1.2 Solving a System of Linear Equations


 Jeffry Wilcox
 1 years ago
 Views:
Transcription
1 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems  Basic De nitions As noticed above, the general form of a linear system of m equations in n variables is of the form >< > a 11 x 1 + a 1 x + + a 1n x n = b 1 a 1 x 1 + a x + + a n x n = b a m1 x 1 + a m x + + a mn x n = b m Some of the coe cients a ij can be zero. In fact, the more coe cients are zero, the easier the system will be to solve. The easiest system to solve is of the form a 11 x 1 = b 1 >< a x = b a x = b > a mn x n = b m Its corresponding augmented matrix is a b 1 0 a 0 b 0 0 a 0 b a mn b m To solve, we simply divide by the coe cient in front of each variable. This is illustrated below. < x 1 = 10 Example 0 Solve x = 9 x = The solution is x 1 = 10 =, x = 9 =, x =. When all the coe cients in the system above are 1, the system in the above example is said to be in reduced rowechelon form. The precise de nition is given below. De nition 1 Consider a system of linear equations. 1. The system is said to be in reduced rowechelon form if it satis es the four properties below
2 CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES (a) If a row does not consist entirely of zeros, then the rst nonzero number is a 1. This number is called the leading 1. (b) The rows consisting entirely of zeros, if there are any, are at the bottom of the system. (c) If two consecutive rows contain a leading 1, then the leading 1 of the higher row is further to the left than the leading 1 of the lower row. (d) Each column that contains a leading 1 contains 0 everywhere else.. The system is said to be in rowechelon form if the properties a  c are satis ed. Thus, a system in reduced rowechelon form is necessarily in rowechelon form. Remark The same de nitions apply to the augmented matrix of a system. Remark Since we can switch back and forth between a system of linear equations and its corresponding augmented matrix, we can either work on a system, or on its corresponding augmented matrix. We will illustrate both. Example The matrices below are in reduced rowechelon form Example The matrices below are in rowechelon form A system in rowechelon form is also easy to solve. We illustrate how with an example.
3 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 9 Example Consider the system < x 1 +x x = 10 x +x = x = Its corresponding augmented matrix is Such a system is also fairly easy to solve. The method used is called backsubstitution. Since we know x =, we can replace x by its value in the second equation to obtain x + = or x =. Now that we know x and x, we can replace in the rst equation to obtain x 1 + = 10 or x 1 =. It is called back substitution because we work backward. We start by nding the value of the last variable, then the next to the last and so on, until we nd the value of the rst variable. At this point, we are done. Remark In order to be able to do backsubstitution, it is not necessary that a ii = 1 in the i th equation. De nition Two systems are said to be equivalent if they have the same solution set. Similarly, two augmented matrices are said to be equivalent if they are the augmented matrices of two systems having the same solution set. You remember for elementary mathematics, to solve a given equation in one variable say x, we transform it into an equation of the form x = S where S is then the solution we are after. To transform it, we use certain transformations which do not change the solution set of the equation, in other words, we obtain equivalent equations until we have one of the form x = S. You will recall that the transformations that can be applied when solving an equation are 1. Add the same number on both sides of the equation.. Multiply both sides by the same nonzero number. We follow a similar procedure to solve systems of equations. There are actually two procedures we can use. They are outlined below, then explored thoroughly in the next two subsections. The rst technique, is called Gaussian elimination. Algorithm 9 (Gaussian Elimination) To solve a system of linear equations, follow the steps below 1. Transform its augmented matrix into an equivalent augmented matrix in rowechelon form.
4 10 CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES. Use backsubstitution to nish solving. The second technique is called GaussJordan elimination Algorithm 0 (GaussJordan Elimination) To solve a system of linear equations, follow the steps below 1. Transform its augmented matrix into an equivalent augmented matrix in reduced rowechelon form.. The solution is given by the last column of the augmented matrix in reduced rowechelon form. With either technique, since the systems are equivalent, they will have the same solution set. Usually, getting from a system to an equivalent system in rowechelon form or reduced rowechelon form requires several steps, each step producing an equivalent system. The question is which transformations can we apply to a system so that it is transformed into a system in row echelon form or reduced rowechelon form. The key is that the new system should be equivalent to the original one, so they have the same solutions. It would not help to get a system which is simpler to solve, but which does not have the same solutions as the original one. There are three transformations which can be applied, to a system that will produce an equivalent system. For simplicity, let us label the equations of a system E 1, E,, E m. We will use the same names for the rows of the corresponding augmented matrix. Proposition 1 Each of the following operations on a system of linear equations produces an equivalent system 1. Equation E i can be multiplied by a nonzero constant, with the result used in place of E i. This operation is denoted (E i )! (E i ). Equation E j can be multiplied by any constant and added to equation E i, with the result used in place of E i. This operation is denoted (E j + E i )! (E i ). Equations E i and E j can be transposed in order. This operation is denoted (E i ) $ (E j ) Remark The same operations can be applied to the augmented matrix of a system. Simply replace the term equation by row in the above proposition. Remark These operations are called elementary row operations. Let us look at some examples to illustrate how these transformations are done.
5 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 11 Example Consider If we perform (E 1 )! (E ), we obtain Example Consider If we perform 1 E! (E ), we obtain This is useful when we want to obtain a leading 1 on a given row. Example Consider If we perform ( E 1 + E )! (E ), we obtain This is useful when we want to make an entry equal to 0. In this case, we made the rst entry of the second row equal to 0. We now look at each method to solve a system in greater detail. 1.. Gaussian Elimination Gaussian elimination was brie y outlined above. It has two main steps. First, we get a system (or its augmented matrix) in rowechelon form. Then, we use back substitution to nish solving. We have already explained back substitution. The rst step is always done in a very orderly fashion. In fact, it is very easy to implement on a computer. It always proceeds as follows (the algorithm below works for both a system and its augmented matrix. The augmented matrix part appears in parentheses)
6 1 CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES Remark Proceed from the rst equation (row) to the last. Look at the rst equation (row), make sure no equation (row) below it has a leading entry further to the left. If one does, switch the two. Eliminate all the entries below the leading entry of the rst equation (row). Repeat the procedure for equation (row), then, In general, we look at the i th equation (row), make sure the coe cient for x i is not 0. If it is, you will have to nd an equation in which it is not, and interchange the two equations. Then, eliminate x i from all the equations below the i th equation. We do this for i = 1; ; ; n 1 if the system has n equations. Recall that Gaussian elimination is a systematic procedure which transforms a system into an equivalent system in rowechelon form. We illustrate it with examples. Example Consider the system E 1 x 1 +x +x = >< E x 1 +x x +x = 1 E x 1 x x +x = > E x 1 +x +x x = (1.) Its corresponding augmented matrix is We follow the algorithm described above. We begin with the rst row. Its leading entry is in column 1. The goal is to set to 0 all the other entries in column 1 and rows . In order to achieve this, we perform (E E 1 )! (E ), (E E 1 )! (E ) and (E + E 1 )! (E ). The resulting augmented matrix is Next, we work on the second row. First, we make its leading entry 1 by performing ( E 1 )! (E 1 ). The resulting augmented matrix is
7 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1 Then we make the entries below the leading entry of the second row equal to 0 by performing (E + E )! (E ) and (E E )! (E ). The resulting augmented matrix is We actually were lucky. In the process, we did part of the next step. Looking at row, we already set the entry below it to 0. We just have 1 to set the leading entry of row to 1 by performing E! E. The resulting augmented matrix is We are on the last row, the only thing to do here is to set its leading entry 1 to 1 by performing 1 E! (E ). The resulting augmented matrix is Now, we see that the matrix is in rowechelon form. The corresponding system is > < > E 1 x 1 +x +x = E x +x +x = E x + 1 x = 1 E x = 1 (1.) The system in (1) has the same solutions as the original system (1). We nish solving the system in (1) using backsubstitution. From E, we get x = 1 We can now use E to nd x as follows x + 1 x = 1 x = 1 (1 x ) x = 0 since x = 1
8 1 CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES We now continue with E. Finally, using E 1 yields x x x = x = (x + x ) x = x 1 + x + x = x 1 = x x + x 1 = 1 The solutions of the system in (1) and therefore of the system in (1) are x 1 = 1; x = ; x = 0; and x = 1 This is an example in which the procedure produced a unique solution, we had a consistent system. Following the same procedure, how will we detect if we have a system with no solutions, or one with an in nite number of solutions? The next two examples illustrate this. To also illustrate that the same procedure can be applied to system as well as augmented matrices, we do the next example using the system. < Example 9 Solve the system x 1 + x + x = x 1 + x + x = x 1 + x + x = To eliminate x 1 from E and E, we perform (E E 1 )! (E ) and (E E 1 )! (E ). The resulting system is < x 1 + x + x = x = x = The next step would be to eliminate x from E. This was done in the previous step. The only thing left is to set the leading coe cient of the second equation to 1 by performing ( E )! (E ). The corresponding system is < x 1 + x + x = x = x = This system has an in nite number of solutions given by x = and x 1 +x = or x 1 = x. The free variable is x. In parametric notation, the solution set is < x 1 = t x = t x =
9 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1 < Example 0 Solve the system x 1 + x + x = x 1 + x + x = x 1 + x + x = To eliminate x 1 from E and E, we perform (E E 1 )! (E ) and (E E 1 )! (E ). The resulting system is < x 1 + x + x = x = x = The next step would be to eliminate x from E. This was done in the previous step. The only thing left is to set the leading coe cient of the second equation to 1 by performing ( E )! (E ). The corresponding system is x 1 + x + x = x = x = This system has no solutions because the last two equations are in contradiction. Remark 1 It is important to notice that the Gaussian elimination procedure is always done in a very orderly fashion. In fact, it is very easy to implement on a computer. It always proceeds as follows Look at the rst equation, make sure the coe cient for x 1 is not 0. If it is, you will have to nd an equation in which it is not, and interchange the two equations. Then, eliminate x 1 from all the equations below the rst equation. Next, look at the second equation, make sure the coe cient for x is not 0. If it is, you will have to nd an equation in which it is not, and interchange the two equations. Then, eliminate x from all the equations below the second equation. The procedure continues the same way. In general, we look at the i th equation, make sure the coe cient for x i is not 0. If it is, you will have to nd an equation in which it is not, and interchange the two equations. Then, eliminate x i from all the equations below the i th equation. We do this for i = 1; ; ; n 1 if the system has n equations. 1.. GaussJordan Elimination Here, we will work with augmented matrices. Recall that GaussJordan elimination is a systematic procedure which transforms a system into an equivalent system in reduced rowechelon form. To obtain a reduced rowechelon form, we rst obtain a rowechelon form, then we go further. We need to make the columns containing the leading entries equal to 0, except for the leading entry of course. We illustrate it with an example.
10 1 CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES Example Solve the system x 1 +x +x = >< x 1 +x x +x = 1 x 1 x x +x = > x 1 +x +x x = Its corresponding augmented matrix is This is the system we did above. We will take it from its rowechelon form Now, we start from the bottom and work our way up. There is nothing to do on the last row. For row, we make the th entry 0 by performing E 1 E! (E ). The resulting augmented matrix is For row, we make the rd and th entry 0. Let s do them one at a time. First, we perform (E E )! (E ). We obtain Next, we perform (E E ) to obtain
11 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1 Finally row 1. We make entries and equal to 0 by performing rst (E 1 E )! (E 1 ), this gives us then (E 1 E )! (E 1 ), this gives us The solution is now read from the last column. Thus, we see that x 1 = 1, x =, x = 0, and x = Homogeneous Linear Systems We nish this section by looking at a special type of system, homogeneous systems. These system as we will see, always have at least one solution. De nition (Homogeneous System) A system of linear equations is said to be homogeneous if the constant terms are always 0. In other words, a homogeneous system is of the form a 11 x 1 + a 1 x + + a 1n x n = 0 >< a 1 x 1 + a x + + a n x n = 0 > a m1 x 1 + a m x + + a mn x n = 0 De nition The solution x i = 0 for i = 1,,, n is called the trivial solution. Other solutions are called nontrivial solutions. Remark A homogeneous system always has at least the trivial solution. So, a homogeneous system is always consistent. Remark Another important fact about homogeneous systems is that the elementary transformations will not alter the last column of their augmented matrix. Thus, when we transform a homogeneous system in rowechelon form or reduced rowechelon form, we still have a homogeneous system. It turns out that a homogeneous system of linear equations either has only the trivial solution, or has in nitely many solutions. We state this result without proof. We will prove it later in the course. If the homogeneous system has more equations than unknowns, then it will have in nitely many solutions.
12 1 CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES Example Solve < x 1 +x +x = 0 x 1 +x x +x = 0 x 1 x x +x = 0 The corresponding augmented matrix is To set the 1st entries in rows and, we perform (E (E E 1 )! E. We obtain E 1 )! E and Set the leading entry of row to 1 by performing ( E )! E. We obtain Set the second entry in row to 0 by performing (E + E )! E. We obtain Set the leading entry in row to 1 by performing 1 E! E. We obtain The corresponding system is < x 1 +x +x = 0 x +x +x = 0 x + 1 x = 0 We can use back substitution. Letting x be the free variable and setting x = t, we get from equation that x = t. From equation, we get 1 x = x x = 1 t t = t
13 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 19 Therefore x 1 = x x = t t = t Thus, the solution in para,etric form is x 1 = >< x = x = > x = t 1.. Concept Review t t 1 t Know what a system in rowechelon form and reduced rowechelon form is. Be able to solve a system in rowechelon form, using backsubstitution. Know the transformations which produce equivalent systems. Be able to perform Gaussian elimination to transform a system of linear equations into an equivalent system in rowechelon form. Be able to perform GaussJordan elimination to transform a system of linear equations into an equivalent system in reduced rowechelon form. Be able to solve a system of linear equation using either Gaussian elimination or GaussJordan elimination. 1.. Problems On pages 191, do # 1,,,,,,, 1, 1, 1, 1, 19, 0,,,,.
160 CHAPTER 4. VECTOR SPACES
160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results
More informationSection 8.2 Solving a System of Equations Using Matrices (Guassian Elimination)
Section 8. Solving a System of Equations Using Matrices (Guassian Elimination) x + y + z = x y + 4z = x 4y + z = System of Equations x 4 y = 4 z A System in matrix form x A x = b b 4 4 Augmented Matrix
More information1 Gaussian Elimination
Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 GaussJordan reduction and the Reduced
More information1 Systems Of Linear Equations and Matrices
1 Systems Of Linear Equations and Matrices 1.1 Systems Of Linear Equations In this section you ll learn what Systems Of Linear Equations are and how to solve them. Remember that equations of the form a
More informationSystems of Linear Equations
A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler Systems of Linear Equations Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION AttributionNonCommercialShareAlike (CC
More informationMath 240: Linear Systems and Rank of a Matrix
Math 240: Linear Systems and Rank of a Matrix Ryan Blair University of Pennsylvania Thursday January 20, 2011 Ryan Blair (U Penn) Math 240: Linear Systems and Rank of a Matrix Thursday January 20, 2011
More information2. Systems of Linear Equations.
2. Systems of Linear Equations 2.1. Introduction to Systems of Linear Equations Linear Systems In general, we define a linear equation in the n variables x 1, x 2,, x n to be one that can be expressed
More information4.2: Systems of Linear Equations and Augmented Matrices 4.3: GaussJordan Elimination
4.2: Systems of Linear Equations and Augmented Matrices 4.3: GaussJordan Elimination 4.2/3.1 We have discussed using the substitution and elimination methods of solving a system of linear equations in
More information1.4 More Matrix Operations and Properties
8 CHAPTER. SYSTEMS OF LINEAR EQUATIONS AND MATRICES. More Matrix Operations Properties In this section, we look at the properties of the various operations on matrices. As we do so, we try to draw a parallel
More information4 Solving Systems of Equations by Reducing Matrices
Math 15 Sec S0601/S060 4 Solving Systems of Equations by Reducing Matrices 4.1 Introduction One of the main applications of matrix methods is the solution of systems of linear equations. Consider for example
More informationSystems of Linear Equations
Systems of Linear Equations S F Ellermeyer May, 9 These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (rd edition) These notes
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More information1. Linear systems of equations. Chapters 78: Linear Algebra. Solution(s) of a linear system of equations. Row operations.
A linear system of equations of the form Sections 75 78 & 8 a x + a x + + a n x n = b a x + a x + + a n x n = b a m x + a m x + + a mn x n = b m can be written in matrix form as AX = B where a a a n x
More informationSolving Systems of Linear Equations; Row Reduction
Harvey Mudd College Math Tutorial: Solving Systems of Linear Equations; Row Reduction Systems of linear equations arise in all sorts of applications in many different fields of study The method reviewed
More informationMAC Module 1 Systems of Linear Equations and Matrices I. Learning Objectives. Upon completing this module, you should be able to:
MAC 03 Module Systems of Linear Equations and Matrices I Learning Objectives Upon completing this module, you should be able to:. Represent a system of linear equations as an augmented matrix.. Identify
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
More information4.6 Null Space, Column Space, Row Space
NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear
More informationReinserting the variables in the last row of this augmented matrix gives
Math 313 Lecture #2 1.2: Row Echelon Form Not Reaching Strict Triangular Form. The algorithm that reduces the augmented matrix of an n n system to that of an equivalent strictly triangular system fails
More information1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form
1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and
More informationLecture 11: Solving Systems of Linear Equations by Gaussian Elimination
Lecture 11: Solving Systems of Linear Equations by Gaussian Elimination Winfried Just, Ohio University February 3, 2016 Review: The coefficient matrix Consider a system of m linear equations in n variables.
More informationMATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0,
MATH 23: SYSTEMS OF LINEAR EQUATIONS Systems of Linear Equations In the plane R 2 the general form of the equation of a line is ax + by = c and that the general equation of a plane in R 3 will be we call
More informationLecture 2. Solving Linear Systems
Lecture. Solving Linear Systems As we discussed before, we can solve any system of linear equations by the method of elimination, which is equivalent to applying a sequence of elementary row reductions
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationLecture 1: Systems of Linear Equations
MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables
More informationLecture 12: Solving Systems of Linear Equations by Gaussian Elimination
Lecture 12: Solving Systems of Linear Equations by Gaussian Elimination Winfried Just, Ohio University September 23, 2015 Review: The coefficient matrix Consider a system of m linear equations in n variables.
More information2.5 Gaussian Elimination
page 150 150 CHAPTER 2 Matrices and Systems of Linear Equations 37 10 the linear algebra package of Maple, the three elementary 20 23 1 row operations are 12 1 swaprow(a,i,j): permute rows i and j 3 3
More informationUsing Matrix Elimination to Solve Three Equations With Three Unknowns
Using Matrix Elimination to Solve Three Equations With Three Unknowns Here we will be learning how to use Matrix Elimination to solve a linear system with three equation and three unknowns. Matrix Elimination
More information1 Systems Of Linear Equations and Matrices
1 Systems Of Linear Equations and Matrices 1.1 Systems Of Linear Equations In this section you ll learn what Systems Of Linear Equations are and how to solve them. Remember that equations of the form a
More informationLinear Systems and Gaussian Elimination
Eivind Eriksen Linear Systems and Gaussian Elimination September 2, 2011 BI Norwegian Business School Contents 1 Linear Systems................................................ 1 1.1 Linear Equations...........................................
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More information4.5 Basis and Dimension of a Vector Space
.. BASIS AND DIMENSION OF A VECTOR SPACE. Basis and Dimension of a Vector Space In the section on spanning sets and linear independence, we were trying to understand what the elements of a vector space
More informationMAT Solving Linear Systems Using Matrices and Row Operations
MAT 171 8.5 Solving Linear Systems Using Matrices and Row Operations A. Introduction to Matrices Identifying the Size and Entries of a Matrix B. The Augmented Matrix of a System of Equations Forming Augmented
More information12. x 1 = 4x 1 + 3x 2 + 4t, x 2 = 6x 1 4x 2 + t x 1 = t2 x 1 tx 2, x 2 = ( sin t)x 1 + x x 1 = e2t x 2, x 2 + (sin t)x 1 = 1.
page 139 3 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 RowEchelon Matrices
More information4.3 Linear Combinations and Spanning Sets
4.. LINEAR COMBINATIONS AND SPANNING SETS 5 4. Linear Combinations and Spanning Sets In the previous section, we looked at conditions under which a subset W of a vector space V was itself a vector space.
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationSolving Systems of Linear Equations Using Matrices
Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.
More informationChapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Section 4.1: Systems of Equations Systems of equations A system of equations consists of two or more equations involving two or more variables { ax + by = c dx + ey = f A solution of such a system is an
More informationLecture 6. Inverse of Matrix
Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that
More informationOverview. Matrix Solutions to Linear Systems. Threevariable systems. Matrices. Solving a threevariable system
Overview Matrix Solutions to Linear Systems Section 8.1 When solving systems of linear equations in two variables, we utilized the following techniques: 1.Substitution 2.Elimination 3.Graphing In this
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationTopic 1: Matrices and Systems of Linear Equations.
Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method
More informationMATH 304 Linear Algebra Lecture 4: Row echelon form. GaussJordan reduction.
MATH 304 Linear Algebra Lecture 4: Row echelon form GaussJordan reduction System of linear equations: a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2
More informationA Quick Introduction to Row Reduction
A Quick Introduction to Row Reduction Gaussian Elimination Suppose we are asked to solve the system of equations 4x + 5x 2 + 6x 3 = 7 6x + 7x 2 + 8x 3 = 9. That is, we want to find all values of x, x 2
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationChapter 1 Matrices and Systems of Linear Equations
Chapter 1 Matrices and Systems of Linear Equations 1.1: Introduction to Matrices and Systems of Linear Equations 1.2: Echelon Form and GaussJordan Elimination Lecture Linear Algebra  Math 2568M on Friday,
More informationSection 6.2 Larger Systems of Linear Equations
Section 6.2 Larger Systems of Linear Equations EXAMPLE: Solve the system of linear equations. Solution: From Equation 3, you know the value z. To solve for y, substitute z = 2 into Equation 2 to obtain
More informationElementary row operations and some applications
Physics 116A Winter 2011 Elementary row operations and some applications 1. Elementary row operations Given an N N matrix A, we can perform various operations that modify some of the rows of A. There are
More informationNotes on Row Reduction
Notes on Row Reduction Francis J. Narcowich Department of Mathematics Texas A&M University May 2016 1 Notation and Terms The row echelon form of a matrix contains a great deal of information, both about
More information3 Gaussian elimination (row reduction)
LINEAR ALGEBRA: THEORY. Version: August 12, 2000 23 3 Gaussian elimination (row reduction) Let A be an n k matrix and b is an n 1 vector. We wish to solve the equation Ax = b (3.1) where x R k. One can
More informationSection 6.2 Larger Systems of Linear Equations
Section 6.2 Larger Systems of Linear Equations Gaussian Elimination In general, to solve a system of linear equations using its augmented matrix, we use elementary row operations to arrive at a matrix
More informationUnit 17 The Theory of Linear Systems
Unit 17 The Theory of Linear Systems In this section, we look at characteristics of systems of linear equations and also of their solution sets. Theorem 17.1. For any system of linear equations A x = b,
More informationJones and Bartlett Publishers, LLC. NOT FOR SALE OR DISTRIBUTION
8498_CH08_WilliamsA.qxd 11/13/09 10:35 AM Page 347 Jones and Bartlett Publishers, LLC. NOT FOR SALE OR DISTRIBUTION C H A P T E R Numerical Methods 8 I n this chapter we look at numerical techniques for
More information5.5. Solving linear systems by the elimination method
55 Solving linear systems by the elimination method Equivalent systems The major technique of solving systems of equations is changing the original problem into another one which is of an easier to solve
More informationMatrix Inverses. Since the linear system. can be written as. where. ,, and,
Matrix Inverses Consider the ordinary algebraic equation and its solution shown below: Since the linear system can be written as where,, and, (A = coefficient matrix, x = variable vector, b = constant
More informationLinear Equations ! 25 30 35$ & " 350 150% & " 11,750 12,750 13,750% MATHEMATICS LEARNING SERVICE Centre for Learning and Professional Development
MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) Topic 4 Module 9 Introduction Systems of to Matrices Linear Equations Income = Tickets!
More informationSystems of Linear Equations
Systems of Linear Equations Systems of Linear Equations. We consider the problem of solving linear systems of equations, such as x 1 2x 2 = 8 3x 1 + x 2 = 3 In general, we write a system of m equations
More information1.5 Elementary Matrices and a Method for Finding the Inverse
.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:
More informationDirect Methods for Solving Linear Systems. Linear Systems of Equations
Direct Methods for Solving Linear Systems Linear Systems of Equations Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University
More informationSystems of Linear Equations
Chapter 1 Systems of Linear Equations 1.1 Intro. to systems of linear equations Homework: [Textbook, Ex. 13, 15, 41, 47, 49, 51, 65, 73; page 11]. Main points in this section: 1. Definition of Linear
More informationRow Reduction and Echelon Forms
MA 2071 A 12 Bill Farr August 22, 2012 1 2 3 4 5 Definition of Pivot Postion An Algorithm for Putting a Matrix in RREF An Linear Systems and Matrices A linear system has two associated matrices, the coefficient
More information2.6 The Inverse of a Square Matrix
200/2/6 page 62 62 CHAPTER 2 Matrices and Systems of Linear Equations 0 0 2 + i i 2i 5 A = 0 9 0 54 A = i i 4 + i 2 0 60 i + i + 5i 26 The Inverse of a Square Matrix In this section we investigate the
More informationAPPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the cofactor matrix [A ij ] of A.
APPLICATIONS OF MATRICES ADJOINT: Let A = [a ij ] be a square matrix of order n. Let Aij be the cofactor of a ij. Then the n th order matrix [A ij ] T is called the adjoint of A. It is denoted by adj
More information1.3 Matrices and Matrix Operations
0 CHAPTER. SYSTEMS OF LINEAR EQUATIONS AND MATRICES. Matrices and Matrix Operations.. De nitions and Notation Matrices are yet another mathematical object. Learning about matrices means learning what they
More informationTransition Maths and Algebra with Geometry
Transition Maths and Algebra with Geometry Tomasz Brengos Lecture Notes Electrical and Computer Engineering Tomasz Brengos Transition Maths and Algebra with Geometry 1/27 Contents 1 Systems of linear equations
More informationPartial Fractions Decomposition
Partial Fractions Decomposition Dr. Philippe B. Laval Kennesaw State University August 6, 008 Abstract This handout describes partial fractions decomposition and how it can be used when integrating rational
More informationMethods for Finding Bases
Methods for Finding Bases Bases for the subspaces of a matrix Rowreduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,
More informationROW REDUCTION AND ITS MANY USES
ROW REDUCTION AND ITS MANY USES CHRIS KOTTKE These notes will cover the use of row reduction on matrices and its many applications, including solving linear systems, inverting linear operators, and computing
More informationLinear algebra. Systems of linear equations
Linear algebra Outline 1 Basic notation Gaussian elimination Cramer s rule 2 Examples 3 List of tasks for students Lucie Doudová (UoD Brno) Linear algebra 2 / 27 Outline 1 Basic notation Gaussian elimination
More informationMatrix Division Je Stuart c 2008
Matrix Division Je Stuart c 008 High School Algebra Revisited The linear system ax = b where a and b are real numbers. Why do we need division? In its earliest form, division must have arisen to answer
More informationDe nitions of Linear Algebra Terms
De nitions of Linear Algebra Terms In order to learn and understand mathematics, it is necessary to understand the meanings of the terms (vocabulary words) that are used This document contains de nitions
More informationLet A be a square matrix of size n. A matrix B is called inverse of A if AB = BA = I.
Matrix Inverses Let A be a square matrix of size n A matrix B is called inverse of A if AB = BA = I If A has an inverse, then we say that A is invertible The inverse of A is unique and we denote it by
More informationSolving Linear Systems, Continued and The Inverse of a Matrix
, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing
More information2 Matrices and systems of linear equations
Matrices and systems of linear equations You have all seen systems of linear equations such as 3x + 4y = x y = 0. ( This system can be solved easily: Multiply the nd equation by 4, and add the two resulting
More information2 Matrices and systems of linear equations
Matrices and systems of linear equations You have all seen systems of linear equations such as 3x + 4y = 5 x y = 0. () This system can easily be solved: just multiply the nd equation by 4, and add the
More informationSUBSPACES. Chapter Introduction. 3.2 Subspaces of F n
Chapter 3 SUBSPACES 3. Introduction Throughout this chapter, we will be studying F n, the set of all n dimensional column vectors with components from a field F. We continue our study of matrices by considering
More informationSECTION 1.1: SYSTEMS OF LINEAR EQUATIONS
SECTION.: SYSTEMS OF LINEAR EQUATIONS THE BASICS What is a linear equation? What is a system of linear equations? What is a solution of a system? What is a solution set? When are two systems equivalent?
More informationRow Echelon Form and Reduced Row Echelon Form
These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for inclass presentation
More information8.2 Systems of Linear Equations: Augmented Matrices
8. Systems of Linear Equations: Augmented Matrices 567 8. Systems of Linear Equations: Augmented Matrices In Section 8. we introduced Gaussian Elimination as a means of transforming a system of linear
More information10.1 Systems of Linear Equations: Substitution and Elimination
10.1 Systems of Linear Equations: Substitution and Elimination What does it mean to be a solution to a system of equations?  It is the set of all ordered pairs (x, y) that satisfy the two equations. You
More informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More informationSystems of Linear Equations
Systems of Linear Equations Recall that an equation of the form Ax + By = C is a linear equation in two variables. A solution of a linear equation in two variables is an ordered pair (x, y) that makes
More information3 Systems of Linear. Equations and Matrices. Copyright Cengage Learning. All rights reserved.
3 Systems of Linear Equations and Matrices Copyright Cengage Learning. All rights reserved. 3.2 Using Matrices to Solve Systems of Equations Copyright Cengage Learning. All rights reserved. Using Matrices
More informationMath 1313 Section 3.2. Section 3.2: Solving Systems of Linear Equations Using Matrices
Math Section. Section.: Solving Systems of Linear Equations Using Matrices As you may recall from College Algebra or Section., you can solve a system of linear equations in two variables easily by applying
More informationDefinition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that
0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c
More informationCHAPTER 9: Systems of Equations and Matrices
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 9: Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations in Three Variables
More information( % . This matrix consists of $ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&
Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important
More information2.4 Solving a System of Linear Equations with Matrices
.4 Solving a System of Linear Equations with Matrices Question : What is a matrix? Question : How do you form an augmented matrix from a system of linear equations? Question : How do you use row operations
More information2. Echelon form It is time to make some more definitions and actually prove that Gaussian elimination works: Definition 2.1. Let m and n Z be two
2. Echelon form It is time to make some more definitions and actually prove that Gaussian elimination works: Definition 2.1. Let m and n Z be two positive integers. Let I = { i Z 0 < i m } and J = { j
More informationPhysics 116A Solving linear equations by Gaussian Elimination (Row Reduction)
Physics 116A Solving linear equations by Gaussian Elimination (Row Reduction) Peter Young (Dated: February 12, 2014) I. INTRODUCTION The general problem is to solve m linear equations in n variables. In
More informationInverses and powers: Rules of Matrix Arithmetic
Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3
More informationSECTION 8.3: THE INVERSE OF A SQUARE MATRIX
(Section 8.3: The Inverse of a Square Matrix) 8.47 SECTION 8.3: THE INVERSE OF A SQUARE MATRIX PART A: (REVIEW) THE INVERSE OF A REAL NUMBER If a is a nonzero real number, then aa 1 = a 1 a = 1. a 1, or
More informationSolving Systems of Linear Equations. Substitution
Solving Systems of Linear Equations There are two basic methods we will use to solve systems of linear equations: Substitution Elimination We will describe each for a system of two equations in two unknowns,
More informationSYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89. by Joseph Collison
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections
More informationB such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix
Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.2 Row Reduction and Echelon Forms ECHELON FORM A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties: 1. All nonzero
More information2.2/2.3  Solving Systems of Linear Equations
c Kathryn Bollinger, August 28, 2011 1 2.2/2.3  Solving Systems of Linear Equations A Brief Introduction to Matrices Matrices are used to organize data efficiently and will help us to solve systems of
More information(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product
More informationBasic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.
Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required
More informationPhysics 116A Solving linear equations by Gaussian Elimination (Row Reduction)
Physics 116A Solving linear equations by Gaussian Elimination (Row Reduction) Peter Young (Dated: February 22, 2013) I. INTRODUCTION The general problem is to solve m linear equations in n variables. In
More informationMath 2331 Linear Algebra
1.1 Linear System Math 2331 Linear Algebra 1.1 Systems of Linear Equations Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math2331 Jiwen He, University
More information