Consider the following linear system: Write the linear system with augmented matrix matrix every one of whose entries is zero, compute AO.

Using the method in Example 12, compute the following columns of AB: How are the linear systems whose augmented matrices Write each of the following linear systems as a linear 9 10 Assembly process combination of the columns of the coefficient matrix. In What making these products, the pollutants sulfur dioxide, if anything can you say about the matrix product AB nitric oxide, and particulate matter are produced. The when: Sulfur Nitric Particulate Plant X Plant Y Let A and B be the following matrices: Business A photography business has a store in each 1 of the following cities: The selling prices of the cameras and flash If 0 y 1 1 a What is the total value of the cameras in New York?

For bit matrices Let x be an n-vector. Show that the product of two scalar matrices is a scalar matrix. Let a, b, and c be n-vectors and let k be a real number. Is row of zeros. Answer the following as true or false. If true, prove Write aB as a linear combination of the rows of B. Let u and v be n-vectors. Exercises T. Repeat the preceding exercise with the following the following matrices: Enter matrices following, if possible. Do not type the period! Explain why this is true for almost all vectors v.

If the vectors ML. Use binprod to solve Exercise Find a value for k so based on the columns of B. Repeat Exercise ML. Many of these properties are similar to familiar properties of the real numbers. However, there will be striking differences between the set of real numbers and the set of matrices in their algebraic behavior under certain operations, for example, under multiplication as seen in Section 1.

Most of the properties will be stated as theorems, whose proofs will be left as exercises. Proof a We omit a general proof here. Exercise T. If p is a positive integer, then we define the powers of a matrix as follows: For nonnegative integers p and q, some of the familiar laws of exponents for the real numbers can also be proved for matrix multiplication of a square matrix A Exercise T.

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We now note two other peculiarities of matrix multiplication. However, this is not true for matrices. That is, we can cancel a out. However, the cancellation law does not hold for matrices, as the following example shows. Remark In Section 1. Each year, S keeps 23 of its customers while 1 3 switch to R. It is assumed that the manufacturer can sell as much of the products as is made. To solve this problem, we let x1 , x2 , and x3 denote the number of tons of products A, B, and C, respectively, to be made.

This linear system has infinitely many solutions. The reader should observe that one solution is just as good as the other. There is no best solution unless additional information or restrictions are given. Key Terms Linear equation Solution to a linear system No solution Unknowns Method of elimination Infinitely many solutions Solution to a linear equation Unique solution Manipulations on a linear system Linear system 1.

Without using the method of elimination, solve the 9. Without using the method of elimination, solve the linear system If there is, find it. A plastics manufacturer makes two types of plastic: If there is, find regular and special. Each ton of regular plastic requires it. Describe the number of points that simultaneously lie in each of the three planes shown in each part of Figure 1. A dietician is preparing a meal consisting of foods A, B, and C.

Each ounce of food A contains 2 units of protein, Describe the number of points that simultaneously lie in 3 units of fat, and 4 units of carbohydrate. Each ounce of each of the three planes shown in each part of Figure 1. Each ounce of food C contains 3 P1 P1 units of protein, 3 units of fat, and 2 units of P3 carbohydrate.

If the meal must provide exactly 25 units P2 of protein, 24 units of fat, and 21 units of carbohydrate, P2 P3 how many ounces of each type of food should be used? A manufacturer makes 2-minute, 6-minute, and a b 9-minute film developers. Each ton of 2-minute developer requires 6 minutes in plant A and 24 minutes in plant B.

Each ton of 6-minute developer requires 12 minutes in plant A and 12 minutes in plant B. Each ton P1 P2 of 9-minute developer requires 12 minutes in plant A and 12 minutes in plant B. If plant A is available 10 hours per day and plant B is available 16 hours per day, how many tons of each type of developer can be P3 produced so that the plants are fully utilized?

An oil refinery produces low-sulfur and high-sulfur fuel. If the blending trusts, with the second trust receiving twice as much as plant is available for 3 hours and the refining plant is the first trust.

How utilized? Theoretical Exercises T. Show that the linear system obtained by interchanging equation in 2 by itself plus a multiple of another two equations in 2 has exactly the same solutions as equation in 2 has exactly the same solutions as 2.

Does the linear system T. Show that the linear system obtained by replacing an always have a solution for any values of a, b, c, and d? Only the numbers in front of the unknowns x1 , x2 ,.

Thus we might think of looking for a way of writing a linear system without having to carry along the unknowns. In this section we define an ob- ject, a matrix, that enables us to do this—that is, to write linear systems in a compact form that makes it easier to automate the elimination method on a computer in order to obtain a fast and efficient procedure for finding solutions.

The use of a matrix is not, however, merely that of a convenient notation. We now develop operations on matrices plural of matrix and will work with ma- trices according to the rules they obey; this will enable us to solve systems of linear equations and solve other computational problems in a fast and efficient manner.

Of course, as any good definition should do, the notion of a matrix provides not only a new way of looking at old problems, but also gives rise to a great many new questions, some of which we study in this book.

For the sake of simplicity, we restrict our attention in this book, except for Appendix A, to matrices all of whose entries are real numbers. However, matrices with complex entries are studied and are important in applications.

For convenience, we focus much of our attention in the illustrative ex- amples and exercises in Chapters 1—7 on matrices and expressions containing only real numbers. Complex numbers will make a brief appearance in Chap- ters 8 and 9. An introduction to complex numbers, their properties, and exam- ples and exercises showing how complex numbers are used in linear algebra may be found in Appendix A. When n is understood, we refer to n-vectors merely as vectors.

In Chapter 4 we discuss vectors at length. The set of all n-vectors with real entries is denoted by R n. Similarly, the set of all n-vectors with complex entries is denoted by C n. As we have already pointed out, in the first seven chapters of this book we will work almost entirely with vectors in R n. For example, plant 2 makes units of product 3 in one week. EXAMPLE 5 The wind chill table that follows shows how a combination of air temperature and wind speed makes a body feel colder than the actual temperature.

A large measure of the effectiveness of the search engine Google c is the manner in which matrices are used to determine which sites are referenced by other sites.

To build the connections proceed as follows. When it is detected that Web site j links to Web site i, set entry ai j equal to one. Since n is quite large, about 3 billion as of December , most entries of the connectivity matrix A are zero. Such a matrix is called sparse. If row i of A contains many ones, then there are many sites linking to site i. Such sites would appear near the top of a list returned by a Google search on topics related to the information on site i.

Since Google updates its connectivity matrix about every month, n increases over time and new links and sites are adjoined to the connectivity matrix. The fundamental technique used by Google c to rank sites uses linear algebra concepts that are somewhat beyond the scope of this course.

Further information can be found in the following sources. Berry, Michael W. Siam, Moler, Cleve. Whenever a new object is introduced in mathematics, we must define when two such objects are equal. For example, in the set of all rational num- bers, the numbers 23 and 46 are called equal although they are not represented in the same manner.

Accordingly, we now have the following definition. We shall now define a number of operations that will produce new matri- ces out of given matrices. These operations are useful in the applications of matrices.

That is, C is obtained by adding corresponding elements of A and B. Thus far, addition of matrices has only been defined for two matrices.

Our work with matrices will call for adding more than two matrices. Theorem 1. Additional properties of matrix addition are considered in Section 1. Each model is partially made in factory F1 in Taiwan and then finished in factory F2 in the United States.

The total cost of each product consists of the manufacturing cost and the shipping cost. That is, B is obtained by multiplying each element of A by r. Using scalar multiplication and matrix addition, we can compute C: Thus, the entries in each row of A T are the entries in the corresponding column of A. Hence computations, like linear combi- nations, are determined using matrix properties and standard arithmetic base However, the continued expansion of computer technology has brought to the forefront the use of binary base 2 representation of information.

In most computer applications like video games, FAX communications, ATM money transfers, satellite communications, DVD videos, or the generation of music CDs, the underlying mathematics is invisible and completely transparent to the viewer or user. Binary coded data is so prevalent and plays such a central role that we will briefly discuss certain features of it in appropriate sections of this book. We begin with an overview of binary addition and multiplication and then introduce a special class of binary matrices that play a prominent role in information and communication theory.

Binary representation of information uses only two symbols 0 and 1. In- formation is coded in terms of 0 and 1 in a string of bits. The coefficients of the powers of 2 determine the string of bits, , which provide the binary representation of 5. Just as there is arithmetic base 10 when dealing with the real and complex numbers, there is arithmetic using base 2; that is, binary arithmetic.

Table 1. We will not digress to review such topics at this time. However, our focus will be on a particular type of ma- trix and vector that contain entries that are single binary digits.

This class of matrices and vectors are important in the study of information theory and the mathematical field of error-correcting codes also called coding theory.

That is, each entry is either 0 or 1. Using the definition of matrix addition 0 1 1 0 and Table 1. Thus the additive inverse of 0 is 0 as usual and the additive inverse of 1 is 1. Hence to compute the difference of bit matrices A and B we proceed as follows: We see that the difference of bit matrices contributes nothing new to the alge- braic relationships among bit matrices. If possible, compute the indicated linear combination: If possible, compute: Justify your answer. Compute each of the following.

The elements above the main diagonal are zero. A T is lower triangular. Make a list of all possible bit 2-vectors. Make a list of all possible bit 3-vectors. Upper triangular matrix T.

## CHEAT SHEET

Make a list of all possible bit 4-vectors. How many are The elements below the main diagonal are zero. How many bit 5-vectors are there? How many bit T. A standard light switch has two positions or states ; n-vectors are there?

You are suppresses the display of the contents of matrix H. For more information on the hilb commands that appear in Sections Use bingen to solve Exercises T. Use bingen to solve Exercise T. Compare the elements of B matrix contains the same number of entries as an from part a with the current display.

Note that n 2 -vector. Reset the format to format short. Solve Exercise 11 using binadd. Unlike matrix addition, matrix multiplication has some properties that distinguish it from multiplication of real numbers. The dot product of vectors in C n is defined in Appendix A. The dot product is an important operation that will be used here and in later sections. It is discussed in detail at the end of this section. The basic properties of matrix multiplication will be considered in the following section.

However, multiplication of matrices requires much more care than their addition, since the algebraic properties of matrix multiplication differ from those satisfied by the real numbers. Part of the problem is due to the fact that AB is defined only when the number of columns of A is the same as the number of rows of B.

What about B A? Four different situations may occur: If AB and B A are both of the same size, they may be equal. If AB and B A are both of the same size, they may be unequal. One might ask why matrix equality and matrix addition are defined in such a natural way while matrix multiplication appears to be much more com- plicated.

Example 11 provides a motivation for the definition of matrix multi- plication. How- ever, some of the pesticide is absorbed by the plant. The pesticides are ab- sorbed by herbivores when they eat the plants that have been sprayed.

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To determine the amount of pesticide absorbed by a herbivore, we proceed as fol- lows. Suppose that we have three pesticides and four plants. Let ai j denote the amount of pesticide i in milligrams that has been absorbed by plant j. If we now have p carnivores such as man who eat the herbivores, we can repeat the analysis to find out how much of each pesticide has been absorbed by each carnivore.

It can be shown Exercise T. This observation will be used in Chapter 3.

## Elementary Linear Algebra Kolman Pdf

Now define the following matrices: The augmented matrix of 5 will be written as A b. Conversely, any matrix with more than one column can be thought of as the augmented matrix of a linear system.

The coefficient and augmented matrices will play key roles in our method for solving linear systems. Of course, the partitioning can be carried out in many different ways. We thus speak of partitioned matrices. Similarly, if A is a partitioned matrix, then the scalar multiple c A is obtained by forming the scalar multiple of each submatrix.

We verify that C11 is this expression as follows: Partitioned matrices can be used to great advantage in dealing with matrices that exceed the memory capacity of a computer.

Thus, in mul- tiplying two partitioned matrices, one can keep the matrices on disk and only bring into memory the submatrices required to form the submatrix products. The latter, of course, can be put out on disk as they are formed. The parti- tioning must be done so that the products of corresponding submatrices are defined. In contemporary computing technology, parallel-processing comput- ers use partitioned matrices to perform matrix computations more rapidly.

Partitioning of a matrix implies a subdivision of the information into blocks or units. The reverse process is to consider individual matrices as blocks and adjoin them to form a partitioned matrix.

The only requirement is that after joining the blocks, all rows have the same number of entries and all columns have the same number of entries. Similarly, results of new laboratory experiments are adjoined to existing data to update a database in a research facility.

At times we shall need to solve several linear systems in which the coefficient matrix A is the same but the right sides of the systems are different, say b, c, and d. See Section 6. The letter i is called the index of summation; it is a dummy variable that can be replaced by another letter. It is not difficult to show Exercise T. It is not difficult to show, in general Exercise T.

If we add up the entries in each row of A and then add the resulting numbers, we obtain the same result as when we add up the entries in each column of A and then add the resulting numbers.

Consider the following linear system: Write the linear system with augmented matrix matrix every one of whose entries is zero, compute AO. Using the method in Example 12, compute the following columns of AB: How are the linear systems whose augmented matrices Write each of the following linear systems as a linear 9 10 Assembly process combination of the columns of the coefficient matrix.

In What making these products, the pollutants sulfur dioxide, if anything can you say about the matrix product AB nitric oxide, and particulate matter are produced. The when: Sulfur Nitric Particulate Plant X Plant Y Let A and B be the following matrices: Business A photography business has a store in each 1 of the following cities: The selling prices of the cameras and flash If 0 y 1 1 a What is the total value of the cameras in New York?

For bit matrices Let x be an n-vector. Show that the product of two scalar matrices is a scalar matrix. Let a, b, and c be n-vectors and let k be a real number. Is row of zeros. Answer the following as true or false. If true, prove Write aB as a linear combination of the rows of B.

Let u and v be n-vectors. Exercises T. Repeat the preceding exercise with the following the following matrices: Enter matrices following, if possible. Do not type the period! Explain why this is true for almost all vectors v. If the vectors ML. Use binprod to solve Exercise Find a value for k so based on the columns of B. Repeat Exercise ML. Many of these properties are similar to familiar properties of the real numbers. However, there will be striking differences between the set of real numbers and the set of matrices in their algebraic behavior under certain operations, for example, under multiplication as seen in Section 1.

Most of the properties will be stated as theorems, whose proofs will be left as exercises. Proof a We omit a general proof here.

Exercise T. If p is a positive integer, then we define the powers of a matrix as follows: For nonnegative integers p and q, some of the familiar laws of exponents for the real numbers can also be proved for matrix multiplication of a square matrix A Exercise T.

We now note two other peculiarities of matrix multiplication. However, this is not true for matrices. That is, we can cancel a out. However, the cancellation law does not hold for matrices, as the following example shows. Remark In Section 1. Each year, S keeps 23 of its customers while 1 3 switch to R. Then, initially, R has 35 k customers, and S has 25 k customers.

When this happens, the distribution of the market is said to be stable. We proceed as follows. Observe that the two equations in 4 are the same. The problem described is an example of a Markov chain. We shall return to this topic in Section 2. The i, jth element of AB T is ciTj. If matrix A is symmetric, then the elements of A are symmetric with respect to the main diagonal of A.

Hence the scalars available are only 0 and 1. Find the additive inverse of A. See also Exercise T. As an example, note the explosive growth of the Internet and the promises of using it to interact with all types of media. Graph theory is an area of applied mathematics that deals with problems such as this one: Consider a local area network consisting of six users denoted by P1 , P2 ,.

On the other hand, Pi may not be able to send a message directly to Pk , but can send it to P j , who will then send it to Pk. Thus A may be Figure 1. Many other problems involving communications can be solved using graph theory. The matrix A above is indeed a bit matrix, but in this situation A is best considered as a matrix in base 10, as will be shown in Section 2.

Verify Theorem 1. Verify a of Theorem 1. Verify b of Theorem 1. Verify a , b , and c of Theorem 1. Verify d of Theorem 1. Verify b and d of Theorem 1. Verify c of Theorem 1.

Approximately what percent of the market was gained by this company? Suppose that in Exercise 20 the matrix A was given by Exercises 18 through 21 deal with Markov chains, an area R S T that will be studied in greater detail in Section 2.

Consider two quick food companies, M and N.

Each 3 year, company M keeps 13 of its customers, while 23 then determine the market distribution after 1 year; switch to N. Each year, N keeps 12 of its customers, after 2 years.

Suppose that in Example 9 there were three rival market share over a long period of time assuming companies R, S, and T so that the pattern of customer that the retention and switching patterns remain the retention and switching is given by the information in same? Approximately what percent of the market the matrix A where was gained by this company? Find then determine the market distribution after 1 year; 1 1 after 2 years. Prove properties b and d of Theorem 1. Show that if A is symmetric, then A T is symmetric.

Prove properties b and c of Theorem 1. Let p and q be nonnegative integers and let A be a square matrix. Show that T. Let A and B be symmetric matrices. Describe all skew symmetric scalar matrices. How many such matrices B are there? If p is a nonnegative integer and c is a scalar, show that Hint: Prove Theorem 1.

If A is a skew symmetric matrix, what type of matrix T. Complete the proof of Theorem 1. What type of matrix is a linear combination of symmetric matrices?

See Section 1. Justify your T. See also Exercise Write a description of the behavior of this matrix The reader should observe that one solution is just as good as the other. A taste of this type of software is provided by the third type of exercises. A standard light switch has two positions or states. Suppose that we have three pesticides and four plants. Thus the instructor can be quite selective both in the choice of material and in the method of study of these applications.

These exer- cises are designed to be solved by an appropriate computer software package.

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