1. The document discusses bases and dimensions for vector spaces. A basis for a subspace enables visualizing the subspace as a k-dimensional hyperplane through the origin in Rn.
2. Examples are provided of determining if sets of vectors form a basis by checking if they are linearly independent. The dimension of solution spaces of homogeneous systems is also determined based on the rank of the systems.
3. Specific examples involve finding bases for solution spaces of systems of linear equations by reducing the coefficient matrices to echelon form and writing the general solutions in terms of the basis vectors.
The document introduces concepts related to vector spaces including vectors, linear independence, and subspaces. It provides examples in R3 involving determining if sets of vectors are linearly dependent or independent, finding representations of vectors as linear combinations of other vectors, and solving homogeneous and nonhomogeneous systems of equations involving vector coefficients. Key concepts are illustrated through a series of problems involving vectors in R3.
The document discusses subspaces of vector spaces. It provides examples of subsets of Rn and determines whether each subset is a subspace by checking if it is closed under vector addition and scalar multiplication. Some subsets are shown to be subspaces, while others are not subspaces because they fail to satisfy one of the closure properties. The document also uses row reduction to determine the solution spaces of homogeneous linear systems, which must always be subspaces.
This document summarizes methods for solving ordinary differential equations (ODEs). It discusses:
1) Types of ODEs including order, degree, linear/nonlinear.
2) Four methods for solving 1st order ODEs: separable variables, homogeneous equations, exact equations, and integrating factors.
3) Solutions to higher order linear ODEs using complementary functions and particular integrals.
4) Finding complementary functions and particular integrals for ODEs with constant coefficients.
The document discusses solving systems of nonlinear equations in two variables. It provides examples of nonlinear systems that contain equations that are not in the form Ax + By = C, such as x^2 = 2y + 10. Methods for solving nonlinear systems include substitution and addition. The substitution method involves solving one equation for one variable and substituting into the other equation. The addition method involves rewriting the equations and adding them to eliminate variables. Examples demonstrate both methods and finding the solution set that satisfies both equations.
The document discusses solving systems of linear equations with two or three variables. There are three possible cases for the solution: 1) a unique solution, 2) infinitely many solutions (a dependent system), or 3) no solution. The document demonstrates solving systems using substitution and elimination methods, and provides examples of each case. Graphically, case 1 corresponds to intersecting lines or planes, case 2 to coinciding lines or intersecting planes, and case 3 to parallel lines or non-intersecting planes.
1. The document discusses concepts related to expectation and variance of random variables including expected value, variance, moments, and examples of calculating these for different probability distributions like uniform, normal, exponential, and Rayleigh.
2. Problems at the end provide examples of computing expected value, variance, and cumulative distribution function for random variables following different distributions. Solutions show the calculations and formulas used.
3. Key formulas introduced include definitions of expected value and variance, the relationship between them, and formulas for calculating moments, expected value, and variance for specific distributions. Examples demonstrate applying the concepts and formulas to problems.
This document describes Picard's method for solving simultaneous first order differential equations numerically. It presents the iterative formula used in Picard's method and applies it to solve four example problems of simultaneous differential equations. The problems are solved over multiple iterations to obtain successive approximations of the solutions at increasing values of x, with the approximations being carried to three or four decimal places.
This document discusses methods for solving numerical equations, including the bisection method, Newton-Raphson method, and method of false position. It provides definitions and step-by-step computations for each method. For the bisection method, it gives an example of finding the positive root of x3 - x = 1. For Newton-Raphson, it gives examples of finding the root of 2x3 - 3x - 6 = 0 and x3 = 6x - 4. The document serves to introduce numerical methods for solving equations.
The document introduces concepts related to vector spaces including vectors, linear independence, and subspaces. It provides examples in R3 involving determining if sets of vectors are linearly dependent or independent, finding representations of vectors as linear combinations of other vectors, and solving homogeneous and nonhomogeneous systems of equations involving vector coefficients. Key concepts are illustrated through a series of problems involving vectors in R3.
The document discusses subspaces of vector spaces. It provides examples of subsets of Rn and determines whether each subset is a subspace by checking if it is closed under vector addition and scalar multiplication. Some subsets are shown to be subspaces, while others are not subspaces because they fail to satisfy one of the closure properties. The document also uses row reduction to determine the solution spaces of homogeneous linear systems, which must always be subspaces.
This document summarizes methods for solving ordinary differential equations (ODEs). It discusses:
1) Types of ODEs including order, degree, linear/nonlinear.
2) Four methods for solving 1st order ODEs: separable variables, homogeneous equations, exact equations, and integrating factors.
3) Solutions to higher order linear ODEs using complementary functions and particular integrals.
4) Finding complementary functions and particular integrals for ODEs with constant coefficients.
The document discusses solving systems of nonlinear equations in two variables. It provides examples of nonlinear systems that contain equations that are not in the form Ax + By = C, such as x^2 = 2y + 10. Methods for solving nonlinear systems include substitution and addition. The substitution method involves solving one equation for one variable and substituting into the other equation. The addition method involves rewriting the equations and adding them to eliminate variables. Examples demonstrate both methods and finding the solution set that satisfies both equations.
The document discusses solving systems of linear equations with two or three variables. There are three possible cases for the solution: 1) a unique solution, 2) infinitely many solutions (a dependent system), or 3) no solution. The document demonstrates solving systems using substitution and elimination methods, and provides examples of each case. Graphically, case 1 corresponds to intersecting lines or planes, case 2 to coinciding lines or intersecting planes, and case 3 to parallel lines or non-intersecting planes.
1. The document discusses concepts related to expectation and variance of random variables including expected value, variance, moments, and examples of calculating these for different probability distributions like uniform, normal, exponential, and Rayleigh.
2. Problems at the end provide examples of computing expected value, variance, and cumulative distribution function for random variables following different distributions. Solutions show the calculations and formulas used.
3. Key formulas introduced include definitions of expected value and variance, the relationship between them, and formulas for calculating moments, expected value, and variance for specific distributions. Examples demonstrate applying the concepts and formulas to problems.
This document describes Picard's method for solving simultaneous first order differential equations numerically. It presents the iterative formula used in Picard's method and applies it to solve four example problems of simultaneous differential equations. The problems are solved over multiple iterations to obtain successive approximations of the solutions at increasing values of x, with the approximations being carried to three or four decimal places.
This document discusses methods for solving numerical equations, including the bisection method, Newton-Raphson method, and method of false position. It provides definitions and step-by-step computations for each method. For the bisection method, it gives an example of finding the positive root of x3 - x = 1. For Newton-Raphson, it gives examples of finding the root of 2x3 - 3x - 6 = 0 and x3 = 6x - 4. The document serves to introduce numerical methods for solving equations.
The document discusses methods for finding the general solution to linear differential equations of second order with constant coefficients. It presents four types of complementary functions depending on whether the roots of the auxiliary equation are real and distinct, real and equal, complex, or surd roots. It also describes four types of particular integrals depending on whether the given function is an exponential, sine, cosine, or contains an exponential term. The document provides examples of solving differential equations of each type and includes multiple choice questions to test understanding of the concepts and methods presented.
This document contains a problem set in quantitative methods with 17 questions covering topics in linear algebra including: solving systems of linear equations using Gauss-Jordan elimination; determining the inverse of matrices; finding the null space and row/column spaces of matrices; determining if sets of vectors are linearly independent/dependent or span vector spaces; and identifying if sets of vectors form bases. The problem set is assigned by Manimay Sengupta for the Monsoon Semester 2012 at South Asian University.
This document discusses partial differential equations (PDEs). It provides examples of how PDEs can be formed by eliminating constants or functions from relations involving multiple variables. It also discusses different types of first-order PDEs and methods for solving them. Several example problems are presented with step-by-step solutions showing how to derive and solve PDEs that model different physical situations. Standard forms and techniques for reducing PDEs to simpler forms are also outlined.
The document discusses systems of non-linear equations and their properties. It covers different types of non-linear equations like absolute value, quadratic, and their graph shapes. It provides tips for graphing systems of non-linear equations on a calculator, such as making sure the view captures all intersections and using trace to locate the intersections when there are multiple. Examples of specific non-linear equation systems are also given.
The document discusses partial differential equations and their solutions. It can be summarized as:
1) A partial differential equation involves a function of two or more variables and some of its partial derivatives, with one dependent variable and one or more independent variables. Standard notation is presented for partial derivatives.
2) Partial differential equations can be formed by eliminating arbitrary constants or arbitrary functions from an equation relating the dependent and independent variables. Examples of each method are provided.
3) Solutions to partial differential equations can be complete, containing the maximum number of arbitrary constants allowed, particular where the constants are given specific values, or singular where no constants are present. Methods for determining the general solution are described.
The document discusses various types of differential equations including ordinary differential equations (ODEs) and partial differential equations (PDEs). It defines key terms like order, degree, and describes several methods for solving common types of differential equations, such as separating variables, exact differentials, linear equations, Bernoulli's equation, and Clairaut's equation. It also includes sample problems and solutions for each method and concludes with multiple choice questions.
Elimination of Systems of Linear EquationSonarin Cruz
The document discusses solving systems of linear equations by elimination. It involves eliminating one variable at a time through addition or subtraction of equations. This leaves an equation with one variable that can be solved for its value, which is then substituted back into the original equations to solve for the other variable. Two examples are provided showing the full process of setting up equations, eliminating variables, solving for values, and checking solutions.
This document provides an overview of engineering mathematics II with a focus on first order ordinary differential equations (ODEs). It explains what first order ODEs are, how to solve separable and reducible first order ODEs, and provides examples of applying first order ODEs to model real-world scenarios like population growth, decay, and radioactive decay. The objectives are to explain first order ODEs, separable equations, and apply the concepts to real life applications.
This document discusses second order differential equations. It defines a second order differential equation as a relationship involving the second derivative of an dependent variable y with respect to an independent variable x. It explains that the characteristic or auxiliary equation is obtained by substituting trial solutions into the original differential equation. The general solution to a second order differential equation is the combination of the complementary function (solution when right hand side is zero) and particular integral (makes right hand side zero). Non-homogeneous second order differential equations can be solved by finding the complementary function and particular integral separately and combining them.
The document discusses solutions to Euler equations, which are differential equations of the form 0][ 2=+′+′′= yyxyxyL βα. It provides the general solutions based on whether the roots r1 and r2 of the characteristic equation are real/distinct, equal, or complex. For initial value problems, the constants in the general solution are determined using the given initial conditions. Near the singular point x=0, the qualitative behavior of the solutions depends on the nature of the roots r1 and r2.
Introduction to Numerical Methods for Differential Equationsmatthew_henderson
The document introduces the Euler method for numerically approximating solutions to initial value problems (IVPs). It defines IVPs and shows an example. The Euler method uses the derivative approximation y(x+h) ≈ y(x) + hf(x,y) to march forward in small steps h to construct a table of approximate y-values. For the example IVP, the Euler method produces values that begin to resemble the exact solution. While not exact, the errors are small. The method is derived from the definition of the derivative and works because it approximates the tangent line at each step.
Constant-Coefficient Linear Differential Equationsashikul akash
This document discusses constant-coefficient linear differential equations. It introduces homogeneous and non-homogeneous equations, and describes how to find the general solution by analyzing the auxiliary polynomial. The roots of the auxiliary polynomial determine the solutions. If there are distinct linear factors, the solutions from each factor combine to form the general solution. Multiple or complex roots require additional solutions involving powers of x or trigonometric functions.
Solving second order ordinary differential equations (boundary value problems) using the Least Squares Technique. Contains one numerical examples from Shah, Eldho, Desai
1) The document discusses second order linear differential equations with constant or variable coefficients.
2) It provides the general form of second order linear differential equations and various methods to solve them including reduction of order, finding independent solutions, and using the characteristic equation.
3) The methods are demonstrated on examples of homogeneous differential equations with constant coefficients, including cases where the roots of the characteristic equation are real, repeated, or complex.
1. The branch and bound algorithm divides the problem into sub-problems by fixing variables to 0 or 1.
2. It bounds sub-problems by relaxing constraints and solving the linear programming relaxation to obtain bounds.
3. Sub-problems are discarded if their bound is less than the best known solution or they are infeasible.
4. The algorithm proceeds by branching on the next variable until no sub-problems remain, leaving the optimal solution.
The document provides examples of solving linear and nonlinear inequalities algebraically and graphing their solution sets. For linear inequalities, the solutions are intervals of real numbers defined by the solutions to the corresponding equalities. For nonlinear inequalities, the solutions are unions of intervals where the factors of the corresponding equalities have the same sign. The document also demonstrates solving compound inequalities and inequalities involving rational expressions.
The document discusses methods for solving first order ordinary differential equations (ODEs). It covers:
1) Finding the integrating factor for exact differential equations.
2) Solving homogeneous first order linear ODEs by making a substitution to reduce it to a separable equation.
3) Solving inhomogeneous first order linear ODEs using an integrating factor.
Examples are provided to demonstrate each method step-by-step.
The document defines and discusses differential equations and their solutions. It begins by classifying differential equations as ordinary or partial based on whether they involve one or more independent variables. Ordinary differential equations are then classified as linear or nonlinear based on their form. The order and degree of a differential equation are also defined.
Solutions to differential equations can be either explicit functions that directly satisfy the equation or implicit relations that define functions satisfying the equation. Picard's theorem guarantees a unique solution through each point for first-order equations. The general solution to a first-order equation is a one-parameter family of curves, with a particular solution corresponding to a specific value of the parameter. An initial value problem specifies both a differential equation and
The document provides step-by-step instructions for factoring polynomials, finding inverse functions, simplifying rational expressions, and graphing rational functions. It includes examples of each type of problem worked out in detail from beginning to end. The examples range from relatively simple to more complex in order to demonstrate a variety of situations that may occur.
This document provides a summary of key concepts in regular perturbation theory. It begins with an introduction and definitions related to regular and singular perturbations. Chapter 1 defines asymptotic sequences, asymptotic expansions, and order symbols like big-oh and little-oh notation. Chapter 2 discusses the fundamental ideas of perturbation like regularly and singularly perturbed problems. Chapter 3 solves sample regular perturbation problems like algebraic equations and differential equations to obtain asymptotic expansions of the solutions in terms of the perturbation parameter.
This document discusses regular perturbation theory and its application to solving algebraic equations. It begins by defining regular and singular perturbations. For regular perturbations, the order of the perturbed and unperturbed problems are the same when the perturbation parameter is set to zero. The document then shows how regular perturbation theory can be used to solve algebraic equations. Specifically, it demonstrates obtaining the series solutions for the quadratic equation x^2-1=ε and the cubic equation x^3-x+ε=0 by assuming power series solutions in ε and solving the equations order-by-order in ε. This yields convergent power series expansions for the roots as functions of the small perturbation parameter ε.
The document discusses methods for finding the general solution to linear differential equations of second order with constant coefficients. It presents four types of complementary functions depending on whether the roots of the auxiliary equation are real and distinct, real and equal, complex, or surd roots. It also describes four types of particular integrals depending on whether the given function is an exponential, sine, cosine, or contains an exponential term. The document provides examples of solving differential equations of each type and includes multiple choice questions to test understanding of the concepts and methods presented.
This document contains a problem set in quantitative methods with 17 questions covering topics in linear algebra including: solving systems of linear equations using Gauss-Jordan elimination; determining the inverse of matrices; finding the null space and row/column spaces of matrices; determining if sets of vectors are linearly independent/dependent or span vector spaces; and identifying if sets of vectors form bases. The problem set is assigned by Manimay Sengupta for the Monsoon Semester 2012 at South Asian University.
This document discusses partial differential equations (PDEs). It provides examples of how PDEs can be formed by eliminating constants or functions from relations involving multiple variables. It also discusses different types of first-order PDEs and methods for solving them. Several example problems are presented with step-by-step solutions showing how to derive and solve PDEs that model different physical situations. Standard forms and techniques for reducing PDEs to simpler forms are also outlined.
The document discusses systems of non-linear equations and their properties. It covers different types of non-linear equations like absolute value, quadratic, and their graph shapes. It provides tips for graphing systems of non-linear equations on a calculator, such as making sure the view captures all intersections and using trace to locate the intersections when there are multiple. Examples of specific non-linear equation systems are also given.
The document discusses partial differential equations and their solutions. It can be summarized as:
1) A partial differential equation involves a function of two or more variables and some of its partial derivatives, with one dependent variable and one or more independent variables. Standard notation is presented for partial derivatives.
2) Partial differential equations can be formed by eliminating arbitrary constants or arbitrary functions from an equation relating the dependent and independent variables. Examples of each method are provided.
3) Solutions to partial differential equations can be complete, containing the maximum number of arbitrary constants allowed, particular where the constants are given specific values, or singular where no constants are present. Methods for determining the general solution are described.
The document discusses various types of differential equations including ordinary differential equations (ODEs) and partial differential equations (PDEs). It defines key terms like order, degree, and describes several methods for solving common types of differential equations, such as separating variables, exact differentials, linear equations, Bernoulli's equation, and Clairaut's equation. It also includes sample problems and solutions for each method and concludes with multiple choice questions.
Elimination of Systems of Linear EquationSonarin Cruz
The document discusses solving systems of linear equations by elimination. It involves eliminating one variable at a time through addition or subtraction of equations. This leaves an equation with one variable that can be solved for its value, which is then substituted back into the original equations to solve for the other variable. Two examples are provided showing the full process of setting up equations, eliminating variables, solving for values, and checking solutions.
This document provides an overview of engineering mathematics II with a focus on first order ordinary differential equations (ODEs). It explains what first order ODEs are, how to solve separable and reducible first order ODEs, and provides examples of applying first order ODEs to model real-world scenarios like population growth, decay, and radioactive decay. The objectives are to explain first order ODEs, separable equations, and apply the concepts to real life applications.
This document discusses second order differential equations. It defines a second order differential equation as a relationship involving the second derivative of an dependent variable y with respect to an independent variable x. It explains that the characteristic or auxiliary equation is obtained by substituting trial solutions into the original differential equation. The general solution to a second order differential equation is the combination of the complementary function (solution when right hand side is zero) and particular integral (makes right hand side zero). Non-homogeneous second order differential equations can be solved by finding the complementary function and particular integral separately and combining them.
The document discusses solutions to Euler equations, which are differential equations of the form 0][ 2=+′+′′= yyxyxyL βα. It provides the general solutions based on whether the roots r1 and r2 of the characteristic equation are real/distinct, equal, or complex. For initial value problems, the constants in the general solution are determined using the given initial conditions. Near the singular point x=0, the qualitative behavior of the solutions depends on the nature of the roots r1 and r2.
Introduction to Numerical Methods for Differential Equationsmatthew_henderson
The document introduces the Euler method for numerically approximating solutions to initial value problems (IVPs). It defines IVPs and shows an example. The Euler method uses the derivative approximation y(x+h) ≈ y(x) + hf(x,y) to march forward in small steps h to construct a table of approximate y-values. For the example IVP, the Euler method produces values that begin to resemble the exact solution. While not exact, the errors are small. The method is derived from the definition of the derivative and works because it approximates the tangent line at each step.
Constant-Coefficient Linear Differential Equationsashikul akash
This document discusses constant-coefficient linear differential equations. It introduces homogeneous and non-homogeneous equations, and describes how to find the general solution by analyzing the auxiliary polynomial. The roots of the auxiliary polynomial determine the solutions. If there are distinct linear factors, the solutions from each factor combine to form the general solution. Multiple or complex roots require additional solutions involving powers of x or trigonometric functions.
Solving second order ordinary differential equations (boundary value problems) using the Least Squares Technique. Contains one numerical examples from Shah, Eldho, Desai
1) The document discusses second order linear differential equations with constant or variable coefficients.
2) It provides the general form of second order linear differential equations and various methods to solve them including reduction of order, finding independent solutions, and using the characteristic equation.
3) The methods are demonstrated on examples of homogeneous differential equations with constant coefficients, including cases where the roots of the characteristic equation are real, repeated, or complex.
1. The branch and bound algorithm divides the problem into sub-problems by fixing variables to 0 or 1.
2. It bounds sub-problems by relaxing constraints and solving the linear programming relaxation to obtain bounds.
3. Sub-problems are discarded if their bound is less than the best known solution or they are infeasible.
4. The algorithm proceeds by branching on the next variable until no sub-problems remain, leaving the optimal solution.
The document provides examples of solving linear and nonlinear inequalities algebraically and graphing their solution sets. For linear inequalities, the solutions are intervals of real numbers defined by the solutions to the corresponding equalities. For nonlinear inequalities, the solutions are unions of intervals where the factors of the corresponding equalities have the same sign. The document also demonstrates solving compound inequalities and inequalities involving rational expressions.
The document discusses methods for solving first order ordinary differential equations (ODEs). It covers:
1) Finding the integrating factor for exact differential equations.
2) Solving homogeneous first order linear ODEs by making a substitution to reduce it to a separable equation.
3) Solving inhomogeneous first order linear ODEs using an integrating factor.
Examples are provided to demonstrate each method step-by-step.
The document defines and discusses differential equations and their solutions. It begins by classifying differential equations as ordinary or partial based on whether they involve one or more independent variables. Ordinary differential equations are then classified as linear or nonlinear based on their form. The order and degree of a differential equation are also defined.
Solutions to differential equations can be either explicit functions that directly satisfy the equation or implicit relations that define functions satisfying the equation. Picard's theorem guarantees a unique solution through each point for first-order equations. The general solution to a first-order equation is a one-parameter family of curves, with a particular solution corresponding to a specific value of the parameter. An initial value problem specifies both a differential equation and
The document provides step-by-step instructions for factoring polynomials, finding inverse functions, simplifying rational expressions, and graphing rational functions. It includes examples of each type of problem worked out in detail from beginning to end. The examples range from relatively simple to more complex in order to demonstrate a variety of situations that may occur.
This document provides a summary of key concepts in regular perturbation theory. It begins with an introduction and definitions related to regular and singular perturbations. Chapter 1 defines asymptotic sequences, asymptotic expansions, and order symbols like big-oh and little-oh notation. Chapter 2 discusses the fundamental ideas of perturbation like regularly and singularly perturbed problems. Chapter 3 solves sample regular perturbation problems like algebraic equations and differential equations to obtain asymptotic expansions of the solutions in terms of the perturbation parameter.
This document discusses regular perturbation theory and its application to solving algebraic equations. It begins by defining regular and singular perturbations. For regular perturbations, the order of the perturbed and unperturbed problems are the same when the perturbation parameter is set to zero. The document then shows how regular perturbation theory can be used to solve algebraic equations. Specifically, it demonstrates obtaining the series solutions for the quadratic equation x^2-1=ε and the cubic equation x^3-x+ε=0 by assuming power series solutions in ε and solving the equations order-by-order in ε. This yields convergent power series expansions for the roots as functions of the small perturbation parameter ε.
This document discusses regular perturbation theory through three chapters:
1. It defines key concepts like asymptotic sequences, asymptotic expansions, and order symbols used in perturbation theory.
2. It explains the fundamental ideas of regular and singular perturbations. Regular perturbations do not change the order of the problem when the perturbation parameter is set to zero, while singular perturbations do.
3. It provides examples of applying regular perturbation theory to solve algebraic equations. The technique involves expanding the solutions in powers of the perturbation parameter and solving the equations order-by-order. This allows approximating solutions even when exact solutions are not available.
The document discusses repeated eigenvalues in systems of linear differential equations. If some eigenvalues are repeated, there may not be n linearly independent solutions of the form x = ξert. Additional solutions must be sought that are products of polynomials and exponential functions. For a double eigenvalue r, the first solution is x(1) = ξert, where ξ satisfies (A-rI)ξ = 0. The second solution has the form x(2) = ξtert + ηert, where η satisfies (A-rI)η = ξ and η is called a generalized eigenvector.
This document discusses methods for solving algebraic and transcendental equations. It begins by defining key terms like roots, simple roots, and multiple roots. It then distinguishes between direct and iterative methods. Direct methods provide exact solutions, while iterative methods use successive approximations that converge to the exact root. The document focuses on iterative methods and describes how to obtain initial approximations, including using Descartes' rule of signs and the intermediate value theorem. It also discusses criteria for terminating iterations. One iterative method described in detail is the method of false position, which approximates the curve defined by the equation as a straight line between two points.
This document discusses methods for solving algebraic and transcendental equations. It begins by defining key terms like roots, simple roots, and multiple roots. It then distinguishes between direct and iterative methods. Direct methods provide exact solutions, while iterative methods use successive approximations that converge to the exact root. The document focuses on iterative methods and describes how to obtain initial approximations, including using Descartes' rule of signs and the intermediate value theorem. It also discusses criteria for terminating iterations. One iterative method described in detail is the method of false position, which approximates the curve defined by the equation as a straight line between two points.
This document provides solutions to problems regarding disturbance rejection and integral control. It summarizes the disturbance models for different types of disturbances, including piecewise exponential, piecewise constant, and piecewise harmonic oscillations. It then designs state feedback and feedforward gains to reject disturbances. Finally, it derives the transfer function from disturbance to output and adds integral feedback to reject constant disturbances.
The document discusses linear combinations and linear independence of vectors and functions. It defines a linear combination of vectors as a vector that can be expressed as a sum of scalar multiples of other vectors. A set of vectors is linearly dependent if one vector can be written as a linear combination of the others. A set is linearly independent if the only solution to the equation involving scalar multiples of the vectors is when all scalars are zero. It also discusses the Wronskian and its use in determining linear independence of functions. Examples are provided to illustrate these concepts.
Advanced Engineering Mathematics Solutions Manual.pdfWhitney Anderson
This document contains 27 multi-part exercises involving differential equations. The exercises cover topics such as determining whether differential equations are linear or nonlinear, solving differential equations, and classifying differential equations by order.
The document discusses Fourier series and their applications. It begins by introducing how Fourier originally developed the technique to study heat transfer and how it can represent periodic functions as an infinite series of sine and cosine terms. It then provides the definition and examples of Fourier series representations. The key points are that Fourier series decompose a function into sinusoidal basis functions with coefficients determined by integrating the function against each basis function. The series may converge to the original function under certain conditions.
The document discusses numerical methods for approximating integrals and solving non-linear equations. It introduces the trapezium rule for approximating integrals and provides examples of using the rule. It then discusses iterative methods like the iteration method and Newton-Raphson method for finding approximate roots of non-linear equations, providing examples of applying each method. The objectives are to enable students to use the trapezium rule and understand solving non-linear equations using iterative methods.
This document discusses homogeneous linear systems with constant coefficients. It begins by defining such a system as x' = Ax, where A is an n×n matrix of real constants. It then explains that the equilibrium solutions are found by solving Ax = 0, and stability is determined by the eigenvalues of A. Examples are provided to illustrate finding the direction field, eigenvalues/eigenvectors, general solution, and phase plane plots for specific 2D systems. Time plots of the solutions are also shown.
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This lecture notes were written as part of the course "Pattern Recognition and Machine Learning" taught by Prof. Dinesh Garg at IIT Gandhinagar. This lecture notes deals with Linear Regression.
The document discusses series solutions to second order linear differential equations near ordinary points. It provides an example of finding the series solution to the differential equation y'' + y = 0 near x0 = 0. The solution is found to be a cosine series which represents the cosine function, a fundamental solution. A second example finds the series solution to Airy's equation near x0 = 0, obtaining fundamental solutions related to Airy functions.
1. The document presents a new approach to proving comparison theorems for stochastic differential equations (SDEs) using differentiation of solutions with respect to initial data.
2. It proves that if the drift term of one SDE is always greater than or equal to the other, and their initial values satisfy the same relation, then the solutions will also satisfy this relation for all time.
3. Two methods are provided: the first uses explicit solutions, the second avoids this by showing the difference process cannot reach zero in finite time based on its behavior.
Interpolation techniques - Background and implementationQuasar Chunawala
This document discusses interpolation techniques, specifically Lagrange interpolation. It begins by introducing the problem of interpolation - given values of an unknown function f(x) at discrete points, finding a simple function that approximates f(x).
It then discusses using Taylor series polynomials for interpolation when the function value and its derivatives are known at a point. The error in interpolation approximations is also examined.
The main part discusses Lagrange interpolation - given data points (xi, f(xi)), there exists a unique interpolating polynomial Pn(x) of degree N that passes through all the points. This is proved using the non-zero Vandermonde determinant. Lagrange's interpolating polynomial is then introduced as a solution.
This document discusses various topics related to integration including:
1. The anti-derivative and how it is the reverse of differentiation.
2. Indefinite integrals which do not have limits and require an arbitrary constant.
3. Definite integrals which do have limits and are used to find the area under a curve between two points.
4. Applications of integration such as using definite integrals to find the area under a curve or revolving an area about an axis to find the volume of a solid.
The document outlines the learning objectives and test questions for a chapter on controlling in organizations. It covers explaining the foundations of control, identifying the phases of the corrective control model, describing primary control methods, and explaining corporate governance issues. The questions are categorized by their learning objective, type (true/false, multiple choice, essay), and difficulty level (easy, moderate, difficult).
This document provides an overview of organizational culture and cultural diversity. It includes learning objectives, test correlation tables, true/false questions, and multiple choice question previews related to:
1) Describing the core elements of organizational culture including symbols, language, values, norms, and narratives.
2) Comparing and contrasting four types of organizational culture: clan, adhocracy, market, and hierarchy.
3) Discussing several types of subcultures that may exist within organizations including departmental, generational, and gender-based subcultures.
4) Describing several activities for successfully managing diversity such as surveys, training, and establishing employee resource groups.
This document contains a chapter on work motivation from a textbook. It includes learning objectives about different theories of motivation, such as the managerial approach, reinforcement theory, expectancy theory, and job characteristics theory. There are also true/false questions and multiple choice questions testing understanding of these motivation theories.
This document provides a test correlation table and learning objectives for a chapter on managing human resources. The table lists the chapter's learning objectives and correlates them with different types of test questions (true/false, multiple choice, essay) at different levels of difficulty (easy, moderate, difficult). It then provides examples of test questions for each objective, including the question, answer, rationale, and difficulty level. The chapter appears to cover topics like the strategic importance of human resources, employment laws and regulations, human resources planning, recruitment and hiring, training and development, performance appraisals, and compensation.
This document provides an overview of managing work teams. It begins with learning objectives about explaining the importance of work teams, identifying types of work teams, stating the meaning and determinants of team effectiveness, describing internal team processes, and explaining how to diagnose and remove barriers to performance. It then provides a correlation table matching questions to these learning objectives at different levels of difficulty. The remainder of the document consists of true/false questions mapping to the learning objectives.
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Sect4 4
1. SECTION 4.4
BASES AND DIMENSION FOR VECTOR SPACES
A basis {v1 , v 2 , , v k } for a subspace W of Rn enables up to visualize W as a k-dimensional
plane (or hyperplane) through the origin in Rn. In case W is the solution space of a
homogeneous linear system, a basis for W is a maximal linearly independent set of solutions of
the system, and every other solution is a linear combination of these particular solutions.
1. The vectors v1 and v2 are linearly independent (because neither is a scalar multiple of
the other) and therefore form a basis for R2.
2. We note that v2 = 2v1. Consequently the vectors v1 , v 2 , v 3 are linearly dependent, and
therefore do not form a basis for R3.
3. Any four vectors in R3 are linearly dependent, so the given vectors do not form a basis
for R3.
4. Any basis for R4 contains four vectors, so the given vectors v1 , v 2 , v 3 do not form a
basis for R4.
5. The three given vectors v1 , v 2 , v 3 all lie in the 2-dimensional subspace x1 = 0 of R3.
Therefore they are linearly dependent, and hence do not form a basis for R3.
6. Det ([ v1 v2 v 3 ]) = −1 ≠ 0, so the three vectors are linearly independent, and hence do
form a basis for R3.
7. Det ([ v1 v2 v 3 ]) = 1 ≠ 0, so the three vectors are linearly independent, and hence do
form a basis for R3.
8. Det ([ v1 v2 v3 v 4 ]) = 66 ≠ 0, so the four vectors are linearly independent, and hence
do form a basis for R4.
9. The single equation x − 2 y + 5 z = 0 is already a system in reduced echelon form, with
free variables y and z. With y = s, z = t , x = 2s − 5t we get the solution vector
( x, y , z ) = (2s − 5t , s, t ) = s (2,1, 0) + t (−5, 0,1).
Hence the plane x − 2 y + 5 z = 0 is a 2-dimensional subspace of R3 with basis consisting
of the vectors v1 = (2,1, 0) and v 2 = (−5, 0,1).
2. 10. The single equation y − z = 0 is already a system in reduced echelon form, with free
variables x and z. With x = s, y = z = t we get the solution vector
( x, y , z ) = (s , t , t ) = s (1, 0, 0) + t (0,1,1).
Hence the plane y − z = 0 is a 2-dimensional subspace of R3 with basis consisting of the
vectors v1 = (1, 0, 0) and v 2 = (0,1,1).
11. The line of intersection of the planes in Problems 9 and 11 is the solution space of the
system
x − 2 y + 5z = 0
y − z = 0.
This system is in echelon form with free variable z = t. With y = t and x = –3t we have
the solution vector (−3t , t , t ) = t (−3,1,1). Thus the line is a 1-dimensional subspace of R3
with basis consisting of the vector v = (–3,1,1).
12. The typical vector in R4 of the form (a, b, c, d ) with a = b + c + d can be written as
v = (b + c + d , b, c, d ) = b (1,1, 0, 0) + c (1, 0,1, 0) + d (1, 0, 0,1).
Hence the subspace consisting of all such vectors is 3-dimensional with basis consisting
of the vectors v1 = (1,1, 0, 0), v 2 = (1, 0,1,0), and v 3 = (1, 0, 0,1).
13. The typical vector in R4 of the form (a, b, c, d ) with a = 3c and b = 4d can be written
as
v = (3c, 4d , c, d ) = c (3, 0,1, 0) + d (0, 4, 0,1).
Hence the subspace consisting of all such vectors is 2-dimensional with basis consisting
of the vectors v1 = (3, 0,1, 0) and v 2 = (0, 4, 0,1).
14. The typical vector in R4 of the form (a, b, c, d ) with a = −2b and c = −3d can be
written as
v = (−2b, b, −3d , d ) = b (−2,1, 0, 0) + d (0, 0, −3,1).
Hence the subspace consisting of all such vectors is 2-dimensional with basis consisting
of the vectors v1 = (−2,1, 0, 0) and v 2 = (0, 0, −3,1).
In Problems 15-26, we show first the reduction of the coefficient matrix A to echelon form E.
Then we write the typical solution vector as a linear combination of basis vectors for the
subspace of the given system.
3. 1 −2 3 1 0 −11
15. A = → 0 1 −7 = E
2 −3 1
With free variable x3 = t and x1 = 11t , x2 = 7t we get the solution vector
x = (11t , 7t , t ) = t (11, 7,1). Thus the solution space of the given system is 1-
dimensional with basis consisting of the vector v1 = (11, 7,1).
1 3 4 1 0 −11
16. A = → 0 1 5 = E
3 8 7
With free variable x3 = t and with x1 = 11t , x2 = −5t we get the solution vector
x = (11t , −5t , t ) = t (11, −5,1). Thus the solution space of the given system is 1-
dimensional with basis consisting of the vector v1 = (11, −5,1).
1 −3 2 −4 1 0 11 11
17. A = → 0 1 3 5 = E
2 −5 7 −3
With free variables x3 = s, x4 = t and with x1 = −11s − 11t , x2 = −3s − 5t we get the
solution vector
x = ( −11s − 11t , −3s − 5t , s, t ) = s ( −11, −3,1, 0) + t (−11, −5, 0,1).
Thus the solution space of the given system is 2-dimensional with basis consisting of the
vectors v1 = (−11, −3,1, 0) and v 2 = (−11, −5,0,1).
1 3 4 5 1 3 0 25
18. A = → 0 0 1 −5 = E
2 6 9 5
With free variables x2 = s, x4 = t and with x1 = −3s − 25t , x3 = 5t we get the solution
vector
x = ( −3s − 25t , s,5t , t ) = s (−3,1, 0, 0) + t (−25, 0,5,1).
Thus the solution space of the given system is 2-dimensional with basis consisting of the
vectors v1 = (−3,1, 0, 0) and v 2 = (−25, 0,5,1).
1 −3 −8 −5 1 0 −3 4
19. 2 1 −4 11 → 0 1 2 3 = E
A =
1 3 3 13
0 0 0 0
With free variables x3 = s, x4 = t and with x1 = 3s − 4t , x2 = −2 s − 3t we get the
solution vector
4. x = (3s − 4t , −2s − 3t , s, t ) = s (3, −2,1, 0) + t ( −4, −3, 0,1).
Thus the solution space of the given system is 2-dimensional with basis consisting of the
vectors v1 = (3, −2,1, 0) and v 2 = (−4, −3, 0,1).
1 −3 −10 5 1 0 −1 2
20. 1 4 11 −2 → 0 1 3 −1 = E
A =
1 3
8 −1
0 0 0 0
With free variables x3 = s, x4 = t and with x1 = s − 2t , x2 = −3s + t we get the solution
vector
x = ( s − 2t , −3s + t , s, t ) = s (1, −3,1, 0) + t (−2,1, 0,1).
Thus the solution space of the given system is 2-dimensional with basis consisting of the
vectors v1 = (1, −3,1, 0) and v 2 = (−2,1, 0,1).
1 −4 −3 −7 1 0 1 5
21. 2 −1 1 7 → 0 1 1 3 = E
A =
1 2 3 11
0 0 0 0
With free variables x3 = s, x4 = t and with x1 = − s − 5t , x2 = − s − 3t we get the solution
vector
x = ( − s − 5t , − s − 3t , s, t ) = s (−1, −1,1, 0) + t ( −5, −3, 0,1).
Thus the solution space of the given system is 2-dimensional with basis consisting of the
vectors v1 = (−1, −1,1, 0) and v 2 = (−5, −3,0,1).
1 −2 −3 −16 1 −2 0 5
22. 2 −4 1 17 → 0 0 1 7 = E
A =
1 −2 3 26
0 0 0 0
With free variables x2 = s, x4 = t and with x1 = 2s − 5t , x3 = −7t we get the solution
vector
x = (2s − 5t , s, −7t , t ) = s (2,1, 0, 0) + t (−5, 0, −7,1).
Thus the solution space of the given system is 2-dimensional with basis consisting of the
vectors v1 = (2,1, 0, 0) and v 2 = (−5, 0, −7,1).
5. 1 5 13 14 1 0 −2 0
23. 2 5 11 12 → 0 1 3 0 = E
A =
2 7 17 19
0 0 0 1
With free variable x3 = s and with x1 = 2s, x2 = −3s, x4 = 0 we get the solution
vector x = (2 s, −3s, s, 0) = s (2, −3,1, 0). Thus the solution space of the given system is
1-dimensional with basis consisting of the vector v1 = (2, −3,1, 0).
1 3 −4 −8 6 1 0 2 1 3
24. 1 0 2
A = 1 → 0 1 −2 −3 1 = E
3
2 7 −10 −19 13
0 0 0 0 0
With free variables x3 = r , x4 = s, x5 = t and with x1 = −2r − s − 3t , x2 = 2r + 3s − t we
get the solution vector
x = ( −2r − s − 3t , 2r + 3s − t , r , s, t ) = r (−2, 2,1, 0, 0) + s (−1,3, 0,1, 0) + t (−3, −1, 0, 0,1).
Thus the solution space of the given system is 3-dimensional with basis consisting of the
vectors v1 = (−2, 2,1, 0, 0), v 2 = (−1,3, 0,1, 0), and v 3 = (−3, −1, 0, 0,1).
1 2 7 −9 31 1 2 0 −2 3
25. 2 4 7 −11 34 → 0 0 1 −1 4 = E
A =
3 6 5 −11 29
0 0 0 0 0
With free variables x2 = r , x4 = s, x5 = t and with x1 = −2r + 2 s − 3t , x3 = s − 4t we get
the solution vector
x = ( −2r + 2 s − 3t , r , s − 4t , s, t ) = r ( −2,1, 0, 0, 0) + s (2, 0,1,1, 0) + t ( −3, 0, −4, 0,1).
Thus the solution space of the given system is 3-dimensional with basis consisting of the
vectors v1 = (−2,1, 0, 0, 0), v 2 = (2, 0,1,1, 0), and v3 = (−3, 0, −4, 0,1).
3 1 −3 11 10 1 0 0 2 −3
26. 5 8 2 −2 7 → 0 1 0 −1 4 = E
A =
2 5 0 −1 14
0 0 1 −2 −5
With free variables x4 = s, x5 = t and with x1 = −2s + 3t , x2 = s − 4t , x3 = 2s + 5t we get
the solution vector
x = ( −2s + 3t , s − 4t , 2 s + 5t , s, t ) = s ( −2,1, 2,1, 0) + t (3, −4,5, 0,1).
6. Thus the solution space of the given system is 2-dimensional with basis consisting of the
vectors v1 = (−2,1, 2,1, 0) and v 2 = (3, −4,5, 0,1).
27. If the vectors v1 , v 2 , , v n are linearly independent, and w is another vector in V, then
the vectors w, v1 , v 2 , , v n are linearly dependent (because no n+1 vectors in the n-
dimensional vector space V are linearly independent). Hence there exist scalars
c, c1 , c2 , , cn not all zero such that
cw + c1 v1 + c2 v 2 + + cn v n = 0.
If c = 0 then the coefficients c1 , c2 , , cn would not all be zero, and hence this equation
would say (contrary to hypothesis) that the vectors v1 , v 2 , , v n are linearly dependent.
Therefore c ≠ 0, so we can solve for w as a linear combination of the vectors
v1 , v 2 , , v n . Thus the linearly independent vectors v1 , v 2 , , v n span V, and
therefore form a basis for V.
28. If the n vectors in S were not linearly independent, then some one of them would be a
linear combination of the others. These remaining n–1 vectors would then span the n-
dimensional vector space V, which is impossible. Therefore the spanning set S is also
linearly independent, and therefore is a basis for V.
29. Suppose cv + c1v1 + c2 v 2 + + ck v k = 0. Then c = 0 because, otherwise, we could
solve for v as a linear combination of the vectors v1 , v 2 , , v k . But this is impossible,
because v is not in the subspace W spanned by v1 , v 2 , , v k . It follows that
c1v1 + c2 v 2 + + ck v k = 0, which implies that c1 = c2 = = ck = 0 also, because the
vectors v1 , v 2 , , v k are linearly independent. Hence we have shown that the k+1
vectors v, v1 , v 2 , , v k are linearly independent.
30. Let S = {v1 , v 2 , , v k } be a linearly independent set of k n vectors in V. If the
vector v k +1 in V is not in W = span(S), then Problem 29 implies that the k+1 vectors
v1 , v 2 , , v k , v k +1 are linearly independent. Continuing in this fashion, we can add one
vector at a time until we have n linearly independent vectors in V, which then form a
basis for V that contains the original basis S.
31. If v k +1 is a linear combination of the vectors v1 , v 2 , , v k , then obviously every linear
combination of the vectors v1 , v 2 , , v k , v k +1 is also a linear combination of
v1 , v 2 , , v k . But the former set of k+1 vectors spans V, so the latter set of k vectors
also spans V.
7. 32. If the spanning set S for V is not linearly independent, then some vector in S is a
linear combination of the others. But Problem 31 says that when we remove this
dependent vector from S, the resulting set of one fewer vectors still spans V.
Continuing in this fashion, we remove one vector at a time from S until we wind up with
a spanning set for V that is also a linearly independent set, and therefor forms a basis for
V that is contained by the original spanning set S.
33. If S is a maximal linearly independent set in V, the we see immediately that every other
vector in V is a linear combination of the vectors in S. Thus S also spans V, and is
therefore a basis for V.
34. If the minimal spanning set S for V were not linearly independent, then (by Problem
28) some vector S would be a linear combination of the others. Then the set obtained
from the minimal spanning set S by deleting this dependent vector would be a smaller
spanning set for S (which is impossible). Hence the spanning set S is also a linearly
independent set, and therefore is a basis for V.
35. Let S = {v1 , v 2 , , v n } be a uniquely spanning set for V. Then the fact, that
0 = 0 v1 + 0 v 2 + + 0 v n
is the unique expression of the zero vector 0 as a linear combination of the vectors in S,
means that S is a linearly independent set of vectors. Hence S is a basis for V.
36. If a1 , a2 , , ak are scalars, then the linear combination c1v1 + c2 v 2 + + ck v k — of the
column vectors of the matrix in Eq. (12) having the k × k identity matrix as its bottom
k × k submatrix — is a vector of the form (*, *, , *, a1 , a2 , , ak ) . Hence this linear
combination can equal the zero vector only if a1 = a2 = = ak = 0. Thus the vectors
v1 , v 2 , , v k are linearly independent.