System of Linear Equations Matrix

9 Key excerpts on "System of Linear Equations Matrix"

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  Elementary Linear Algebra with Supplemental Applications

  • Howard Anton, Chris Rorres(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  1 C H A P T E R 1 Systems of Linear Equations and Matrices CHAPTER CONTENTS 1.1 Introduction to Systems of Linear Equations 2 1.2 Gaussian Elimination 9 1.3 Matrices and Matrix Operations 21 1.4 Inverses; Algebraic Properties of Matrices 31 1.5 Elementary Matrices and a Method for Finding A −1 42 1.6 More on Linear Systems and Invertible Matrices 49 1.7 Diagonal, Triangular, and Symmetric Matrices 54 1.8 Applications of Linear Systems 60 • Network Analysis (Traffic Flow) 60 • Electrical Circuits 62 • Balancing Chemical Equations 64 • Polynomial Interpolation 67 1.9 Leontief Input-Output Models 70 INTRODUCTION Information in science, business, and mathematics is often organized into rows and columns to form rectangular arrays called “matrices” (plural of “matrix”). Matrices often appear as tables of numerical data that arise from physical observations, but they occur in various mathematical contexts as well. For example, we will see in this chapter that all of the information required to solve a system of equations such as 5x + y = 3 2x − y = 4 is embodied in the matrix  5 2 1 −1 3 4  and that the solution of the system can be obtained by performing appropriate operations on this matrix. This is particularly important in developing computer programs for solving systems of equations because computers are well suited for manipulating arrays of numerical information. However, matrices are not simply a notational tool for solving systems of equations; they can be viewed as mathematical objects in their own right, and there is a rich and important theory associated with them that has a multitude of practical applications. It is the study of matrices and related topics that forms the mathematical field that we call “linear algebra.” In this chapter we will begin our study of matrices.

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  Discrete Mathematics

  Proofs, Structures and Applications, Third Edition

  • Rowan Garnier, John Taylor(Authors)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  The system 2 x 1 -x 2 + x 3 -x 4 = 0 x 2 = 2 x 1 -4 x 3 -x 4 3 x 1 + 5 x 2 -2 x 3 = 0 is recognizable as a homogeneous system of three linear equations in the four variables x 1 , x 2 , x 3 and x 4 once the second equation has been written in standard form: 2 x 1 -x 2 -4 x 3 -x 4 = 0 . A solution of a system of linear equations is an ordered n -tuple defining values of the variables which satisfy each equation in the system. For instance, the system 3 x -2 y + z = -3 x + y + z = 5 x -2 y -z = -9 y + z = 6 has a solution ( -1 , 2 , 4) . 334 Systems of Linear Equations In general, just as for the single linear equation ax = b , a system of linear equations may have none, one or many solutions. A system which has no solution is called inconsistent . A system which has one or many solutions is called consistent . A convenient way to represent a system of linear equations is in matrix form. Consider the following general system of m equations in the n variables x 1 , x 2 , . . . , x n : a 11 x 1 + a 12 x 2 + · · · + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + · · · + a 2 n x n = b 2 . . . . . . . . . . . . a m 1 x 1 + a m 2 x 2 + · · · + a mn x n = b m . This system can be represented by the equivalent matrix equation:      a 11 a 12 . . . a 1 n a 12 a 22 . . . a 2 n . . . . . . . . . a m 1 a m 2 . . . a mn           x 1 x 2 . . . x n      =      b 1 b 2 . . . b m      . Multiplying together the matrices on the left-hand side of the equation gives a matrix of dimension m × 1 whose elements are the left-hand side of the system of equations. Equating the elements of this matrix with those in the matrix on the right-hand side of the matrix equation gives each of the m equations in the system. If we let A = [ a ij ] x =      x 1 x 2 . . . x n      and b =      b 1 b 2 . . . b m      then we can write the matrix equation as A x = b . The matrix A is often referred to as the matrix of coefficients .

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  Linear Algebra

  An Inquiry-Based Approach

  • Jeff Suzuki(Author)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  all equations are in standard form, we’ll be able to work with the vectors as easily as we could have worked with the equations.

  For example, the system of equations

  3 x + 5 y

  = 11

  2 x − 7 y

  = 4

  could be represented as the vectors

  〈 3 , 5 , 11 〉

  and

  〈 2 , − 7 , 4 〉

  , where the first component is the coefficient of x; the second component is the coefficient of y; and the last component is the constant.

  Actually, if we write our two equations as two vectors, we might forget that they are in fact related. To reinforce their relationship, we’ll represent them as a single object by throwing them inside a set of parentheses. We might represent the evolution of our notation as

  {

  3 x + 5 y

  = 11

  2 x − 7 y

  = 4

  →

  〈 3 , 5 , 11 〉

  〈 2 , − 7 , 4 〉

  →

  (

  3 5 11

  2

  − 7

  4

  )

  As a general rule, mathematicians are terrible at coming up with new words, so we appropriate existing words that convey the essential meaning of what we want to express.

  In geology, when precious objects like gemstones or fossils are embedded in a rock, the rock is called a matrix. The term is also used in biology for the structure that holds other precious objects—the internal organs—in place, and in construction for the mortar that holds bricks in place. Thus in 1850, the mathematician James Joseph Sylvester (1814–1897) appropriated the term for the structure that keeps the coefficients and constants of a linear equation in place.

  Since the matrix shown includes both the coefficients of our equation and the constants, we can describe it as the coefficient matrix augmented by a column of constants or, more simply, the augmented coefficient matrix

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  Mathematics

  An Applied Approach

  • Michael Sullivan, Abe Mizrahi(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  65. Write a brief paragraph outlining your strategy for solving a system of two linear equations containing two variables. 66. Do you prefer the method of substitution or the method of elimination for solving a system of two linear equations con- taining two variables? Give reasons. Systems of Linear Equations: Matrix Method 65 The systematic approach of the method of elimination for solving a system of linear equations provides another method of solution that involves a simplified notation using a matrix. A matrix is defined as a rectangular array of numbers, enclosed by brackets. The numbers are referred to as the entries of the matrix. A matrix is further identified by naming its rows and columns. Some examples of matrices are Column 1 Column 2 Column 1 Column 2 Column 3 Column 1 Column 2 Row 1 [4 3] (a) (b) (c) Matrix Representation of a System of Linear Equations Consider the following two systems of two linear equations containing two variables and We observe that, except for the symbols used to represent the variables, these two sys- tems are identical. As a result, we can dispense altogether with the letters used to sym- bolize the variables, provided we have some means of keeping track of them. A matrix serves us well in this regard. When a matrix is used to represent a system of linear equations, it is called the aug- mented matrix of the system. For example, the system can be represented by the augmented matrix Column 1 Column 2 Column 3 x y right-hand side Here it is understood that column 1 contains the coefficients of the variable x, column 2 contains the coefficients of the variable y, and column 3 contains the numbers to the right of the equal sign. Each row of the matrix represents an equation of the system. Although not required, it has become customary to place a vertical bar in the matrix as a reminder of the equal sign.

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  Algebra and Trigonometry

  • Sheldon Axler(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  Now we turn to another way to express a system of linear equations using matrices. We will make the coefficients of each equation into one row of a matrix, but now we will not include the constant term. Instead, the constant terms will form their own matrix with one column. The variables will also be put into a matrix with one column. The next example shows how these ideas are used to rewrite a system of linear equations into a matrix equation. Instead of calling the variables x, y , and z, the variables here have been called x 1 , x 2 , and x 3 so that you will become comfortable with the subscript notation that is used when dealing with a large number of variables. example 14 Consider the following system of equations: x 1 + 2x 2 + 3x 3 = 9 5x 1 - x 2 + 6x 3 = -4 7x 1 - 2x 2 + 8x 3 = 6. (a) Rewrite the system of equations above as a matrix equation. (b) Use an inverse matrix to solve the matrix equation and find the values of x 1 , x 2 , and x 3 . solution (a) Form a matrix A whose rows consist of the coefficients on the left side of the equations above. Form another matrix X with one column consisting of the variables. Finally, form another matrix B with one column consisting of the constant terms from the right side of the equations above. In each of these matrices, keep everything in the proper order as listed above. In other words, we have A =    1 2 3 5 -1 6 7 -2 8    , X =    x 1 x 2 x 3    , and B =    9 -4 6    . The system of linear equations above can now be expressed as the single This matrix equation provides another rea- son why the definition of matrix multiplica- tion, which at first seem strange, turns out to be so useful. matrix equation AX = B. To verify that the matrix equation above is the same as the original system of equations, use the definition of matrix multiplication. For example, the entry in section 7.4 Matrix Algebra 437 row 1, column 1 of AX equals row 1 of A times column 1 of X (which has only one column).

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  Dynamical Systems and Geometric Mechanics

  An Introduction

  • Jared Maruskin(Author)
  • 2018(Publication Date)
  • De Gruyter

    (Publisher)

  1 Linear Systems Linear systems constitute an important hallmark of dynamical systems theory. Not only are they among the rare class of systems in which exact, analytic solutions can actually be obtained and computed, they have also become a cornerstone even in the analysis of nonlinear systems, including systems on manifolds. We will see later in this book that the flow near a fixed point can be locally understood by its associated lin-earization . The different solution modes of a linear system may be analyzed individu-ally using linearly independent solutions obtained from eigenvalues and eigenvectors or matrixwise using operators and operator exponentials. We begin our study of linear systems with a discussion of both approaches and some fundamental consequences. 1.1 Eigenvector Approach Once the early pioneers of linear dynamical systems realized they could represent these systems in matrix form, it was not long before they realized that, when expressed relative to an eigenbasis of the constant coefficient matrix, a spectacular decoupling results, yielding the solution almost trivially. We begin our study of linear systems with a review of this method, paying particularly close attention to the case of distinct eigenvalues. We will consider the case of repeated eigenvalues later in the chapter. Linear Systems and the Principle of Superposition Throughout this chapter, we will consider linear systems of first-order ordinary differ-ential equations of the form ̇ x = Ax , (1.1) x ( 0 ) = x 0 , (1.2) where x : ℝ → ℝ n is the unknown solution to the above initial value problem, x 0 ∈ ℝ n is the initial condition , and A ∈ ℝ n × n is a real-valued matrix called the coefficient ma-trix . The solution , or flow , of system (1.1) is a smooth function φ : ℝ × ℝ n → ℝ n that satisfies dφ dt = Aφ ( t ; x 0 ), for all t ∈ ℝ, (1.3) φ ( 0 ; x 0 ) = x 0 , for all x 0 ∈ ℝ n .

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  College Algebra

  • Sheldon Axler(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  Now we turn to another way to express a system of linear equations using matrices. We will make the coefficients of each equation into one row of a matrix, but now we will not include the constant term. Instead, the constant terms will form their own matrix with one column. The variables will also be put into a matrix with one column. The next example shows how these ideas are used to rewrite a system of linear equations into a matrix equation. Instead of calling the variables x, y , and z, the variables here have been called x 1 , x 2 , and x 3 so that you will become comfortable with the subscript notation that is used when dealing with a large number of variables. example 14 Consider the following system of equations: x 1 + 2x 2 + 3x 3 = 9 5x 1 - x 2 + 6x 3 = -4 7x 1 - 2x 2 + 8x 3 = 6. (a) Rewrite the system of equations above as a matrix equation. (b) Use an inverse matrix to solve the matrix equation and find the values of x 1 , x 2 , and x 3 . solution (a) Form a matrix A whose rows consist of the coefficients on the left side of the equations above. Form another matrix X with one column consisting of the variables. Finally, form another matrix B with one column consisting of the constant terms from the right side of the equations above. In each of these matrices, keep everything in the proper order as listed above. In other words, we have A =    1 2 3 5 -1 6 7 -2 8    , X =    x 1 x 2 x 3    , and B =    9 -4 6    . The system of linear equations above can now be expressed as the single This matrix equation provides another rea- son why the definition of matrix multiplica- tion, which at first seem strange, turns out to be so useful. matrix equation AX = B. To verify that the matrix equation above is the same as the original system of equations, use the definition of matrix multiplication. For example, the entry in section 7.4 Matrix Algebra 437 row 1, column 1 of AX equals row 1 of A times column 1 of X (which has only one column).

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  Linear Algebra

  Ideas and Applications

  • Richard C. Penney(Author)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  1.41 Suppose that A is the matrix of a dominance relationship. Explain why a ij a ji = 0. 1.42 ✓We say that two points A and B of a directed graph are two-step connected if there is a point C such that A → C → B. Thus, for example, in the route map in Figure 1.7, A and C are two-step connected, but D and C are not. Also A is two-step connected with itself. Give the two-step route route matrix for the route map in Figure 1.7. 1.43 Figure 1.10 shows the end-of-season results from an athletic conference with teams A–D. The arrows indicate which team beat which. (a) Find the matrix A for the graph in Figure 1.10. (b) Compute the win loss record of team C. 1.2 SYSTEMS An equation in variables x 1 , x 2 , … , x n is a linear equation if and only if it is express- ible in the form a 1 x 1 + a 2 x 2 + · · · + a n x n = b (1.7) 28 SYSTEMS OF LINEAR EQUATIONS where a i and b are all scalars. By a solution to equation (1.7), we mean a column vector [x 1 , x 2 , … , x n ] t of values for the variables that make the equation valid. Thus, X = [1, 2, −1] t is a solution of the equation 2x + 3y + z = 7 because 2(1) + 3(2) + (−1) = 7 More generally, a set of linear equations in a particular collection of variables is called a linear system of equations. Thus, the general system of linear equations in the variables x 1 , x 2 , … , x n may be written as follows: 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 + · · · + a mn x n = b m (1.8) A solution to the system is a column vector that is a solution to each equation in the system. The set of all column vectors that solve the system is the solution set for the system. In particular, x + 2y + z = 1 3x + y + 4z = 0 2x + 2y + 3z = 2 (1.9) is a linear system in the variables x, y, and z.

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  Signals and Systems Analysis In Biomedical Engineering

  • Robert B. Northrop(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  2 -1 2 Review of Linear Systems Theory 2.1 Linearity, Causality, and Stationarity In this chapter, we describe the important attributes of simple, linear , time-invariant (LTI) systems. The art of solving the ordinary differential equations (ODEs) that describe the input and output behavior of continuous, LTI systems is introduced. Matrix algebra and matrix operations are reviewed in Section 2.3, and the state variable (SV) formalism for describing the behavior of high-order, continuous dynamic systems is described in Section 2.3.4. Section 2.4 treats the mathematical tools used to characterize LTI sys-tems and the concepts of system impulse response , real convolution, transient response, and steady-state sinusoidal (SSS) frequency response, including Bode and Nyquist plots, are introduced. Section 2.5 treats discrete LTI (numerical) systems . Difference equa-tions replace ODEs in describing system behavior in the time domain and use of the z -transform is introduced as a means of solving discrete-state equations. Finally, in Section 2.5, we consider factors affecting the stability of LTI systems. Although the real world is fraught with thousands of examples of nonlinear systems, engineers and applied mathematicians have almost exclusively devoted their attention to developing mathematical tools to describe and analyze linear systems . One reason for this specialization appears to be that certain mechanical and biological systems, and electrical/electronic circuits, can be treated as linear systems, that is, they can be lin-earized. Therefore, the powerful mathematical tools of linear systems analysis can be applied. A system is said to be linear if it obeys all of the properties below (Northrop, 2000).

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