Linear Systems

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  An Introduction to Linear Algebra

  • Xuan LIU, Zhi ZHAO, Wei-Hui LIU, Xiao-Qing JIN(Authors)
  • 2022(Publication Date)
  • EDP Sciences

    (Publisher)

  Chapter 1 Linear Systems and Matrices “No beginner’s course in mathematics can do without linear algebra.” — Lars Gårding “Matrices act. They don’t just sit there.” — Gilbert Strang Solving Linear Systems (a system of linear equations) is the most important problem of linear algebra and possibly of applied mathematics as well. Usually, information in a linear system is often arranged into a rectangular array, called a “matrix”. The matrix is particularly important in developing computer programs to solve Linear Systems with huge sizes because computers are suitable to manage numerical data in arrays. Moreover, matrices are not only a simple tool for solving Linear Systems but also mathematical objects in their own right. In fact, matrix theory has a variety of applications in science, engineering, and mathematics. Therefore, we begin our study on Linear Systems and matrices in the first chapter. 1.1 Introduction to Linear Systems and Matrices Let denote the set of real numbers. We now introduce linear equations, Linear Systems, and matrices. 1.1.1 Linear equations and Linear Systems We consider a 1 x 1 + a 2 x 2 + · · · + a n x n = b, where a i ∈ (i = 1, 2, . . . , n) are coecients, x i (i = 1, 2, . . . , n) are variables (unknowns), n is a positive integer, and b ∈ is a constant. An equation of this form is called a linear equation, in which all variables occur to the first power. When b = 0, the linear equation is called a homogeneous linear equation. A 2 Chapter 1 Linear Systems and Matrices sequence of numbers s 1 , s 2 , . . . , s n is called a solution of the equation if x 1 = s 1 , x 2 = s 2 , . . . , x n = s n such that a 1 s 1 + a 2 s 2 + · · · + a n s n = b. The set of all solutions of the equation is called the solution set of the equation. In the book, we always use example(s) to make our points clear. Example We consider the following linear equations: (a) x + y = 1.

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  Elementary Linear Algebra

  • Howard Anton, Chris Rorres, Anton Kaul(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  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. 1.1 Introduction to Systems of Linear Equations Systems of linear equations and their solutions constitute one of the major topics that we will study in this course. In this first section we will introduce some basic terminology and discuss a method for solving such systems. Linear Equations Recall that in two dimensions a line in a rectangular xy-coordinate system can be repre- sented by an equation of the form ax + by = c (a, b not both 0) and in three dimensions a plane in a rectangular xyz-coordinate system can be represented by an equation of the form ax + by + cz = d (a, b, c not all 0) These are examples of “linear equations,” the first being a linear equation in the variables x and y and the second a linear equation in the variables x, y, and z. More generally, we define a linear equation in the n variables x 1 , x 2 , . . . , x n to be one that can be expressed in the form a 1 x 1 + a 2 x 2 + ⋅ ⋅ ⋅ + a n x n = b (1) where a 1 , a 2 , . . . , a n and b are constants, and the a’s are not all zero. In the special cases where n = 2 or n = 3, we will often use variables without subscripts and write linear equa- tions as a 1 x + a 2 y = b (2) a 1 x + a 2 y + a 3 z = b (3) In the special case where b = 0, Equation (1) has the form a 1 x 1 + a 2 x 2 + ⋅ ⋅ ⋅ + a n x n = 0 (4) which is called a homogeneous linear equation in the variables x 1 , x 2 , . . . , x n . EXAMPLE 1 | Linear Equations Observe that a linear equation does not involve any products or roots of variables. All vari- ables occur only to the first power and do not appear, for example, as arguments of trigono- metric, logarithmic, or exponential functions.

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  Elementary Linear Algebra with Supplemental Applications

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

    (Publisher)

  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. 2 Chapter 1 Systems of Linear Equations and Matrices 1.1 Introduction to Systems of Linear Equations Systems of linear equations and their solutions constitute one of the major topics that we will study in this course. In this first section we will introduce some basic terminology and discuss a method for solving such systems. Linear Equations Recall that in two dimensions a line in a rectangular xy -coordinate system can be repre- sented by an equation of the form ax + by = c (a, b not both 0) and in three dimensions a plane in a rectangular xyz-coordinate system can be represented by an equation of the form ax + by + cz = d (a, b, c not all 0) These are examples of “linear equations,” the first being a linear equation in the variables x and y and the second a linear equation in the variables x , y , and z. More generally, we define a linear equation in the n variables x 1 , x 2 , . . . , x n to be one that can be expressed in the form a 1 x 1 + a 2 x 2 + · · · + a n x n = b (1) where a 1 , a 2 , . . . , a n and b are constants, and the a’s are not all zero. In the special cases where n = 2 or n = 3, we will often use variables without subscripts and write linear equations as a 1 x + a 2 y = b (a 1 , a 2 not both 0) (2) a 1 x + a 2 y + a 3 z = b (a 1 , a 2 , a 3 not all 0) (3) In the special case where b = 0, Equation (1) has the form a 1 x 1 + a 2 x 2 + · · · + a n x n = 0 (4) which is called a homogeneous linear equation in the variables x 1 , x 2 , . . . , x n . c EXAM PLE 1 Linear Equations Observe that a linear equation does not involve any products or roots of variables. All variables occur only to the first power and do not appear, for example, as arguments of trigonometric, logarithmic, or exponential functions.

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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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  Elementary Differential Equations and Boundary Value Problems

  • William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  288 CHAPTER 7 Systems of First-Order Linear Equations Many physical problems involve a number of separate but interconnected components. For example, the current and voltage in an electrical network, each mass in a mechanical system, each element (or compound) in a chemical system, or each species in a biological system have this character. In these and similar cases, the corresponding mathematical problem consists of a system of two or more differential equations, which can always be written as first-order differential equations. In this chapter we focus on systems of first-order linear differential equations and, in particular, differential equations having constant coefficients, utilizing some of the elementary aspects of linear algebra to unify the presentation. In many respects this chapter follows the same lines as the treatment of second-order linear differential equations in Chapter 3. 7.1 Introduction Systems of simultaneous ordinary differential equations arise naturally in problems involving several dependent variables, each of which is a function of the same single independent variable. We will denote the independent variable by  and will let  1 ,  2 ,  3 , ... represent dependent variables that are functions of . Differentiation 1 with respect to  will be denoted by, for example,  1  or  ′ 1 . Let us begin by considering the spring – mass system in Figure 7.1.1. The two masses move on a frictionless surface under the influence of external forces  1 () and  2 (), and they are also constrained by the three springs whose constants are  1 ,  2 , and  3 , respectively. We regard motion and displacement to the right as being positive. k 1 F 1 (t) F 2 (t) k 2 m 1 m 2 x 1 x 2 k 3 FIGURE 7.1.1 A two-mass, three-spring system.

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  Differential Equations: From Calculus to Dynamical Systems

  Second Edition

  • Virginia W. Noonburg(Author)
  • 2019(Publication Date)
  • American Mathematical Society

    (Publisher)

  (b) What is the dimension of the system? (c) Is the system linear? 4.2 Matrix Algebra To proceed any further in solving Linear Systems, it is necessary to be able to write them in matrix form. If you have taken a course in linear algebra, this section will be a review. Otherwise, you should read this section very carefully. It contains all of the material needed for the rest of this chapter and also for Chapter 5. Definition 4.2. A matrix A is a rectangular array containing elements arranged in ? rows and ? columns. When working with differential equations it can be assumed that these elements (also called scalars ) are either numbers or functions. The notation we will use for a matrix is ? = ⎛ ⎜ ⎜ ⎝ ? 11 ? 12 … ? 1? ? 21 ? 22 … ? 2? ⋱ ? ?1 ? ?2 … ? ?? ⎞ ⎟ ⎟ ⎠ . A matrix with ? rows and ? columns, for some positive integers ? and ? , is said to be of size ? by ? (written as ? × ? ). In the above notation, ? 𝑖𝑗 represents the element in the 𝑖 th row and 𝑗 th column of ? . It will sometimes be convenient to use the notation ? = (? 𝑖𝑗 ) which shows the form of the general element in ? . An ? × ? matrix with ? = 1 has a single column and is called a column vector ; similarly, an ? × ? matrix with ? = 1 is called a row vector . An ? × ? matrix with ? = ? is called a square matrix of size ? . The zero matrix , of any size, denoted by 0 , is a matrix with the elements in every row and every column equal to 0 . Definition 4.3. Two matrices ? = (? 𝑖𝑗 ) and ? = (? 𝑖𝑗 ) are said to be equal if they are of the same size ? × ? , and ? 𝑖𝑗 = ? 𝑖𝑗 for 1 ≤ 𝑖 ≤ ?, 1 ≤ 𝑗 ≤ ? . The following three basic algebraic operations are defined for matrices: addition, scalar multiplication, and matrix multiplication. Addition of Matrices: If ? = (? 𝑖𝑗 ) and ? = (? 𝑖𝑗 ) are both the same size, say ? × ? , then their sum ? = ? + ? is defined, and ? = (? 𝑖𝑗 ) with ? 𝑖𝑗 = ? 𝑖𝑗 + ? 𝑖𝑗 for 1 ≤ 𝑖 ≤ ?, 1 ≤ 𝑗 ≤ ? . Example. ( 4 −6 3 1 ) + ( 2 2 0 −3 ) = ( 4 + 2 −6 + 2 3 + 0 1 − 3 ) = ( 6 −4 3 −2 ) .

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  Boyce's Elementary Differential Equations and Boundary Value Problems

  • William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  C H A P T E R 7 Systems of First-Order Linear Equations Many physical problems involve a number of separate but interconnected components. For example, the current and voltage in an electrical network, each mass in a mechanical system, each element (or compound) in a chemical system, or each species in a biological system have this character. In these and similar cases, the corresponding mathematical problem consists of a system of two or more differential equations, which can always be written as first-order differential equations. In this chapter we focus on systems of first-order linear differential equations and, in particular, differential equations having constant coefficients, utilizing some of the elementary aspects of linear algebra to unify the presentation. In many respects this chapter follows the same lines as the treatment of second-order linear differential equations in Chapter 3. 7.1 Introduction Systems of simultaneous ordinary differential equations arise naturally in problems involving several dependent variables, each of which is a function of the same single independent variable. We will denote the independent variable by t and will let x 1 , x 2 , x 3 , . . . represent dependent variables that are functions of t. Differentiation 1 with respect to t will be denoted by, for example, dx 1 dt or x ′ 1 . Let us begin by considering the spring – mass system in Figure 7.1.1. The two masses move on a frictionless surface under the influence of external forces F 1 (t) and F 2 (t), and they are also constrained by the three springs whose constants are k 1 , k 2 , and k 3 , respectively. We regard motion and displacement to the right as being positive. k 1 F 1 (t) F 2 (t) k 2 m 1 m 2 x 1 x 2 k 3 FIGURE 7.1.1 A two-mass, three-spring system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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  Lumped and Distributed Passive Networks

  A Generalized and Advanced Viewpoint

  • M. Ronald Wohlers, Henry G. Booker, Nicholas Declaris(Authors)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  On the other hand, we will say that a sequence of distributions converges to zero in the topology of, say, 2' if lim n _ + 00 §f n (pdt = 0 for all φ e C 0 °°. With this brief introduction we will now proceed with our discussion of Linear Systems. 1.1. T H E A X I O M S O F L I N E A R S Y S T E M T H E O R Y Let us assume that in a physical phenomenon under investigation we can identify a certain number of variables—to be specific, we assume them to be functions of time—as independent parameters and another group of variables as dependent parameters. It is to be understood that the independent param-eters, which we will designate as inputs, may be varied at our discretion, but that the dependent parameters, which we designate as outputs, are deter-mined by the physical phenomenon once the inputs are specified. If we use the notation Γ/ι(0ΐ f = .urn to specify a collection of η parameters, then we model the physical phenom-enon by an operator Τ which maps inputs f into outputs g and we may state the following definition of a system : 4 I. Linear Systems Definition 1. A system is a unique (single-valued) mapping between a collec-tion of inputs and the corresponding collection of outputs. Thus it is assumed that there exists some operator Τ such that for all inputs f belonging to the set Z> T (the so-called domain of the operator) there exists a single g contained in R T (the range of the operator) where g = Γ [ ί ] , We will have occasion to consider operator whose domain D T consists of all f such that each / ) ( / ) e C 0 °°, and the corresponding outputs g (there may in general be more or less outputs than inputs) are such that each g { e Q)'. However, depending upon our immediate purpose these domains and ranges may vary. In any event the physical content of Definition 1 is clear: we will always assume that we have identified all the possible independent param-eters of the system (the inputs) so that our outputs are uniquely determined.

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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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