nito/shutterstock, Inc. Jones The & Montjuïc Bartlett Communications Learning, LLC Tower, or Torre Telefónica, NOT FOR was SALE built in the OR center DISTRIBUTION of the Olympic Park in Barcelona, Spain, for the 199 Olympic Games. The tower, built by Spanish architect Santiago Calatrava, was designed to carry coverage of the Olympic Games to broadcast stations around the world. The structure was designed to Jones & Bartlett Learning, represent an LLC athlete holding up an Olympic torch. Jones & Bartlett Learning, LLC..
PART Linear Equations, Vectors, NOT FOR and SALE OR Matrices DISTRIBUTION 1 1 3 Linear Equations and Vectors Jones & Bartlett Matrices Learning, and Linear LLC Transformations Determinants and Eigenvectors 3..
okeyphotos/getty Images NOT Informally FOR SALE referred OR to DISTRIBUTION as the Gherkin, 30 St. Mary Axe in London, England, is located in London s financial district. The building employs energy-saving methods, such as maximizing the use of natural light and ventilation, which allow it to use half the power a similar structure would typically consume...
C H A P T E R Linear Equations and Vectors 1 Mathematics is, of course, a discipline in its own right. It is, however, more than that it is a tool used in many other fields. Linear algebra is a branch NOT FOR SALE OR of DISTRIBUTION mathematics that plays a central role NOT in modern FOR mathematics, SALE OR DISTRIBUTION and also is of importance to engineers and physical, social, and behavioral scientists. In this course the reader will learn mathematics, will learn to think mathematically, and will be instructed in the art of applying mathematics. The course is a blend of theory, numerical techniques, and interesting applications. When mathematics is used to solve a problem it often becomes necessary to find a solution to a so-called system of linear equations. Historically, linear algebra developed from studying methods for solving such equations. This chapter introduces methods for solving Jones systems & of Bartlett linear equations Learning, and looks LLCat some of the properties of Jones the solutions. & Bartlett Learning, LLC It is important NOT FOR to SALE know not OR only DISTRIBUTION what the solutions to a given system NOT of equations FOR SALE are OR DISTRIBUTION but why they are the solutions. If the system describes some real-life situation, then an understanding of the behavior of the solutions can lead to a better understanding of the Jones & circumstances. Bartlett Learning, The solutions LLCform subsets of spaces called Jones vector & Bartlett spaces. We Learning, develop LLC NOT FOR the SALE basic OR algebraic DISTRIBUTION structure of vector spaces. We shall NOT discuss FOR two SALE applications OR DISTRIBUTION of systems of linear equations. We shall determine currents through electrical networks and analyze traffic flows through road networks. 1.1 Matrices and NOT Systems FOR of SALE Linear OR Equations DISTRIBUTION An equation in the variables x and y that can be written in the form ax 1 by 5 c, where a, b,andcare real constants (a and b not both zero), is called a linear equation. The graph of such an equation is a straight line in the xy-plane. Consider the system of two linear equations, x 1 y 5 5 x y 5 4 5..
6 CHAPTER 1 Linear Equations and Vectors A pair of values of x and y that NOT satisfies FOR both SALE equations OR DISTRIBUTION is called a solution. It can be seen by substitution that x 5 3, y 5 is a solution to this system. A solution to such a system will be a point at which the graphs of the two equations intersect. The following examples, Figures 1.1, 1., and 1.3, illustrate that three possibilities can arise for such systems of equations. There can be a unique solution, no solution, or many solutions. We use the Jones point/slope & Bartlett form y 5 Learning, mx 1 b, where LLC m is the slope and b is the y-intercept, Jones & Bartlett to graph these Learning, LL NOT lines. FOR SALE OR DISTRIBUTION Unique solution x + y = 5 x y = 4 Write as y = x + 5 and y = x 4. The lines have slopes 1 and, and y-intercepts 5 and 4. They intersect at a point, the solution. There is a unique solution, x = 3, y =. y NOT FOR x + y = SALE 5 OR x DISTRIBUTION y = 4 No solution x + y = 3 4x + y = Write as y = x + 3 and y = x + 1. The lines have slope, and y-intercepts 3 and 1. They are parallel. There is no point of intersection. No solution. y Many solutions 4x y = 6 6x 3y = 9 Each equation can be written as y = x 3. The graph of each equation is a line with slope and y-intercept 3. Any point on the line is a solution. Many solutions. y (3, ) x + y = 3 4x + y = x x 4x y = 6 6x 3y = 9 x Figure 1.1 Figure 1. Figure 1.3 Our aim in this chapter is to analyze larger systems of linear equations. A linear equation in n variables x 1, x, x 3, c, x n is one that can be written in the form a 1 x 1 1 a x 1 a 3 x 3 1 c 1 a n x n 5 b, where the coefficients a 1, a NOT, c, FOR a n and SALE b are constants. OR DISTRIBUTION The following is an example of a system of three linear equations. 1x 1 1 1x 1 1x 3 5 Jones & Bartlett Learning, x LLC 1 1 3x 1 x 3 5 3 1x 1 1x x 3 56 It can be seen on substitution that x 1 51, x 5 1, x 3 5 is a solution to this system. (We arrive at this solution in Example 1 of this section.) A linear equation in three variables corresponds to a plane in three-dimensional space. Solutions to a system of three such equations will be points that lie on all three planes. As for systems of two equations, there can be a unique solution, no solution, or many solutions. We illustrate some of the various possibilities in Figure 1.4. As the number of variables increases, a geometrical interpretation of such a system of equations becomes increasingly complex. Each equation will represent a space embedded in a larger space. Solutions will Jones be points & that Bartlett lie on all Learning, the embedded LLC spaces. While a general geometrical way of thinking NOT about FOR a problem SALE is OR often DISTRIBUTION useful, we rely on algebraic meth- ods for arriving at and interpreting the solution. We introduce a method for solving systems..
1.1 Matrices and Systems of Linear Equations 7 NOT FOR Unique SALE solution OR DISTRIBUTION B B A P A P C C Three planes A, B, and C intersect at a single point P. P corresponds to a unique solution. Jones A & Bartlett Learning, LLC No solution A B C B C Many solutions Planes A, B, and C have no points in common. There is no solution. B A B C A Q P Three planes A, B, and C intersect in a line PQ. Any point on the line is a solution. C Three equations represent the same plane. Any point on the plane is a solution. Figure 1.4 of linear equations called Jones Gauss-Jordan & Bartlett elimination. Learning, 1 This LLC method involves systematically eliminating variables NOT from FOR equations. SALE OR In this DISTRIBUTION section, we shall see how this method applies to systems of equations that have a unique solution. In the following section, we shall extend the method to more general systems of linear equations. We shall use rectangular arrays of numbers called matrices to describe systems of linear equations. Jones At & this Bartlett time we Learning, introduce the LLC necessary terminology. 1Carl Friedrich Gauss (1777 1855) was one of the greatest mathematical scientists ever. Among his discoveries was a way to calculate the orbits of asteroids. He taught for forty-seven years at the University of Göttingen, Germany. He made contributions to many areas of mathematics, including number theory, probability, and statistics. Gauss has been described as not really a physicist in the sense of searching for new phenomena, but rather a mathematician who attempted to formulate in exact mathematical terms the experimental results of others. Gauss had a turbulent personal life, suffering financial and political problems because of revolutions in Germany. Wilhelm Jordan (184 1899) taught geodesy at the Technical College of Karlsruhe, Germany. His most important work was a handbook on geodesy that contained his research on systems of equations. Jordan was recognized as being a master teacher and an excellent writer...
8 CHAPTER 1 Linear Equations and Vectors NOT FOR DEFINITION SALE OR DISTRIBUTION A matrix is a rectangular array of numbers. NOT The FOR numbers SALE in the OR array DISTRIBUTION are called the elements of the matrix. Matrices are usually denoted by capital letters. Examples of matrices in standard notation are& Bartlett Learning, LLC Jones 7 1 3 5 6 A 5 c 3 4, B 5 0 5, C 5 0 5 7 5 1 d 8 3 8 9 1 Jones & Bartlett Rows Learning, and Columns LLC Matrices consist of rows and Jones columns. & Bartlett Rows are Learning, labeled from LLC the top NOT FOR SALE OR of the DISTRIBUTION matrix, columns from the left. The following NOT matrix FOR has SALE two rows OR DISTRIBUTION and three columns. c 3 4 7 5 1 d The rows are: The columns are: NOT 3 FOR 3 44, SALE 37OR 5DISTRIBUTION 14 row 1 row Jones & Bartlett Learning, c,, c 4 7 d LLC c3 5 d 1 d column 1 column column 3 Submatrix A submatrix of a given matrix is an array obtained by deleting certain rows and columns of the matrix. For example, consider the following matrix A. The matrices P, Jones & Bartlett Q, and Learning, R are submatrices LLC of A. 1 7 4 1 7 7 A 5 3 0 P 5 3 Q 5 3 R 5 c 1 4 5 d 5 1 5 1 1 matrix A submatrices of A Size and Type The size of NOT a matrix FOR is described SALE OR by DISTRIBUTION specifying the number of rows and columns in the matrix. For example, a matrix having two rows and three columns is said to be a 3 3 matrix; the first number indicates the number of rows, and the second indicates the number of columns. When the number of rows is equal to the number of columns, the matrix is said to be a square matrix. A matrix consisting of one row is called a row matrix. Jones A matrix & consisting Bartlett Learning, of one column LLC is a column matrix. The following Jones matrices & Bartlett are of Learning, the LL NOT stated FOR sizes SALE and types. OR DISTRIBUTION 5 7 8 1 0 3 c 4 5 d 9 0 1 34 3 8 54 3 3 5 8 3 3matrix 3 3 3matrix NOT FOR 1 3SALE 4 matrix OR DISTRIBUTION 3 3 1 matrix a square matrix a row matrix a column matrix Location The location of an element in a matrix is described by giving the row and column in which the element lies. For example, consider the following matrix. Jones c & Bartlett 3 4 7 5 1 d Learning, LLC The element 7 is in row, column 1. We say that it is in location (, 1)...
1.1 Matrices and Systems of Linear Equations 9 NOT FOR SALE The element OR DISTRIBUTION in location (1, 3) is 4. Note that the NOT convention FOR SALE is to give OR the DISTRIBUTION row in which the element lies, followed by the column. Identity Matrices An identity matrix is a square matrix with 1s in the diagonal locations (1, 1), (, ), (3, 3), etc., and zeros elsewhere. We write I n for the n 3 n identity matrix. The following matrices Jones are identity & Bartlett matrices. Learning, LLC 1 0 0 I 5 c 1 0, I 3 5 0 1 0 0 1 d 0 0 1 We are now Jones ready & to Bartlett continue Learning, the discussion LLC of systems of linear equations. Jones We use & matrices to describe systems of linear equations. There are two important matrices associated Bartlett Learning, LLC with every system of linear equations. The coefficients of the variables form a matrix called the matrix of coefficients of the system. The coefficients, together with the constant terms, form a matrix called the augmented matrix of the system. For example, the matrix of coefficients and the augmented matrix of the following system of linear equations are as shown: NOT FOR SALE x OR 1 1DISTRIBUTION x 1 x 3 5 1 1 1NOT FOR SALE 1 1OR DISTRIBUTION 1 x 1 1 3x 1 x 3 5 3 3 1 3 1 3 x 1 x x 3 56 1 1 1 1 6 matrix of coefficients augmented matrix Observe that the matrix NOT of coefficients FOR SALE is a OR submatrix DISTRIBUTION of the augmented matrix. The augmented matrix completely describes the system. Transformations called elementary transformations can be used to change a system of linear equations into another system of linear equations that has the same solution. These transformations are used to solve systems of linear equations by eliminating variables. In practice Jones it is simpler & Bartlett to work Learning, in terms of matrices LLC using analogous transformations Jones & called Bartlett Learning, LLC elementary NOT FOR row operations. SALE OR It DISTRIBUTION is not necessary to write down the variables NOT FOR x 1, x, SALE x 3, at OR DISTRIBUTION each stage. Systems of linear equations are in fact described and manipulated on computers in terms of such matrices. These transformations are as follows. Elementary Transformations Elementary Row Operations Jones & 1. Bartlett Interchange Learning, two equations. LLC 1. Interchange Jones two rows & Bartlett of a matrix. Learning, LLC. Multiply both sides of an equation. Multiply the elements of a row by by a nonzero constant. a nonzero constant. 3. Add a multiple of one equation to 3. Add a multiple of the elements of another equation. one row to the corresponding Jones & Bartlett Learning, elements of LLC another row. Systems of equations that are related through elementary transformations are called equivalent systems. Matrices that are related through elementary row operations are called row equivalent matrices. The symbol < is used to indicate equivalence in both cases. Elementary transformations preserve solutions since the order of the equations does not affect Jones the solution, & Bartlett multiplying Learning, an equation LLC throughout by a nonzero constant Jones does & Bartlett not Learning, LLC change NOT the FOR truth of SALE the equality, OR DISTRIBUTION and adding equal quantities to both sides NOT of FOR an equality SALE OR DISTRIBUTION results in an equality. The method of Gauss-Jordan elimination uses elementary transformations to eliminate variables in a systematic manner, until we arrive at a system that gives the solution. We Jones & illustrate Bartlett Gauss-Jordan Learning, elimination LLC using equations and the Jones analogous & Bartlett matrix implementation SALE of the OR method DISTRIBUTION side by side in the following example. NOT The FOR reader SALE should OR note DISTRIBUTION the way Learning, LLC NOT FOR in which the variables are eliminated in the equations in the left column. At the same time,..
10 CHAPTER 1 Linear Equations and Vectors observe how this is accomplished NOT in FOR terms SALE of matrices OR in DISTRIBUTION the right column by creating zeros in certain locations. We shall henceforth be using the matrix approach. EXAMPLE 1 Solve the system of linear equations NOT FOR SALE OR 1x 1 DISTRIBUTION 1 1x 1 1x 3 5 x 1 1 3x 1 x 3 5 3 SOLUTION Equation NOT Method FOR SALE OR DISTRIBUTION Initial System 1x 1 1x x 3 56 Analogous NOT Matrix FOR Method SALE OR DISTRIBUTION Augmented Matrix x 1 1 x 1 x 3 5 x 1 1 3x 1 x 3 5 3 NOT FOR SALE OR x DISTRIBUTION 1 x x 3 56 Eliminate x 1 from nd and 3rd equations. 1 1 1 3 1 3 Jones & Bartlett Learning, 1 1 6 LLC Create zeros in column 1. < Eq 1 1Eq1 Eq3 1 11Eq1 x 1 1 x 1 x 3 5 < Jones & x Bartlett x 3 51 Learning, LLC R 1 1R1 R3 1 11R1 NOT FOR x 1 x SALE 3x 3 58 OR DISTRIBUTION 1 1 1 0Jones 1 1 & Bartlett 1 Learning, LL 0 3 8 Eliminate Eliminate x x 3 from 1st and 3rd equations. < x 1 1 x 1 1 x 3 5 03 Eq1 1 11Eq x x 3x 3 5 01 Eq3 1 1Eq x 3 5 x 3 5x 3 510 Make coefficient of x 3 in 3rd equation 1 (i.e., solve for x 3 ). x 1 1 x 1 1 x 3 5 3 Jones & Bartlett < Learning, LLC NOT FOR SALE x x x 3 51 11/5Eq3 OR DISTRIBUTION x 3 5 x 3 5 x 3 5 from 1st and nd equations. Create appropriate zeros in column. < 1 0 3 R1 1 11R Jones & Bartlett 0 1Learning, 1 1 LLC R3NOT 1 1R FOR SALE 0 OR 0 DISTRIBUTION 5 10 Make the (3, 3) element 1 (called normalizing the element). 1 0 3 < Jones & Bartlett Learning, 0 LLC 1 1 1 NOT FOR SALE 11/5R3 OR DISTRIBUTION 0 0 1 Create zeros in column 3. < Eq1 1 1Eq3 Eq 1 Eq3 Jones x 1 5& x 1 Bartlett 5 x 1 51 Learning, LLC < NOT FOR x 5 xsale 5 x 5OR 1DISTRIBUTIONR1 1 1R3 R 1 R3 x 3 5 x 3 5 x 3 5 1Jones 0 & 0 Bartlett 1 Learning, LL NOT 0 FOR 1 0SALE 1 OR DISTRIBUT 0 0 1 The solution is x 1 51, x 5 1, x 3 5. Matrix corresponds to the system. Jones x 1 5 & x Bartlett 1 5 x 1 51 Learning, LLC x 5 x 5 x 5 1 x 3 5 x 3 5 x 3 5 The solution is x 1 51, x 5 1, x 3 5...
1.1 Matrices and Systems of Linear Equations 11 NOT FOR Geometrically, SALE OR DISTRIBUTION each of the three original equations in NOT this example FOR SALE represents OR a DISTRIBUTION plane in three-dimensional space. The fact that there is a unique solution means that these three planes intersect at a single point. The solution 11, 1, gives the coordinates of this point where the three planes intersect. We now give another example to reinforce the method. EXAMPLE Solve NOT the following FOR SALE system OR of linear DISTRIBUTION equations. Jones & Bartlett Learning, x 1 1 3xLLC 3x 3 5 8 SOLUTION Start with the augmented matrix and use the first row to create zeros in the first column. (This corresponds to using the first equation to eliminate x 1 from the second and third Jones & equations.) Bartlett Learning, LLC 1 4 1 < 1 4 1 1 5 18 R 1 1R1 0 3 3 6 1 3 3 8 R3 1 R1 0 1 1 4 1 Next multiply row by Jones 3 to make & the Bartlett (, ) element Learning, 1. (This LLC corresponds to making the coefficient of in the NOT second FOR equation SALE 1.) OR DISTRIBUTION 1 4 1 < 1 1 3 R 0 1 1 Jones & Bartlett Learning, 0LLC1 1 4 Create zeros in the second column as follows. (This corresponds to using the second equation to eliminate from the first and third equations.) < R1 1 1R R3 1 11R x 1 x x 1 x 1 4x 3 5 x 1 x 1 5x 3 5 1 0 8 0 1 1 Jones & Bartlett Learning, LLC 0 0 NOT FOR 6 SALE OR DISTRIBUTION Multiply row 3 by. (This corresponds to making the coefficient of x 3 in the third equation 1.) 1 18 Jones & Bartlett 1 Learning, 0 8 LLC < NOT FOR 0 1 1 1 1 R3 SALE OR DISTRIBUTION 0 0 1 3 Finally, create zeros in the third column. (This corresponds to using the third equation to eliminate x from the first and second equations.) Jones 3 & Bartlett Learning, LLC < 1 0 0 R1 1 1R3 0 1 0 1 R 1 R3 0 0 1 3..
1 CHAPTER 1 Linear Equations and Vectors This matrix corresponds to the system x 1 5 x 1 5 x 1 5 x 5 x 5 x 5 1 x 3 5 x 3 5 x 3 5 3 The solution is x 1 5, x 5 1, x 3 5 3. NOT FOR SALE OR This DISTRIBUTION Gauss-Jordan method of solving a system NOT of FOR linear equations SALE OR using DISTRIBUTION matrices involves creating 1s and 0s in certain locations of matrices. These numbers are created in a systematic manner, column by column. The following example illustrates that it may be necessary to interchange two rows at some stage in order to proceed in the preceding manner. EXAMPLE 3 Solve the system 4x 1 1 8x 1x 3 5 44 3x 1 1 6x 18x 3 5 3 x 1 8x 13x 3 5 7 SOLUTION We start with the augmented matrix and proceed as follows. (Note the use of zero in the augmented matrix as the coefficient of the missing variable x 3 in the third Jones & Bartlett equation.) Learning, LLC 4 81 44 NOT FOR SALE 1 OR 3 DISTRIBUTION 11 < 3 6 8 3 3 6 8 3 1 1 4 1 0 7 R1 1 0 7 At this stage we need a nonzero element in the location (, ) in order to continue. To achieve this we interchange the second row with the third row (a later row) and then Jones proceed. & Bartlett Learning, LLC 1 3 11 1 3 11 < < 0 3 6 15 1 1 3 R 0 1 5 R 4 R3 0 0 1 1 0 0 1 1 1 0 1 1 NOT FOR < SALE OR 1 DISTRIBUTION 0 0 < 0 1 5 R1 1 11R3 0 1 0 3 R1 1 1R 0 0 1 1 R 1 1R3 0 0 1 1 The solution is x 1 5, x 5 3, x 3 51. < 1 3 11 R 1 13R1 0 0 1 1 R3 1 1R1 0 3 6 15..
NOT FOR Summary SALE OR DISTRIBUTION 1.1 Matrices and Systems of Linear Equations 13 We now summarize the method of Gauss-Jordan elimination for solving a system of n linear equations in n variables that has a unique solution. The augmented matrix is made up of a matrix of coefficients A and a column matrix of constant terms B. Let us write 3A : B4 for this matrix. Use row Jones operations & to Bartlett gradually Learning, transform this LLCmatrix, column by column, into a matrix 3I n : X4, NOT where FOR is the SALE identity OR n 3DISTRIBUTION n matrix. This final matrix 3I n : X4 is called the reduced echelon form of the original augmented matrix. Jones The matrix & Bartlett of coefficients Learning, of the final LLC system of equations is I n and Jones X is the column & Bartlett Learning, LLC matrix NOT of constant FOR SALE terms. This OR implies DISTRIBUTION that the elements of X are the unique NOT solution. FOR Observe SALE OR DISTRIBUTION that as 3A : B4 is being transformed to 3I n : X4, A is being changed to I n. Thus: If A is the matrix of coefficients of a system of n equations in n variables that has a unique solution, then it is row equivalent to I n. NOT FOR If SALE 3A : B4 OR cannot DISTRIBUTION be transformed in this manner into a matrix NOT FOR of the SALE form 3I n OR : X4, DISTRIBUTION the system of equations does not have a unique solution. More will be said about such systems in the next section. Many Systems Certain applications involve NOT FOR solving SALE a number OR DISTRIBUTION of systems of linear equations, all having the same square matrix of coefficients A. Let the systems be The constant Jones terms & Bartlett B 1, B, c, Learning, B k, might for LLC example be test data, and one Jones wants to & know Bartlett Learning, LLC the solutions NOT FOR that would SALE lead OR to DISTRIBUTION these results. The situation often dictates NOT that the FOR solutions SALE OR DISTRIBUTION be unique. One could of course go through the method of Gauss-Jordan elimination for each system, solving each system independently. This procedure would lead to the reduced echelon forms 3I n : X 1 4, 3I n : X 4, c, 3I n : XJones k 4 & Bartlett Learning, LLC and the solutions would be X 1, X, c, X k. However, the same reduction of A to I n would be repeated for each system; this involves a great deal of unnecessary duplication. The systems can be represented by one large augmented matrix 3A : B 1 B c Bk 4, and the Gauss- Jordan method can be applied Jones to & this Bartlett one matrix. Learning, We would LLC get NOT FOR 3A : B 1 B csale Bk 4 OR < c DISTRIBUTION < 3I n : X 1 X c Xk 4 leading to the solutions X 1, X, c, X k. I n 3A : B4 < c < 3I n : X4 3A : B 1 4, 3A : B 4, c, 3A : B k 4 EXAMPLE 4 Solve the following three systems of linear equations, all of which have the same matrix of coefficients. x 1 x 1 3x 3 5 b 1 b 1 8 0 3 x 1 x 1 4x 3 5 b for b 5 11, 1, 3 in turn. Jones & Bartlett x 1 Learning, 1 x 4x 3 LLC 5 b 3 b 3 11 Jones & 4 Bartlett Learning, LLC..
14 CHAPTER 1 Linear Equations and Vectors SOLUTION Construct the large augmented matrix that describes all three systems and determine the reduced echelon form as follows. 1 1 3 8 0 3 < 1 1 3 8 0 3 1 4 11 1 3 R 1 1R1 0 1 5 1 3 1 4 11 4 R3 1 R1 0 1 1 3 1 < 1 0 1 3 1 0 R1 1 Jones R & 0Bartlett 1 Learning, 5 1 3LLC R3 1NOT 11R FOR 0SALE 0 OR 1 DISTRIBUTION 1 < 1 0 0 1 0 R1 1 11R3 0 1 0 1 3 1 R 1 R3 0 0 1 1 The solutions to the three systems NOT FOR of equations SALE are OR given DISTRIBUTION by the last three columns of the reduced echelon form. They are x 1 5 1, x 51, x 3 5 Jones & Bartlett Learning, x 1 LLC 5 0, x 5 3, x 3 5 1 x 1 5, x 5 1, x 3 5 In this section we have limited our discussion to systems of n linear equations in n variables Learning, that have a LLC unique solution. In the following Jones section, & Bartlett we shall extend Learning, the method LLCof Gauss-Jordan elimination to accommodate other systems that have a unique solution, and Jones & Bartlett also to include systems that have many solutions or no solutions. EXERCISE SET 1.1* NOT FOR Matrices SALE OR DISTRIBUTION NOT. FOR Give SALE the (1, 1), OR (, DISTRIBUTION ), (3, 3), (1, 5), (, 4), (3, ) elements 1. Give the sizes of the following matrices. of the following matrix. 1 3 0 9 1 3 0 1 (a) 0 1 (b) 6 4 4 5 3 6 4 5 3 Jones 3& Bartlett Learning, LLC 5 8 9 Jones 3 & Bartlett Learning, LL 7 3. Give the (, 3), (3, ), (4, 1), (1, 3), (4, 4), (3, 1) elements (c) c 1 3 0 (d) 4 of the following matrix. 1 4 5 d 3 (e) (f) 1 9 8 7 4Jones 5& Bartlett 7 Learning, LLC NOT 4 6FOR 4 SALE 0 0OR DISTRIBUTION 3 3 4 74 1 7 0 1 4 5 Jones 3& Bartlett 5 0 1Learning, LLC NOT FOR 6 SALE 9 0 OR DISTRIBUTION 4. Write down the identity matrix I 4. * Answers to exercises marked in red are provided in the back of the book...
1.1 Matrices and Systems of Linear Equations 15 NOT FOR Matrices SALE and OR Systems DISTRIBUTION of Equations NOT FOR Elementary SALE Row OR Operations DISTRIBUTION 5. Determine the matrix of coefficients and augmented matrix of each of the following systems of equations. 7. In the following exercises you are given a matrix followed by an elementary row operation. Determine each resulting (a) x 1 1 3x 5 7 matrix. x 1 5x 53 6 4 0 Jones < & Bartlett Learning, LL (a) 1 3 6 (b) 5x 1 1 R1 1 1 x 4x NOT 3 5 8FOR SALE OR DISTRIBUTION 8 3 5 5x 1 1 3x 1 6x 3 5 4 0 8 4 3 4x < 1 1 6x 9x 3 5 7 (b) 7 5 1 R1 4 R (c) 3 5 8 9 Jones x 1 1 & 3x Bartlett 5x 3 53 Learning, LLC 1 3 1 NOT xfor < 1 xsale 1 4x 3 OR 5 8 DISTRIBUTION (c) 1 1 7 1 R 1 R1 x 1 1 3x 1 3x 5 6 4 5 3 R3 1 1R1 (d) 5x 1 1 4x 5 9 1 3 4 < x 1 8x 54 (d) 0 1 1 R1 1 1R 0 4 3 5 R3 1 14R NOT FOR SALE x OR 1 1 DISTRIBUTION x 5 3 1 0 4 3 < (e) 5x 1 1 x 4x 3 5 8 (e) 0 1 3 R1 1 14R3 4x 1 4x 1 3x 3 5 0 0 0 1 5 R 1 13R3 5x 1 5x 1 5x 3 5 7 1 0 7 Jones < & Bartlett Learning, LL (f) x 1 1 3x (f) 0 1 5 3 NOT 9x 3 54 FOR SALE OR DISTRIBUTION NOT FOR 1 1 0 0 8 R3 SALE OR DISTRIBUT x 1 1x 1 4x 3 5 1 11 x 1 1 8x 1 8x 5 1 8. Interpret each of the following row operations as a stage in arriving at the reduced echelon form of a matrix. Why have (g) x 1 x 1 x 1 53 the indicated Jones operations & Bartlett been selected? Learning, What particular LLC aims x x x 5 1 1 do they accomplish in terms of the systems of linear equations that are described by the matrices? x 3 x 3 x 3 5 8 (a) (h) 4x 1 1 x 9x 3 1 4x 4 51 1 4 3 5 < 1 4 3 5 1x 1 1 6x 8x 3 7x 4 5 1 15 1 7 5 R 1 1R1 0 7 13 15 Jones & Bartlett Learning, x 1 3x 3 LLC 5x 4 5 0 Jones & 4Bartlett 0 3Learning, 6 R3 1LLC 14R1 0 16 15 14 NOT FOR SALE 6. Interpret OR the DISTRIBUTION following matrices as augmented matrices NOT of FOR SALE OR DISTRIBUTION 1 4 7 1 4 7 systems of equations. Write down each system of equations. < (b) 0 3 9 6 1 1 3 (a) (b) 0 4 7 8 0 1 3 c 7 9 8 0 4 7 8 4 5 6 d 6 4 3 d 1 3 4 5 1 3 4 5 Jones < & Bartlett Learning, LL 8 7 5 1 (c) 0 0 6 0 1 3 8 R 4 R3 (c) c 1 9 3 (d) 4 6 4 0 1 3 8 0 0 6 5 0 d 9 3 7 6 (d) 3 6 4 0 4 1 5 0 < 1 0 1 6 (e) 7 5 3 (f) 5 7 3 0 1 3 R1 1 1R 0 1 3 Jones & Bartlett 0 4 0 6 0 8 0 3 1 R3 1 13R Learning, LLC 0 0 711 1 0 0 3 1 1 6 (g) 0 1 0 8 (h) 0 1 4 5 0 0 1 4 0 0 1..
16 CHAPTER 1 Linear Equations and Vectors NOT FOR 9. Interpret SALE each OR of DISTRIBUTION the following row operations as a stage in NOT 11. FOR The following SALE OR systems DISTRIBUTION of equations all have unique solutions. Solve these systems using the method of Gauss-Jordan arriving at the reduced echelon form of a matrix. Why have these operations been selected? elimination with matrices. 1 0 6 < 1 0 0 (a) x 1 1 x 1 3x 3 5 1 14 (a) 0 1 1 3 R1 1 1R3 0 1 0 5 x 1 1 5x 1 8x 3 5 1 36 0 0 1 R 1 R3 0 0 1 x 1 x x 54 (b) 0 4 1 4 3 8 (b) x 1 x x 3 51 < 4 3 8 0 4 1 R1 4 R x 1 1 6x 1 10x 3 5 1 14 5 7 1 5 7 1 x 1 1 x 1 6x 3 5 9 1 Jones 0 3& Bartlett 7 Learning, 1 0 3LLC7 < (c) x (c) 1 1 x 4x 3 5 1 14 NOT 0 1FOR 4 SALE 1OR 1 R3 DISTRIBUTION 0 1 4 3x 1 1 x 1 x 3 5 1 8 0 0 6 0 0 1 3 x 1 x 1 x 3 51 (d) 1 0 4 < 1 0 0 (d) x 1 x 1 4x 3 5 8 Jones & Bartlett 0 1 3Learning, 4 R1 1LLC 1R3 0 1 0 5 Jones & xbartlett 1 1 1Learning, x 5 6 LLC NOT FOR SALE 0 0 OR 1 DISTRIBUTION 3 R 1 13R3 0 0 1 3 NOT FOR x SALE 1 1 xor 1 x DISTRIBUTION 3 5 5 Solving Systems of Linear Equations 10. The following systems of equations all have unique solutions. (e) x 1 x 1 x 3 5 3 x 1 1 x 1 1 x 3 58 Solve these systems using the method of Gauss-Jordan 3x 1 1 x x 3 5 0 elimination with matrices. 1. The following systems of equations all have unique solutions. Solve these systems using (a) 1x 1 x 5 18 NOT the FOR method SALE of Gauss-Jordan OR DISTRIBUT (b) x 1 3x 511 x 1 1 x 5 4 elimination with matrices. 3 (a) x 1 1 3x 3 1 3x 3 5 15 3x 1 1 x 5 3 x 1 1 7x 9x 3 545 x 1 1 5x 3 1 5x 3 5 (c) xnot 1 1 xfor 1 1 3xSALE 3 5 3 OR DISTRIBUTION x x x 3 54 (b) 3x 1 6x 15x 3 53 x 3 x x 3 5 5 x 1 1 3x 1 19x 3 5 1 4x 1 7x 17x 3 54 (d) x 1 1 x 1 3x 3 5 6 Jones & Bartlett x 1 1 xlearning, 1 4x 3 5 9 LLC Jones (c)& 3xBartlett 1 1 6x Learning, x 4 3x 4 5 13 LLC NOT FOR SALE x 1 1OR x 1DISTRIBUTION 6x 3 5 11 NOT FOR 1x SALE 1 1 3xOR xdistribution 3 4x 4 51 (e) x 1 x 1 3x 3 5 3 x 1 x 1 x 3 1 x 4 5 18 x 1 1 3x 1 x 1 3x 5 18 x 1 x 1 x 3 5 3x (d) x 1 1 x 1 x 3 1 5x 4 5 11 1 1 x x 3 5 3 Jones & Bartlett Learning, LLC x 1 1 4x 1 x 3 1 4 5 14 (f) x 1 1 x x 3 NOT 5 FOR SALE OR DISTRIBUTIONx 1 1 3x 1 4x 3 1 8xNOT 4 5 19FOR SALE OR DISTRIBUT 3x 1 1 x 1 x 3 5 1 10 x 1 x 1 x 3 1 x 3 5 1 4x 1 1 x 1 3x 3 5 1 14 (e) 1x 1 1 1x 1 x 3 1 16x 4 5 11 x 1 1 Jones 3x 1 6x& 3 1Bartlett 19x 4 5 36 Learning, LLC x 1 3x 1 4x 3 1 15x 4 5 8 1x 1 1x 1x 3 16x 4 51..
1. Gauss-Jordan Elimination 17 NOT FOR 13. SALE The following OR DISTRIBUTION exercises involve many systems of linear NOT FOR (c) SALE x 1 OR x DISTRIBUTION 1 3x 3 5 b 1 equations with unique solutions that have the same matrix x 1 x 1 x 3 5 b of coefficients. Solve the systems by applying the method x 1 3x 1 6x 3 5 b 3 of Gauss-Jordan elimination to a large augmented matrix that describes many systems. b 1 6 5 4 (a) 1x 1 1 x 5 b 1 Jones & Bartlett Learning, LLC for b 5 5, Jones 3, 3& Bartlett in turn. Learning, LL 3x 1 1 5x 5 b b 3 14 NOT 8FOR 9 SALE OR DISTRIBUT for cb 1 d 5 c 3 c 3 in turn. b 7 d 8 d, c4 9 d, (b) Jones x 1 1 x& 5Bartlett b 1 Learning, LLC NOT x 1 FOR 1 3x SALE 5 b OR DISTRIBUTION for cb 1 d 5 c 0, c 1 in turn. b d 1 d, c 5 13 d (d) x 1 1 x x 3 5 b 1 x 1 x 1 x 3 5 b 3x Jones 1 1 7x & x Bartlett 3 5 b 3 Learning, LLC NOT b 1 FOR SALE 1 OR 6 DISTRIBUTION 0 for b 5 1, 4, in turn. b 3 1 18 4 1. Gauss-Jordan Jones Elimination & Bartlett Learning, LLC In the previous section we used the method of Gauss-Jordan elimination to solve systems of n equations in n variables that had a unique solution. We shall now discuss the method in its more general setting, where the number of equations can differ from the number of variables and where there can be a unique solution, many solutions, or no solutions. Our approach Jones again will & Bartlett be to start Learning, from the augmented LLC matrix of the given system Jones and & to Bartlett perform NOT a sequence FOR SALE of elementary OR DISTRIBUTION row operations that will result in a simpler NOT FOR matrix SALE (the OR DISTRIBUTION Learning, LLC reduced echelon form), which leads directly to the solution. We now give the general definition of reduced echelon form. The reader will observe that the reduced echelon forms discussed in the previous section all conform to this definition. DEFINITION A matrix is in reduced echelon form if: 1. Any rows consisting entirely of zeros are grouped at the bottom of the matrix.. The first nonzero element of each other row is 1. This element is called a leading 1. 3. The leading 1 of each row after the first is positioned to the right of the leading 1 of the previous row. 4. All Jones other elements & Bartlett in a column Learning, that contains LLC a leading 1 are zero. The following matrices are all in reduced echelon form. 108 1007 Jones & Bartlett 01 Learning, 0103 LLC NOT FOR SALE 000OR DISTRIBUTION 0019 1400 0010 0001 130 0001 Jones & Bartlett Learning, LLC 0000 105008 10304 017009 00107 000105 000016 000014 000000..
18 CHAPTER 1 Linear Equations and Vectors The following matrices NOT are not FOR in reduced SALE echelon OR DISTRIBUTION form for the reasons stated. 1 7 0 8 1 0 4 1 0 3 0 1 0 0 0 1 0 3 0 0 0 0 0 0 3 4 0 0 0 1 4 0 0 1 Jones & 0Bartlett 0 1 3Learning, 0 0 LLC 0 0 1 0 1 0 3 0 0 0 0 NOT FOR SALE Row of zeros OR DISTRIBUTION First nonzero Leading 1 innot FOR Nonzero SALE OR DISTRIBUT not at bottom element in row row 3 not to the element above of matrix is not 1 right of leading leading 1 in 1 in row row Jones & Bartlett Learning, There are usually LLC many sequences of row operations Jones & that Bartlett can be Learning, used to transform LLC a NOT FOR SALE OR given DISTRIBUTION matrix to reduced echelon form they NOT all, however, FOR SALE lead to OR the same DISTRIBUTION reduced echelon form. We say that the reduced echelon form of a matrix is unique. The method of Gauss- Jordan elimination is an important systematic way (called an algorithm) for arriving at the reduced echelon form. It can be programmed on a computer. We now summarize the method, then give examples of its implementation. Gauss-Jordan Elimination 1. Write down the augmented matrix of the system of linear equations.. Derive the reduced echelon form of the augmented matrix using elementary row operations. This is done by creating leading 1s, then zeros above and below each Jones leading & Bartlett 1, column Learning, by column, LLC starting with the first column. NOT 3. FOR Write SALE down OR the DISTRIBUTION system of equations corresponding to NOT the reduced FOR echelon SALE OR form. DISTRIBUT This system gives the solution. We stress the importance of mastering this algorithm. Not only is getting the correct solution important, the method of arriving at the solution is important. We shall, for example, Learning, be interested LLC in the efficiency of this algorithm Jones (the number & Bartlett of additions Learning, and multipli- LLC Jones & Bartlett NOT FOR SALE OR cations DISTRIBUTION used) and the comparison of it with other NOT algorithms FOR SALE that can OR be used DISTRIBUTION to solve systems of linear equations. EXAMPLE 1 Use the method of Gauss-Jordan elimination to find the reduced echelon form of the following matrix. NOT 0FOR 0 SALE OR DISTRIBUTION 3 3 3 9 1 4 4 11 1 SOLUTION NOT Step FOR 1 SALE Interchange OR DISTRIBUTION rows, if necessary, to bring a nonzero element NOT FOR to the top SALE of the OR first DISTRIBUT nonzero column. This nonzero element is called a pivot. pivot 3 3 3 9 1 < 0 0Jones & Bartlett Learning, LLC R1 4 R 4 NOT 4 FOR 11SALE 1 OR DISTRIBUTION 1 Step Create a 1 in the pivot location by multiplying the pivot row by 1 1 1 3 4 < 1 1 3 R1 0 0 4 4 11 1 pivot...
1. Gauss-Jordan Elimination 19 NOT FOR SALE Step 3 Create OR DISTRIBUTION zeros elsewhere in the pivot column by adding suitable multiples of the pivot row to all other rows of the matrix. 1 1 1 3 4 < Jones & Bartlett 0 Learning, 0 LLC R3 1 14R1 NOT FOR SALE OR 0 DISTRIBUTION 0 1 4 Step 4 Cover the pivot row and all rows above it. Repeat Steps 1 and for the remaining submatrix. Repeat Step 3 for the whole matrix. Continue thus until the reduced echelon form is reached. NOT FOR 1 SALE 1 1OR 3DISTRIBUTION 4 1 1NOT 1FOR 3 SALE 4 OR DISTRIBUTION 0 0 5 0 0 0 0 1 4 0 0 1 4 first nonzero column of the submatrix 1 1 1 3 4 < 1 1 R 0 0 1 1 1 0 0 1 4 < R1 1 1R3 R 1 R3 This NOT matrix FOR is the SALE reduced OR echelon DISTRIBUTION form of the given matrix. pivot 1 1 0 5 0 0 1 1 1 0 0 0 1 6 < R1 1 R R3 1 1R pivot 1 1 0 0 17 0 0 1 0 5 0 0 0 1 6 We now illustrate how this method is used to solve various systems of equations. The Jones & following Bartlett example Learning, illustrates LLChow to solve a system of linear Jones equations & Bartlett that has many Learning, solutions. SALE The OR reduced DISTRIBUTION echelon form is derived. It then becomes NOT FOR necessary SALE to OR interpret DISTRIBUTION the LLC NOT FOR reduced echelon form, expressing the many solutions in a clear manner. EXAMPLE Solve, if possible, the system of equations Jones & 3xBartlett 1 1Learning, 3x 3 5 9 LLC x 1 3x 1 4x 3 5 7 3x 1 5x 3x 3 5 7 SOLUTION Start with the augmented matrix and follow the Gauss-Jordan algorithm. Pivots and leading ones are circled. 3 3 3 9 1 4 7 Jones & Bartlett Learning, 3 5 LLC 1 7 1 1 1 3 < 1 1 3 R1 1 4 7 3 5Jones 1 & 7Bartlett Learning, LLC..
0 CHAPTER 1 Linear Equations and Vectors < R 1 1R1 R3 1 13R1 1 NOT 1 FOR 1 SALE 3 OR DISTRIBUTION < 1 0 3 4 0 1 1 R1 1 R 0 1 1 0 4 R3 1 1R 0 0 0 0 Jones We have & Bartlett arrived at Learning, the reduced echelon LLC form. The corresponding Jones system & of Bartlett equations Learning, is LL x 1 1 x 1 1 3x 3 5 4 x 1 x 1 x 3 5 1 There are many values of x 1, x, and x 3 that satisfy these equations. This is a system of Jones & Bartlett equations Learning, that has LLCmany solutions. x 1 is called Jones the leading & Bartlett variable Learning, of the first LLC equation DISTRIBUTION and x is the leading variable of the second NOT equation. FOR SALE To express OR DISTRIBUTION these many solu- NOT FOR SALE OR tions, we write the leading variables in each equation in terms of the remaining variables, called free variables. We get x 1 53x 3 1 4 NOT FOR x 5x SALE 3 1OR 1 DISTRIBUTION Let us assign the arbitrary value r to x 3. The general solution to the system is x 1 53r 1 4, x 5r 1 1, x 3 5 r Jones As r & ranges Bartlett over the Learning, set of realllc numbers, we get many solutions. Jones r is called & Bartlett a parameter. FOR We SALE can get OR specific DISTRIBUTION solutions by giving r different values. NOT For FOR example, SALE OR Learning, LL NOT DISTRIBUT r 5 1 gives x 1 5 11, x 51, x 3 5 1 r 5 gives x 1 5 10, x 5 5, x 3 5 EXAMPLE 3 This example illustrates that the general solution can involve a number of parameters. Solve the system of equations Jones x 1 1 x & Bartlett 3x 3 1 Learning, 3x 4 5 4 LLC NOT xfor 1 1 4xSALE x 3 OR 1 7x DISTRIBUTION 4 5 10 x 1 x 1 4x 3 4x 4 56 SOLUTION On applying the Gauss-Jordan algorithm, we get 1 1 3 4 < 1 1 3 4 4 7 10 R 1 1R1 0 0 0 1 1 1 4 6 R3 1 R1 0 0 0 1 < 1 NOT 1 FOR 0SALE OR DISTRIBUTION R1 1 13R 0 0 0 1 R3 1 R 0 0 0 0 0..
1. Gauss-Jordan Elimination 1 NOT FOR We SALE have OR arrived DISTRIBUTION at the reduced echelon form. The corresponding system of equations is x 1 1 x x 3 x 3 5 x 4 5 Expressing the leading NOT variables FOR SALE in terms OR of the DISTRIBUTION remaining variables, we get x 1 5x 1 x 3, x 4 5 Let us assign the arbitrary values r to and s to x 3. The general solution is Jones & Bartlett x 1 5r Learning, 1 s, LLC x 5 r, x 3 5 s, x 4 5 Specific NOT solutions FOR SALE can be OR obtained DISTRIBUTION by giving r and s various values. x EXAMPLE 4 This example illustrates a system that has no solution. Let us try to solve Jones & the Bartlett system Learning, LLC x 1 1 x 1 5x 3 5 3 x 1 3x 1 3x 3 51 x 1 1 x 1 8x 3 5 3 SOLUTION Starting with the augmented matrix, we get 1 1 5 3 0 1 3 1 Jones & Bartlett Learning, 1 8LLC 3 < R3 1 11R1 NOT FOR SALE < OR 1DISTRIBUTION 0 4 < R1 1 11R 0 1 3 1 R1 1 14R3 R3 1 11R 0 0 0 1 R 1 R3 1 1 5 3 0 1 3 1 0 Jones 1 3 & 0Bartlett Learning, LLC 1NOT 0 FOR SALE 0 OR DISTRIBUTION 0 1 3 0 0 0 0 1 The last row of this reduced echelon form gives the equation 0x 1 1 0x 1 0x 3 5NOT 1 FOR SALE OR DISTRIBUTION This equation cannot be satisfied for any values of x 1, x, and x 3. Thus the system has no solution. (This information was in fact available from the next-to-last matrix.) Homogeneous Systems of Linear Equations A system of linear equations is said to be homogeneous if all the constants are zero. As we proceed in the course, we shall find that homogeneous systems of linear equations have many interesting properties and play a key role in our discussions. The following system is a homogeneous system of linear equations. x 1 1 x x 4 5 0 x 1 1 x x 3 1 x 4 5 0 x 1 x 1 x 3 3x 4 5 0 Observe that x 1 5 0, x 5 0, x 3 5 0, x 4 5 0, is a solution to this system. It is apparent, by Jones & letting Bartlett all the Learning, variables be LLC zero, that this result can be extended Jones as & follows Bartlett to any Learning, homogeneous SALE system OR of DISTRIBUTION equations. NOT FOR SALE OR LLC NOT FOR DISTRIBUTION..
CHAPTER 1 Linear Equations and Vectors THEOREM 1.1 A homogeneous system of linear equations in n variables always has the solution x 1 5 0, x 5 0, c, x n 5 0. This solution is called the trivial solution. NOT FOR Let us SALE see if OR the preceding DISTRIBUTION homogeneous system has any other NOT solutions. FOR SALE We solve OR the DISTRIBUT system using Gauss-Jordan elimination. 1 1 0 1 0 < 1 1 0 1 0 1 1 0 R 1 11R1 0 1 1 3 0 1 3 0 Jones R3 1 R1 & Bartlett 0 1 Learning, 4 LLC 0 < 1 0 1 4 0 < 1 0 0 3 0 R1 1 11R 0 1 1 3 0 R1 1 11R3 0 1 0 0 R3 1 R 0 0 1 1 0 R 1 R3 0 0 1 1 0 This reduced echelon form gives NOT the FOR system SALE OR DISTRIBUTION x 1 3x 4 5 0 x 1 x 4 5 0 x 3 x 4 5 0 Expressing the leading variables in terms of the remaining free variable, we get x 1 5 3x 4, x 5x 4, x 3 5 x 4 Letting x 4 r, we see that the system has many solutions, x 1 5 3r, x 5r, NOT x 3 5FOR r, x 4 SALE 5 r OR DISTRIBUTION Observe that the solution x 1 5 0, x 5 0, x 3 5 0, x 4 5 0, is obtained by letting r 5 0. Note that in this example the homogeneous system had more variables (4) than equations (3). This led to free variables in the general solution, implying many solutions. Guided by this thinking, we now consider Jones a general & Bartlett homogeneous Learning, system LLC of m linear equations in n variables with n > m the NOT number FOR of variables SALE OR is greater DISTRIBUTION than the number of equations. The reduced echelon form will have at most m nonzero rows (Gauss-Jordan elimination may have created some zero rows). The corresponding system of equations has fewer equations than variables. There will thus be free variables, leading to many solutions. Jones THEOREM & Bartlett 1. Learning, LLC NOT A FOR homogeneous SALE OR system DISTRIBUTION of linear equations that has more variables NOT FOR than SALE equations OR hasdistribut many solutions. One of these solutions is the trivial solution. Jones & Bartlett Learning, More will be LLC said about the existence and uniqueness Jones & Bartlett of solutions Learning, to other classes LLCof NOT FOR SALE OR linear DISTRIBUTION equations as we continue in the course. In the first two sections of this chapter, we introduced the method of Gauss-Jordan elimination for solving systems of linear equations. As we proceed in the course, we shall introduce other methods and compare the merits of the methods. There is another popular elimination method for solving systems of linear equations, for example, called Gaussian elimination...
1. Gauss-Jordan Elimination 3 NOT FOR We SALE introduce OR that DISTRIBUTION method in Section 1 of the Numerical NOT Methods FOR chapter. SALE The OR following DISTRIBUTION discussion reveals some of the numerical concerns when solving systems of equations. Numerical Considerations In practice, systems of linear Jones equations & Bartlett are solved Learning, on computers. LLCNumbers are represented on computers in the form NOT 60. FOR a 1 ca SALE n 3OR 10 r, DISTRIBUTION where a 1, c, a n are integers betweennot 0 FOR SALE OR DISTRIBUT and9andr is an integer (positive or negative). Such a number is called a floating-point number. The quantity a 1, c, a n is called the mantissa, and r is the exponent. For example, the number 15.6 is written in floating-point form as 0.156 3 10 3. An arithmetic operation Jones of multiplication, & Bartlett division, Learning, addition, LLC or subtraction on floating-point Jones numbers & Bartlett is Learning, LLC called a floating-point operation, or flop. Computers can handle only a limited number of integers in the mantissa of a number. The mantissa is rounded to a certain number of places during each operation, and consequently errors called round-off errors occur in methods such as Gauss-Jordan elimination. These errors are propagated and magnified during computation. The fewer flops Jones & that Bartlett are performed Learning, during LLC computation, the faster and more Jones accurate & Bartlett the result Learning, will be. LLC NOT FOR (Ways SALE of OR minimizing DISTRIBUTION these errors are discussed in the NOT Numerical FOR Methods SALE OR chapter.) DISTRIBUTION To compute the reduced echelon form of a system of n equations in n variables, the method 1 of Gauss-Jordan elimination requires n 3 1 1 n multiplications and 1 n 3 1 n additions (Section 1 of the Numerical Methods chapter). The number of multiplications required to solve a system of, say, Jones ten equations & Bartlett in ten Learning, variables 1n LLC 5 10 is 550, and the number of additions is 495. The total number of flops is the sum of these, namely 1045. Algorithms are usually measured and compared using such data. EXERCISE SET 1. Reduced Echelon Form of a Matrix 1 0 3 1 0 0 4 1. Determine whether the following matrices are in reduced (a) 0 0 1 8 (b) 0 0 1 0 6 echelon form. If a matrix is not in reduced echelon form, 0 1 4 9 0 0 0 1 5 give a reason. (a) c 1 0 (b) 0 1 3 d c1 0 4 1 5 0 0 1 0 4 6 0 0 1 7 d 0 0 1 9 0 0 1 3 4 Jones &(c) Bartlett Learning, (d) LLC 0 0 0 0 1 0 0 0 1 NOT FOR SALE (c) c 1 OR DISTRIBUTION 5 6 (d) 0 1 3 7 d c1 4 0 5 0 0 9 d 0 0 0 0 0 0 0 0 0 1 1 0 0 3 1 0 4 0 0 1 0 0 1 5 0 0 0 0 0 0 0 1 0 0 (e) (f) (e) 0 1 0 (f) 0 0 1 0 1 0 7 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 3 0 0 0 0 1 1 0 0 4 1 0 0 3 1 0 0 5 3 0 0 1 0 4 (g) 0 1 0 5 (h) 0 0 6 1 (g) 0 0 1 0 3 (h) 0 0 0 1 5 0 0 1 9 0 0 1 3 0 1 3 7 0 1 0 0 3 Jones 1 0& Bartlett 3 0 Learning, LLC 1Jones 5 3& Bartlett 0 7 Learning, LLC NOT (i) 0FOR 1 SALE 6 0 OR DISTRIBUTION (i) NOT 0 FOR 0 0 SALE 1 4OR DISTRIBUTION 0 0 0 1 0 0 0 0 0. Determine whether the following matrices are in reduced 3. Each of the following matrices is the reduced echelon form echelon form. If a matrix is not in reduced echelon form, of the augmented matrix of a system of linear equations. Jones & Bartlett give a reason. Learning, LLC Jones & Give Bartlett the solution Learning, (if it exists) LLC to each system of equations. NOT FOR SALE OR DISTRIBUTION Gaussian elimination can in fact be used in place of Gauss-Jordan elimination as the standard method for this course if so desired...
4 CHAPTER 1 Linear Equations and Vectors NOT FOR SALE 1 OR 0 DISTRIBUTION 0 1 0 3 4 NOT FOR (e) 1x SALE 1 1xOR 1 1x DISTRIBUTION 3 5 3 (a) 0 1 0 4 (b) 0 1 8 x 1 1x 1 4x 3 5 7 0 0 1 3 0 0 0 0 3x 1 5x 1x 3 5 7 1 3 0 6 1 0 5 0 (f) (c) 0 0 1 (d) 0 1 7 0 13x 1 3x 1 9x 3 5 4 0 0 0 0NOT FOR 0 SALE 0 0 OR 1 DISTRIBUTION1x 1 x 1 7x 3 5 17 1 0 0 5 3 x 1 1 x 4x 3 511 (e) 0 1 0 6 6. Solve (if possible) each of the following systems of three 0 0 1 4 equations in three variables using the method of Gauss- 1 Jones 3 0& Bartlett 0 Learning, LLC Jordan elimination. (f) NOT 0 0FOR 1 SALE 0 4 OR DISTRIBUTION (a) 3xNOT 1 1 6xFOR 3x 3 SALE 5 6 OR DISTRIBUTION 0 0 0 1 5 x 1 4x 3x 3 51 4. Each of the following matrices is the reduced echelon form of the augmented matrix of a system of linear equations. Jones & Give Bartlett the solution Learning, (if it exists) LLC to each system of equations. 3x 1 1 6x x 3 5 1 10 Jones (b)& 1xBartlett 1 1 x 1Learning, 1x 3 5 17 LLC NOT FOR SALE 1 OR 0 DISTRIBUTION 4 1 NOT FOR 1x SALE 1 1 xor 1 x DISTRIBUTION 3 5 11 (a) 0 1 3 5 6 x 1 1 4x 1 3x 3 5 18 0 0 0 0 0 (c) 1x 1 1 x 1x 3 5 3 (b) 1 3 0 4 x 1 1 4x x 3 5 6 0 0 0 1 7 Jones & Bartlett Learning, LLC 3x 1 1 6x 1 x 3 51 0 0 0 0NOT 0 FOR SALE OR DISTRIBUTION 1 0 3 0 4 (d) 1x 1 1 x 1 3x 3 5 18 (c) 0 0 1 0 9 3x 1 1 7x 1 9x 3 5 6 0 0 0 0 1 8 x 1 1 x 1 1 6x 3 5 11 1 0 0 3 6 (e) x (d) 1NOT 1x 1FOR x 3 5SALE 15 OR DISTRIBUTION 0 1 5 0 4 7 0 0 0 1 9 3 x 1 1 x 1 5x 3 5 13 x 1 1 x 3 1 x 3 5 4 Solving Systems of Linear Equations (f) 1x 1 1 x 1 18x 3 5 17 Jones 5. & Solve Bartlett (if possible) Learning, each of the LLC following systems of three NOT FOR x equations SALE OR in three DISTRIBUTION variables using the method of Gauss- NOT FOR SALE 1 1 4xOR 1 16x DISTRIBUTION 3 5 14 Jordan elimination. 1x 1 1x 1 13x 3 5 14 (a) x 1 1 4x 1 3x 3 5 1 7. Solve (if possible) each of the following systems of equations using the method of Gauss-Jordan elimination. 1 1 8x 1 11x 3 5 7 x x (a) 1x 1 1 1x 3x 3 5 10 Jones & Bartlett Learning, LL 1 1 6x 1 7x 3 5 3 3x 1 x 1 4x 3 54 (b) x 1 1 x 1 4x 3 5 15 x 1 1 4x 1 9x 3 5 33 (b) x 1 6x 14x 3 5 38 x 3x 1 1 7x 1 15x 3 537 1 1 3x 1 5x 3 5 0 (c) (c) x 1 1 x x 3 x 4 5 0 11x NOT 1 1 FOR 1x 1 SALE 1x 3 5 17 OR DISTRIBUTION x 1 1 x 1 x 4 1 x 4 5 4 1x 1 1 3x 1 1x 3 5 18 x 1 1 1x 3x 3 5 11 x 1 x 1 x 3 1 4x 4 5 5 (d) x (d) 1 1 x 1 4x 4 1 4x 4 5 0 1x 1 1 4x 1 x 3 5 Jones & x Bartlett 1 4x Learning, 1 3x 3 x 4 5LLC 0 1x 1 1 x x 3 5 0 (A homogeneous system) x 1 1 6x 1 x 5 3..
Understanding Systems of Linear Equations 9. Construct examples of the following: Jones & Bartlett (a) A system Learning, of linear equations LLC with more variables than equations, having no solution. 1. Gauss-Jordan Elimination 5 NOT FOR SALE (e) 1x OR DISTRIBUTION 1x 3x 3 1 1x 4 5 0 NOT FOR (b) SALE A system OR of DISTRIBUTION linear equations with more equations 1x than variables, having a unique solution. 1 1 1x 1x 3 1 4x 4 5 0 x 10. The reduced echelon forms of the matrices of systems of 1 x 1 x 3 8x 4 5 0 two equations in two variables, and the types of solutions (A homogeneous system) they represent can be classified as follows. ( corresponds 8. Solve (if possible) each the following systems of equations to possible nonzero elements.) using the method of Gauss-Jordan elimination. (a) 11x c1 0 c1 0 c 1 0 1 d 0 0 1 d 0 0 0 d 1 1 1x 1 x 3 1x 4 53 1x 1 1 3x 1 x 3 5x 4 59 Jones 11x 1 1 3x & Bartlett x 3 6x 4 Learning, 57 LLC unique solution no solutions many solutions NOT xfor 1 SALE x 3 OR x 5DISTRIBUTION 1 Classify NOT in a FOR similar SALE manner OR the reduced DISTRIBUTION echelon forms of the matrices, and the types of solutions they represent, of (b) x 1 1x 1 1x 3 5 17 (a) systems of three equations in two variables, x 1 x 16x 3 518 (b) systems of three equations in three variables. Jones & Bartlett x 1 Learning, 1x 1x 3 LLC 5 15 NOT FOR SALE 1x OR 1 DISTRIBUTION 5x 15x 3 546 NOT FOR 11. Consider SALE OR the homogeneous DISTRIBUTION system of linear equations (c) 1x 1 4x 1 16x 3 14x 4 5 1 10 ax 1 by 5 0 x 1 1 5x 17x 3 1 19x 4 5 cx 1 dy 5 0 x 1 3x 1 11x Jones 3 11x& 4 5Bartlett 4 Learning, LLC (a) Show that if x 5 x 0, y 5 y 0 is a solution, then 13x 1 4x 1NOT 18x 3 FOR 13x 4 5SALE 1 17 OR DISTRIBUTION x 5 kx 0, y 5 ky 0, NOT is also FOR a solution, SALE for any OR value DISTRIBUT of the constant k. (d) 1x 1 x 1 x 3 1 x 3 5 17 (b) Show that if x 5 x 0, y 5 y 0, and x 5 x 1, y 5 y 1, are x 1 x 1 x 3 14x 4 5 1 any two solutions, then x 5 x 0 1 x 1, y 5 y 0 1 y 1, is Jones 1x 1 1& 1xBartlett 3 1 Learning, 1x 4 5 14 LLC also Jones a solution. & Bartlett Learning, LLC NOT 3x FOR 1 1 1xSALE 8x 3 OR 10x DISTRIBUTION 4 59 1. Show that x 5 0, y 5 0 is a solution to the homogeneous (e) 1x 1 1 16x 1x 3 14x 4 5 0 system of linear equations x 1 1x 1 5x 3 1 17x 4 5 0 ax 1 by 5 0 3x 1 1 18x 1x 3 16x 4 5 0 Jones & Bartlett Learning, cx 1 dyllc 5 0 (A homogeneous system) Prove that this is the only solution if and only if (f) 4x 1 1 8x 1x 3 5 8 ad bc 0. 1x 1 x 1 13x 3 5 17 13. Consider two systems of linear equations having augmented x 1 1 4x 18x 3 5 16 matrices 3A : B 1 4 and 3A : B 4, where the matrix of coefficients of both systems is the same 3 3 3 matrix A. 3x 1 6x 1NOT 19x 3 51 FOR SALE OR DISTRIBUTION (a) Is it possible for 3A : B 1 4 to have a unique solution (g) 1x 1 1 1x 5 and 3A : B 4 to have many solutions? x 1 1 3x 5 3 (b) Is it possible for 3A : B 1 4 to have a unique solution 1x 1 1 3x 5 0 and Jones 3A : B 4 & to have Bartlett no solutions? Learning, LLC NOT 1x 1 FOR 1 x SALE 5 1 OR DISTRIBUTION (c) Is it possible for 3A : B 1 4 to have many solutions and 3A : B 4 to have no solutions?..