2.3 Terminology for Systems of Linear Equations

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1 page 133 e 2t sin 2t 44 A(t) = t 2 5 te t, a = 0, b = 1 sec 2 t 3t sin t 45 The matrix function A(t) in Problem 39, with a = 0 and b = 1 Integration of matrix functions given in the text was done with definite integrals, but one can naturally compute indefinite integrals of matrix functions as well, by performing indefinite integrals for each element of the matrix function 23 Terminology for Systems of Linear Equations 133 In Problems 46 49, evaluate the indefinite integral A(t) dt for the given matrix function You may assume that the constants of all indefinite integrations are zero 46 A(t) = [ 2t 3t 2 ] 47 The matrix function A(t) in Problem The matrix function A(t) in Problem The matrix function A(t) in Problem Terminology for Systems of Linear Equations As we mentioned in Section 21, a main aim of this chapter is to apply matrices to determine the solution properties of any system of linear equations We are now in a position to pursue that aim We begin by introducing some notation and terminology DEFINITION 231 The general m n system of linear equations is of the form a 11 x 1 + a 12 x a 1n x n = b 1, a 21 x 1 + a 22 x a 2n x n = b 2, a m1 x 1 + a m2 x a mn x n = b m, (231) where the system coefficients a ij and the system constants b j are given scalars and x 1,x 2,,x n denote the unknowns in the system If b i = 0 for all i, then the system is called homogeneous; otherwise it is called nonhomogeneous DEFINITION 232 By a solution to the system (231) we mean an ordered n-tuple of scalars, (c 1,c 2,,c n ), which, when substituted for x 1,x 2,,x n into the left-hand side of system (231), yield the values on the right-hand side The set of all solutions to system (231) is called the solution set to the system Remarks 1 Usually the a ij and b j will be real numbers, and we will then be interested in determining only the real solutions to system (231) However, many of the problems that arise in the later chapters will require the solution to systems with complex coefficients, in which case the corresponding solutions will also be complex 2 If (c 1,c 2,,c n ) is a solution to the system (231), we will sometimes specify this solution by writing x 1 = c 1,x 2 = c 2,,x n = c n For example, the ordered pair of numbers (1, 2) is a solution to the system x 1 + x 2 = 3, 3x 1 2x 2 = 1, and we could express this solution in the equivalent form x 1 = 1,x 2 = 2

2 page CHAPTER 2 Matrices and Systems of Linear Equations At this point, we pause to introduce some important notation that will be used frequently throughout the remainder of the text Notation 233 The set of all ordered n-tuples of real numbers (c 1,c 2,,c n ) will be denoted by R n Therefore, the set of all real solutions to the linear system (231) forms a subset of R n In like manner, the set of all ordered n-tuples of complex numbers will be denoted by C n, and the solution set for a linear system (231) containing complex coefficients can be viewed as a subset of C n Notice that when we restrict all scalar values to be real, we have a natural correspondence between elements of R n,rown-vectors, and column n-vectors: x 2 (x 1,x 2,,x n ) [x 1 x 2 x n ] x n Therefore, we may use the operations of addition, subtraction, and scalar multiplication of row n-vectors and column n-vectors to naturally equip R n with these same operations Therefore, just as we can perform addition and scalar multiplication of row or column vectors, so too can we perform these operations on n-tuples of scalars In fact, we will often treat ordered n-tuples of scalars, row n-vectors, and column n-vectors as if they were just different representations of the same basic object Of course, if we allow all scalars in question to assume complex values, then the correspondence is between elements of C n,rown-vectors, and column n-vectors We will have much more to say about the sets R n and C n in Chapter 4 Returning to the general discussion of system (231), we will consider some fundamental questions: 1 Does the system (231) have a solution? 2 If the answer to question 1 is yes, then how many solutions are there? 3 How do we determine all of the solutions? To obtain an idea of the answer to questions 1 and 2, consider the special case of a system of three equations in three unknowns The linear system (231) then reduces to a 11 x 1 + a 12 x 2 + a 13 x 3 = b 1, a 21 x 1 + a 22 x 2 + a 23 x 3 = b 2, a 31 x 1 + a 32 x 2 + a 33 x 3 = b 3, which can be interpreted as defining three planes in space An ordered triple (c 1,c 2,c 3 ) is a solution to this system if and only if it corresponds to the coordinates of a point of intersection of the three planes There are precisely four possibilities: 1 The planes have no intersection point 2 The planes intersect in just one point 3 The planes intersect in a line 4 The planes are all identical x 1

3 page Terminology for Systems of Linear Equations 135 In case 1, the corresponding system has no solution, whereas in case 2, the system has just one solution Finally, in cases 3 and 4, every point on the line or plane (respectively) is a solution to the linear system and hence the system has an infinite number of solutions Cases 1,2 and 3 are illustrated in Figure 231 Three parallel planes (no intersection): no solution No common intersection: no solution Planes intersect at a point: a unique solution Planes intersect in a line: an infinite number of solutions Figure 231: Possible intersection points for three planes in space We have therefore proved, geometrically, that there are precisely three possibilities for the solutions of a system of three equations in three unknowns The system either has no solution, it has just one solution, or it has an infinite number of solutions In Section 25, we will establish that these are the only possibilities for the general m n system (231) DEFINITION 234 A system of equations that has at least one solution is said to be consistent, whereas a system that has no solution is called inconsistent Our problem will be to determine whether a given system is consistent and then, if it is, to find its solution set DEFINITION 235 Naturally associated with the system (231) are the following two matrices: a 11 a 12 a 1n a 21 a 22 a 2n 1 The matrix of coefficients A = a m1 a m2 a mn a 11 a 12 a 1n b 1 2 The augmented matrix A # a 21 a 22 a 2n b 2 = a m1 a m2 a mn b m

4 page CHAPTER 2 Matrices and Systems of Linear Equations The augmented matrix completely characterizes a system of equations, since it contains all of the system coefficients and system constants We will see in the subsequent sections that the relationship between A and A # determines the solution properties of a linear system Notice that the matrix of coefficients is the matrix consisting of the first n columns of A # Example 236 Write the system of equations with the following augmented matrix: Solution: The appropriate system is x 1 + 2x 2 + 9x 3 x 4 = 1, 2x 1 3x 2 + 7x 3 + 4x 4 = 2, x 1 + 3x 2 + 5x 3 = 1 Vector Formulation We next show that the matrix product described in the preceding section can be used to write a linear system as a single equation involving the matrix of coefficients and column vectors For example, the system x 1 + 3x 2 4x 3 = 1, 2x 1 + 5x 2 x 3 = 5, x 1 + 6x 3 = 3 can be written as the vector equation x 1 x 2 x 3 1 = 5, 3 since this vector equation is satisfied if and only if x 1 + 3x 2 4x 3 1 2x 1 + 5x 2 x 3 = 5 ; x 1 + 6x 3 3 that is, if and only if each equation of the given system is satisfied Similarly, the general m n system of linear equations can be written as the vector equation a 11 x 1 + a 12 x a 1n x n = b 1, a 21 x 1 + a 22 x a 2n x n = b 2, a m1 x 1 + a m2 x a mn x n = b m, Ax = b,

5 page Terminology for Systems of Linear Equations 137 where A is the m n matrix of coefficients and x 1 b 1 x 2 b 2 x = and b = x n b m We will refer to the column n-vector x as the vector of unknowns, and to the column m-vector b as the right-hand-side vector Assuming that all elements in the system are real, we can view b as an element of R m and x as an element of R n We can denote these statements by b R m and x R n, respectively 3 Therefore, the set of all real solutions to the system Ax = b is S ={x R n : Ax = b}, which is a subset of R n Example 237 It can be shown, using the techniques of the next two sections, that the solution set of the linear system x 1 + x 2 + 2x 3 x 4 = 0, 3x 1 2x 2 + x 3 + 2x 4 = 0, 5x 1 + 3x 2 + 3x 3 2x 4 = 0, is the subset of R 4 defined by S ={( t,4t,t,5t) : t R} A similar vector formulation for systems of differential equations can be used not only in developing the theory for such systems, but also in deriving solution techniques As an example of this formulation, consider the system of differential equations dx 1 = 3tx 1 + 9x 2 + 6e t, dt dx 2 = 2x 1 7x 2 + 3e t dt Using matrix and vector functions, this system can be written as the vector equation where x(t) = x1 (t), x 2 (t) dx dt = dx dt = A(t)x(t) + b(t), dx1 /dt, A = dx 2 /dt 3t 9 6e t, and b(t) = 2 7 3e t In this formulation, the basic unknown is the column 2-vector function x(t) Example 238 Give the vector formulation for the system of equations x 1 = 3x 1 + (sin t)x 2 + e t, x 2 = 7tx 1 + t 2 x 2 4e t 3 The symbol is the set-theoretic notation declaring membership in a set; it will be often encountered in the text

6 page CHAPTER 2 Matrices and Systems of Linear Equations Solution: That is, where We have x(t) = [ x 1 3 sint x1 e x 2 = 7t t 2 + t ] x 2 4e t x (t) = A(t)x(t) + b(t), [ x1 (t) 3 sint e, A(t) = x 2 (t) 7t t 2, b(t) = t ] 4e t Exercises for 23 Key Terms System coefficients, System constants, Homogeneous system, Nonhomogeneous system, Solution, Solution set, Consistent system, Inconsistent system, Matrix of coefficients, Augmented matrix, Vector of unknowns, Right-hand-side vector Skills Be able to write a linear system of equations as a vector equation, and identify the matrix of coefficients, the right-hand-side vector, and the augmented matrix Given a matrix of coefficients and a right-hand-side vector, or an augmented matrix, be able to write the corresponding linear system Understand the geometric difference between a consistent linear system and an inconsistent one Be able to verify that the components of a given vector provide a solution to a linear system Be able to give the vector formulation for a system of differential equations True-False Review For Questions 1 6, decide if the given statement is true or false, and give a brief justification for your answer If true, you can quote a relevant definition or theorem from the text If false, provide an example, illustration, or brief explanation of why the statement is false 1 If a linear system of equations has an m n augmented matrix, then the system has m equations and n unknowns 2 A linear system that contains three distinct planes can have at most one solution 3 If the matrix of coefficients of a linear system is an m n matrix, then the right-hand-side vector must have n components 4 It is impossible for a linear system of equations to have exactly two solutions 5 If a linear system has an m n coefficient matrix, then the augmented matrix for the linear system is m (n + 1) 6 If A is an n n matrix, then the linear systems Ax = 0 and A T x = 0 have the same solution set Problems For Problems 1 2, verify that the given triple of real numbers is a solution to the given system 1 (1, 1, 2); 2 (2, 3, 1); 2x 1 3x 2 + 4x 3 = 13, x 1 + x 2 x 3 = 2, 5x 1 + 4x 2 + x 3 = 3 x 1 + x 2 2x 3 = 3, 3x 1 x 2 7x 3 = 2, x 1 + x 2 + x 3 = 0, 2x 1 + 2x 2 4x 3 = 6

7 page Verify that for all values of t, (1 t,2 + 3t,3 2t) is a solution to the linear system x 1 + x 2 + x 3 = 6, x 1 x 2 2x 3 = 7, 5x 1 + x 2 x 3 = 4 24 Elementary Row Operations and Row-Echelon Matrices A = 4 1 2, b = Consider the m n homogeneous system of linear equations Ax = 0 (232) 4 Verify that for all values of s and t, (s, s 2t,2s + 3t,t) is a solution to the linear system x 1 + x 2 x 3 + 5x 4 = 0, 2x 2 x 3 + 7x 4 = 0, 4x 1 + 2x 2 3x x 4 = 0 5 By making a sketch in the xy-plane, prove that the following linear system has no solution: 2x + 3y = 1, 2x + 3y = 2 For Problems 6 8, determine the coefficient matrix, A, the right-hand-side vector, b, and the augmented matrix A # of the given system x 1 + 2x 2 3x 3 = 1, 2x 1 + 4x 2 5x 3 = 2, 7x 1 + 2x 2 x 3 = 3 x + y + z w = 3, 2x + 4y 3z + 7w = 2 x 1 + 2x 2 x 3 = 0, 2x 1 + 3x 2 2x 3 = 0, 5x 1 + 6x 2 5x 3 = 0 For Problems 9 10, write the system of equations with the given coefficient matrix and right-hand-side vector A = , b = (a) If x =[x 1 x 2 x n ] T and y =[y 1 y 2 y n ] T are solutions to (232), show that z = x + y and w = cx are also solutions, where c is an arbitrary scalar (b) Is the result of (a) true when x and y are solutions to the nonhomogeneous system Ax = b? Explain For Problems 12 15, write the vector formulation for the given system of differential equations 12 x 1 = 4x 1 + 3x 2 + 4t, x 2 = 6x 1 4x 2 + t 2 13 x 1 = t2 x 1 tx 2, x 2 = ( sin t)x 1 + x 2 14 x 1 = e2t x 2, x 2 + (sin t)x 1 = 1 15 x 1 = ( sin t)x 2 + x 3 + t, x 2 = et x 1 + t 2 x 3 + t 3, x 3 = tx 1 + t 2 x For Problems verify that the given vector function x defines a solution to x = Ax + b for the given A and b [ e 4t 16 x(t) = 2e 4t ], A = [ 4e 17 x(t) = 2t ] + 2 sin t 3e 2t, A = cos t 2(cos t + sin t) b(t) = 7sint + 2 cos t 2 1 0, b(t) = , Elementary Row Operations and Row-Echelon Matrices In the next section we will develop methods for solving a system of linear equations These methods will consist of reducing a given system of equations to a new system that has the same solution set as the given system but is easier to solve In this section we introduce the requisite mathematical results