This document discusses systems of equations and inequalities. It covers evaluating functions of two variables, solving systems of equations using substitution and elimination methods, graphing systems of equations, and solving linear inequalities symbolically and graphically. The document is a module on these topics, with learning objectives, examples, and explanations of key concepts.
This document discusses consistency criteria for systems of linear equations. It explains that a system can have:
1) A unique solution if the rank of the coefficient matrix equals the rank of the augmented matrix.
2) Infinitely many solutions if the ranks are equal but the system is rectangular or the rank of the coefficient matrix is less than the number of variables.
3) No solution if the ranks are unequal, meaning the system is inconsistent.
It provides examples of consistent and inconsistent systems to demonstrate these criteria.
This document provides a tutorial on basic MATLAB commands for creating, manipulating, and operating on vectors and matrices. It describes how to create vectors and matrices, change their entries, perform matrix multiplication and inversion, extract submatrices, and create special matrices like identity and diagonal matrices. Examples are provided to illustrate various commands like eye, inv, backslash, and how to input vectors, matrices, and create M-files for functions and scripts.
This document discusses two applications of matrices: 1) Solving systems of linear equations by manipulating matrices and 2) Finding eigenvalues and eigenvectors by solving the characteristic equation of a matrix. It provides an example of using matrices to solve a system of 3 equations with 3 unknowns. It also gives an example of finding the eigenvalues (-1, -2) and eigenvectors of a 2x2 matrix and using MatLab functions to calculate them.
An echelon matrix has the following properties:
(1) All zero rows are below non-zero rows.
(2) The first non-zero entry in each non-zero row is 1 and is located to the right of the leading 1 in the row above.
The document provides examples of matrices in echelon and reduced echelon form, and demonstrates how elementary row operations can be used to reduce a matrix to reduced echelon form through successive operations.
Aplicaciones y subespacios y subespacios vectoriales en laemojose107
se enfoca en la enseñanza del Álgebra Lineal en carreras de ingeniería. Los conceptos vinculados a esta rama de las matemáticas se estudian en los cursos básicos de los primeros años de los planes de estudio en esas carreras. Se estudian conceptos tales como vectores, matrices, sistemas de ecuaciones lineales, espacios vectoriales, transformaciones lineales, valores y. vectores propios, y diagonalización de matrices.
UNIT-V-FPGA &CPLD ARCHITECTURES AND APPLICATIONSDr.YNM
This document discusses state machine charts (SM charts), which are used to describe the behavior of digital systems and state machines. SM charts are similar to flowcharts but have specific rules for their construction. The key components of an SM chart are state boxes, decision boxes, and conditional output boxes. An example SM chart for a dice game is presented and realized using programmable logic devices. Alternative realization methods using input multiplexers and microprogramming are also described. The concept of linked state machines to simplify the design of complex systems is introduced.
This document discusses several methods for designing sequential circuits, including state table reduction, state assignment, derivation of flip-flop input equations, and realization using logic gates. It provides an example of designing a comparator circuit using an iterative approach with identical cells. The document also describes implementing sequential circuits using ROMs, PLAs, CPLDs and FPGAs, giving examples of a code converter and parallel adder circuit designs for each method.
This document discusses consistency criteria for systems of linear equations. It explains that a system can have:
1) A unique solution if the rank of the coefficient matrix equals the rank of the augmented matrix.
2) Infinitely many solutions if the ranks are equal but the system is rectangular or the rank of the coefficient matrix is less than the number of variables.
3) No solution if the ranks are unequal, meaning the system is inconsistent.
It provides examples of consistent and inconsistent systems to demonstrate these criteria.
This document provides a tutorial on basic MATLAB commands for creating, manipulating, and operating on vectors and matrices. It describes how to create vectors and matrices, change their entries, perform matrix multiplication and inversion, extract submatrices, and create special matrices like identity and diagonal matrices. Examples are provided to illustrate various commands like eye, inv, backslash, and how to input vectors, matrices, and create M-files for functions and scripts.
This document discusses two applications of matrices: 1) Solving systems of linear equations by manipulating matrices and 2) Finding eigenvalues and eigenvectors by solving the characteristic equation of a matrix. It provides an example of using matrices to solve a system of 3 equations with 3 unknowns. It also gives an example of finding the eigenvalues (-1, -2) and eigenvectors of a 2x2 matrix and using MatLab functions to calculate them.
An echelon matrix has the following properties:
(1) All zero rows are below non-zero rows.
(2) The first non-zero entry in each non-zero row is 1 and is located to the right of the leading 1 in the row above.
The document provides examples of matrices in echelon and reduced echelon form, and demonstrates how elementary row operations can be used to reduce a matrix to reduced echelon form through successive operations.
Aplicaciones y subespacios y subespacios vectoriales en laemojose107
se enfoca en la enseñanza del Álgebra Lineal en carreras de ingeniería. Los conceptos vinculados a esta rama de las matemáticas se estudian en los cursos básicos de los primeros años de los planes de estudio en esas carreras. Se estudian conceptos tales como vectores, matrices, sistemas de ecuaciones lineales, espacios vectoriales, transformaciones lineales, valores y. vectores propios, y diagonalización de matrices.
UNIT-V-FPGA &CPLD ARCHITECTURES AND APPLICATIONSDr.YNM
This document discusses state machine charts (SM charts), which are used to describe the behavior of digital systems and state machines. SM charts are similar to flowcharts but have specific rules for their construction. The key components of an SM chart are state boxes, decision boxes, and conditional output boxes. An example SM chart for a dice game is presented and realized using programmable logic devices. Alternative realization methods using input multiplexers and microprogramming are also described. The concept of linked state machines to simplify the design of complex systems is introduced.
This document discusses several methods for designing sequential circuits, including state table reduction, state assignment, derivation of flip-flop input equations, and realization using logic gates. It provides an example of designing a comparator circuit using an iterative approach with identical cells. The document also describes implementing sequential circuits using ROMs, PLAs, CPLDs and FPGAs, giving examples of a code converter and parallel adder circuit designs for each method.
This document discusses solving systems of linear equations using matrices and conditional statements in MATLAB. It provides an overview of relevant commands like rank(), rref(), inv(), and linsolve(). It then outlines a procedure to check the consistency of a system of linear equations based on the rank of the coefficient matrix A and augmented matrix [A B]. If the ranks are equal, it determines if the solution is unique, infinite, or inconsistent. It presents a MATLAB function that implements this procedure to solve a system of linear equations and return the solution or consistency status.
This document summarizes a seminar presentation on detecting outliers using the Ueda's method. The presentation defined outliers, described Ueda's methodology which uses the Akaike Information Criterion to identify outliers. It provided steps to calculate the test statistic Ut and interpret the results. Examples were shown applying the method to simulated normally distributed data with outliers added. The conclusion noted the method focuses on distribution symmetry and is effective for large samples.
The document discusses digital system design using state machine charts or ASM (algorithmic state machine) charts. It describes the basic components of an ASM chart including state boxes, decision boxes, and conditional output boxes. It provides examples of converting a state graph to an equivalent ASM chart and deriving an ASM chart for a binary multiplier. The document also discusses using hardware description languages like VHDL to model state machines behaviorally and provides examples of VHDL code for a 4-bit multiplier and a serial adder.
Certain Algebraic Procedures for the Aperiodic Stability Analysis and Countin...Waqas Tariq
To evaluate the performance of a linear time-invariant system, various measures are available. In this paper employing Routh’s table, two geometrical criteria for the aperiodic stability analysis of linear time-invariant systems having real coefficients are formulated. The proposed algebraic stability criteria check whether the given linear system is aperiodically stable or not.The additional significance of the two criteria is , it can be used to count the number of complex roots of a system having real coefficients which is not possible by the use of original Routh’s Table. These procedures can also be used for the design of linear systems. In the proposed methods , the characteristic equation having real coefficients are first converted to complex coefficient equations using Romonov’s transformation. These complex coefficients are used in two different ways to form the Modified Routh’s tables for the two schemes named as Sign Pair Criterion I (SPC I) and Sign Pair Criterion II (SPC II). It is found that the proposed algorithms offer computational simplicity compared to other algebraic methods and is illustrated with suitable examples. The developed MATLAB program make the analysis most simple.
This document provides an overview of matrix algebra concepts for business students. It defines key terms like matrix, order, types of matrices including identity, diagonal and triangular matrices, and matrix operations such as addition, subtraction and multiplication. It also explains determinants, which evaluate whether a system of linear equations has a unique solution. Determinants are calculated by taking the difference of products of diagonal elements of a square matrix. This document serves as a basic introduction and recap of matrix algebra.
The document provides an introduction and roadmap to learning MATLAB for students in electronic engineering. It includes 12 chapters that cover basic MATLAB skills like working with matrices, arithmetic and logical operations, plotting, and applications in areas like signal processing and control systems. The roadmap aims to teach MATLAB programming to students within 240 microyears (approximately 7.5 months). It emphasizes that MATLAB can help engineering students learn technical topics even if they don't know how to program initially.
This document provides a summary of a course on introduction to MATLAB. The course includes 7 lectures covering topics like variables, operations, plotting, visualization, programming, solving equations and advanced methods. It will have problem sets to be submitted after each lecture and requirements to pass include attending all lectures and completing all problem sets. The course materials provide an overview of MATLAB including getting started, creating and manipulating variables, and basic plotting.
This document discusses the rank of matrices and how it relates to the solvability of linear systems of equations. It contains the following key points:
1) The rank of a matrix is the number of leading entries in its row-reduced form and determines the number of independent variables in a linear system with that matrix as its coefficient matrix.
2) The rank of the coefficient matrix and augmented matrix determine whether a linear system has no solution, a unique solution, or infinitely many solutions.
3) Homogeneous systems always have at least one solution (the trivial solution of all zeros) and the rank of the coefficient matrix determines if that is the only solution or if there are infinitely many solutions.
This document discusses two methods for graphing piecewise-defined functions on a graphing calculator: the division method and the multiplication method. The division method graphs each section of the function separately, while the multiplication method can graph the entire function as one statement. The multiplication method connects the separate sections together when graphed in connected mode, while the division method does not connect the sections. Both methods assign a value of 0 when the input is not in the defined interval, but this results in different graph appearances depending on whether the sections are connected or not connected.
This document discusses creating and manipulating arrays in MATLAB. It covers:
- Creating one-dimensional arrays (vectors) using known values, spacing, or number of elements.
- Creating two-dimensional arrays (matrices) by specifying rows and ensuring each row has the same number of columns.
- Basic array commands like zeros, ones, and eye to create arrays filled with zeros, ones, or an identity matrix.
- Transposing arrays with the ' operator.
- Accessing and assigning values to individual elements or ranges of elements using indexing and colon notation.
- Adding elements to existing arrays by assigning values to undefined indices or appending rows/columns.
This document introduces eigenvalues and eigenvectors. It provides three key points:
1. Eigenvalues are found by setting the determinant of A - λI equal to 0, where λ is the eigenvalue and I is the identity matrix. This yields a characteristic equation with degree n for an n×n matrix A.
2. Each eigenvalue λ yields an eigenvector x by solving (A - λI)x = 0.
3. Eigenvalues and eigenvectors reveal important properties about how a matrix transforms vectors under multiplication. Vectors that are unchanged are eigenvectors, and the scaling factor is the corresponding eigenvalue.
The document defines key terms related to functions including univariate and bivariate data, independent and dependent variables, domain and range, and linear, exponential, quadratic, and step functions. It provides examples of evaluating various functions and finding linear and quadratic models to describe relationships between variables from sets of data points. The overall content describes different types of mathematical functions and how to analyze and model real-world data using functions.
Dummy variables are used to represent qualitative or categorical variables that take on only two values, usually 0 and 1. A dummy variable indicates the presence or absence of a particular attribute. For example, a dummy variable could represent gender where 1 = male and 0 = female. Dummy variables allow qualitative variables to be used in regression models. However, there is a "dummy variable trap" where including dummy variables for all categories of a qualitative variable leads to perfect multicollinearity. To avoid this, only n-1 dummy variables should be included where there are n categories.
The document provides an overview of analysis of variance (ANOVA) techniques, including:
- One-way ANOVA to evaluate differences between three or more group means and the assumptions of one-way ANOVA.
- Partitioning total variation into between-group and within-group components.
- Computing test statistics like the F-ratio to test for differences between group means.
- Interpreting one-way ANOVA results including rejecting the null hypothesis of no difference between means.
- An example one-way ANOVA calculation and interpretation using golf club distance data.
This document discusses linear regression. It defines linear regression as a statistical measure that determines the strength of the relationship between a dependent variable (Y) and one or more independent variables (X). Linear regression finds the line of best fit to forecast Y from X. It explains key aspects of linear regression including slope, intercept, the linear regression equation (Y=b0+b1X+e), and using scatter plots to visualize the linear relationship between two variables. An example of using linear regression to study the relationship between blood pressure (Y) and factors like age, weight, and sex (X) is provided.
This document discusses various types of regression modeling and linear regression. It provides examples of linear regression analysis on fraud data and discusses assessing goodness of fit. It also briefly covers non-linear regression, problem areas like heteroskedasticity and collinearity, and model selection methods. Linear regression is presented geometrically and the assumptions and computations of ordinary least squares regression are explained.
Eigenvalues for HIV-1 dynamic model with two delaysIOSR Journals
This document presents a new approach to solve the characteristic equation of an HIV-1 infection dynamical system with two delays. The authors develop a series expansion to approximate the eigenvalues (roots) of the nonlinear characteristic equation. They derive the characteristic equation for the linearized HIV-1 model and nondimensionalize the equation. This allows them to express the eigenvalues as a perturbation of the logarithm of a parameter and derive an equation for the perturbation term. The goal is to make the truncated series more computationally efficient for evaluating the eigenvalues.
1. The document examines exponential functions, which have the independent variable in the exponent.
2. Exponential functions always have a positive range, no x-intercepts, a y-intercept of (0,1), and are always increasing. They also have a horizontal asymptote at y=0 as x approaches negative infinity.
3. The special base "e", also called the natural base, is important for modeling natural phenomena and is approximately equal to 2.718281828.
4. When two exponential functions with the same base are set equal, the exponents must be equal, allowing equations with exponential functions to be solved by equating exponents.
This document presents an application of a system of linear equations with five variables to solve an engineering problem. It begins by introducing the problem and objectives, then provides theoretical background on linear systems. Next, it defines the variables, operations, and equations of the specific problem. The document then describes how linear systems are applied in civil engineering and the Gaussian elimination method is used to solve the system. Finally, the problem is coded and solved using MATLAB software.
Chaos synchronization in a 6-D hyperchaotic system with self-excited attractorTELKOMNIKA JOURNAL
This paper presented stability application for chaos synchronization using a 6-D hyperchaotic system of different controllers and two tools: Lyapunov stability theory and Linearization methods. Synchronization methods based on nonlinear control strategy is used. The selecting controller's methods have been modified by applying complete synchronization. The Linearization methods can achieve convergence according to the of complete synchronization. Numerical simulations are carried out by using MATLAB to validate the effectiveness of the analytical technique.
This document discusses solving systems of linear equations using matrices and Gaussian elimination. It begins with an introduction to representing systems of linear equations as augmented matrices. It then explains how to use Gaussian elimination or Gauss-Jordan elimination to transform the augmented matrix into row-echelon form or reduced row-echelon form in order to solve the system. Several examples are provided to demonstrate solving systems with 2, 3, and an inconsistent number of variables using this process of transforming the augmented matrix.
This document describes an experimental evaluation of combinatorial preconditioners for solving linear systems. It compares Vaidya's algorithm for constructing combinatorial preconditioners to newer algorithms presented by Spielman, including a low-stretch spanning tree constructor and tree augmentation approach. The algorithms were implemented in Java and experimentally evaluated using a test framework on various matrices. The main results found that the new augmentation algorithm did not consistently outperform Vaidya's algorithm, though it did sometimes have significantly better performance. Using low-stretch trees as a basis for augmentation provided a consistent but modest improvement over Vaidya.
This document discusses solving systems of linear equations using matrices and conditional statements in MATLAB. It provides an overview of relevant commands like rank(), rref(), inv(), and linsolve(). It then outlines a procedure to check the consistency of a system of linear equations based on the rank of the coefficient matrix A and augmented matrix [A B]. If the ranks are equal, it determines if the solution is unique, infinite, or inconsistent. It presents a MATLAB function that implements this procedure to solve a system of linear equations and return the solution or consistency status.
This document summarizes a seminar presentation on detecting outliers using the Ueda's method. The presentation defined outliers, described Ueda's methodology which uses the Akaike Information Criterion to identify outliers. It provided steps to calculate the test statistic Ut and interpret the results. Examples were shown applying the method to simulated normally distributed data with outliers added. The conclusion noted the method focuses on distribution symmetry and is effective for large samples.
The document discusses digital system design using state machine charts or ASM (algorithmic state machine) charts. It describes the basic components of an ASM chart including state boxes, decision boxes, and conditional output boxes. It provides examples of converting a state graph to an equivalent ASM chart and deriving an ASM chart for a binary multiplier. The document also discusses using hardware description languages like VHDL to model state machines behaviorally and provides examples of VHDL code for a 4-bit multiplier and a serial adder.
Certain Algebraic Procedures for the Aperiodic Stability Analysis and Countin...Waqas Tariq
To evaluate the performance of a linear time-invariant system, various measures are available. In this paper employing Routh’s table, two geometrical criteria for the aperiodic stability analysis of linear time-invariant systems having real coefficients are formulated. The proposed algebraic stability criteria check whether the given linear system is aperiodically stable or not.The additional significance of the two criteria is , it can be used to count the number of complex roots of a system having real coefficients which is not possible by the use of original Routh’s Table. These procedures can also be used for the design of linear systems. In the proposed methods , the characteristic equation having real coefficients are first converted to complex coefficient equations using Romonov’s transformation. These complex coefficients are used in two different ways to form the Modified Routh’s tables for the two schemes named as Sign Pair Criterion I (SPC I) and Sign Pair Criterion II (SPC II). It is found that the proposed algorithms offer computational simplicity compared to other algebraic methods and is illustrated with suitable examples. The developed MATLAB program make the analysis most simple.
This document provides an overview of matrix algebra concepts for business students. It defines key terms like matrix, order, types of matrices including identity, diagonal and triangular matrices, and matrix operations such as addition, subtraction and multiplication. It also explains determinants, which evaluate whether a system of linear equations has a unique solution. Determinants are calculated by taking the difference of products of diagonal elements of a square matrix. This document serves as a basic introduction and recap of matrix algebra.
The document provides an introduction and roadmap to learning MATLAB for students in electronic engineering. It includes 12 chapters that cover basic MATLAB skills like working with matrices, arithmetic and logical operations, plotting, and applications in areas like signal processing and control systems. The roadmap aims to teach MATLAB programming to students within 240 microyears (approximately 7.5 months). It emphasizes that MATLAB can help engineering students learn technical topics even if they don't know how to program initially.
This document provides a summary of a course on introduction to MATLAB. The course includes 7 lectures covering topics like variables, operations, plotting, visualization, programming, solving equations and advanced methods. It will have problem sets to be submitted after each lecture and requirements to pass include attending all lectures and completing all problem sets. The course materials provide an overview of MATLAB including getting started, creating and manipulating variables, and basic plotting.
This document discusses the rank of matrices and how it relates to the solvability of linear systems of equations. It contains the following key points:
1) The rank of a matrix is the number of leading entries in its row-reduced form and determines the number of independent variables in a linear system with that matrix as its coefficient matrix.
2) The rank of the coefficient matrix and augmented matrix determine whether a linear system has no solution, a unique solution, or infinitely many solutions.
3) Homogeneous systems always have at least one solution (the trivial solution of all zeros) and the rank of the coefficient matrix determines if that is the only solution or if there are infinitely many solutions.
This document discusses two methods for graphing piecewise-defined functions on a graphing calculator: the division method and the multiplication method. The division method graphs each section of the function separately, while the multiplication method can graph the entire function as one statement. The multiplication method connects the separate sections together when graphed in connected mode, while the division method does not connect the sections. Both methods assign a value of 0 when the input is not in the defined interval, but this results in different graph appearances depending on whether the sections are connected or not connected.
This document discusses creating and manipulating arrays in MATLAB. It covers:
- Creating one-dimensional arrays (vectors) using known values, spacing, or number of elements.
- Creating two-dimensional arrays (matrices) by specifying rows and ensuring each row has the same number of columns.
- Basic array commands like zeros, ones, and eye to create arrays filled with zeros, ones, or an identity matrix.
- Transposing arrays with the ' operator.
- Accessing and assigning values to individual elements or ranges of elements using indexing and colon notation.
- Adding elements to existing arrays by assigning values to undefined indices or appending rows/columns.
This document introduces eigenvalues and eigenvectors. It provides three key points:
1. Eigenvalues are found by setting the determinant of A - λI equal to 0, where λ is the eigenvalue and I is the identity matrix. This yields a characteristic equation with degree n for an n×n matrix A.
2. Each eigenvalue λ yields an eigenvector x by solving (A - λI)x = 0.
3. Eigenvalues and eigenvectors reveal important properties about how a matrix transforms vectors under multiplication. Vectors that are unchanged are eigenvectors, and the scaling factor is the corresponding eigenvalue.
The document defines key terms related to functions including univariate and bivariate data, independent and dependent variables, domain and range, and linear, exponential, quadratic, and step functions. It provides examples of evaluating various functions and finding linear and quadratic models to describe relationships between variables from sets of data points. The overall content describes different types of mathematical functions and how to analyze and model real-world data using functions.
Dummy variables are used to represent qualitative or categorical variables that take on only two values, usually 0 and 1. A dummy variable indicates the presence or absence of a particular attribute. For example, a dummy variable could represent gender where 1 = male and 0 = female. Dummy variables allow qualitative variables to be used in regression models. However, there is a "dummy variable trap" where including dummy variables for all categories of a qualitative variable leads to perfect multicollinearity. To avoid this, only n-1 dummy variables should be included where there are n categories.
The document provides an overview of analysis of variance (ANOVA) techniques, including:
- One-way ANOVA to evaluate differences between three or more group means and the assumptions of one-way ANOVA.
- Partitioning total variation into between-group and within-group components.
- Computing test statistics like the F-ratio to test for differences between group means.
- Interpreting one-way ANOVA results including rejecting the null hypothesis of no difference between means.
- An example one-way ANOVA calculation and interpretation using golf club distance data.
This document discusses linear regression. It defines linear regression as a statistical measure that determines the strength of the relationship between a dependent variable (Y) and one or more independent variables (X). Linear regression finds the line of best fit to forecast Y from X. It explains key aspects of linear regression including slope, intercept, the linear regression equation (Y=b0+b1X+e), and using scatter plots to visualize the linear relationship between two variables. An example of using linear regression to study the relationship between blood pressure (Y) and factors like age, weight, and sex (X) is provided.
This document discusses various types of regression modeling and linear regression. It provides examples of linear regression analysis on fraud data and discusses assessing goodness of fit. It also briefly covers non-linear regression, problem areas like heteroskedasticity and collinearity, and model selection methods. Linear regression is presented geometrically and the assumptions and computations of ordinary least squares regression are explained.
Eigenvalues for HIV-1 dynamic model with two delaysIOSR Journals
This document presents a new approach to solve the characteristic equation of an HIV-1 infection dynamical system with two delays. The authors develop a series expansion to approximate the eigenvalues (roots) of the nonlinear characteristic equation. They derive the characteristic equation for the linearized HIV-1 model and nondimensionalize the equation. This allows them to express the eigenvalues as a perturbation of the logarithm of a parameter and derive an equation for the perturbation term. The goal is to make the truncated series more computationally efficient for evaluating the eigenvalues.
1. The document examines exponential functions, which have the independent variable in the exponent.
2. Exponential functions always have a positive range, no x-intercepts, a y-intercept of (0,1), and are always increasing. They also have a horizontal asymptote at y=0 as x approaches negative infinity.
3. The special base "e", also called the natural base, is important for modeling natural phenomena and is approximately equal to 2.718281828.
4. When two exponential functions with the same base are set equal, the exponents must be equal, allowing equations with exponential functions to be solved by equating exponents.
This document presents an application of a system of linear equations with five variables to solve an engineering problem. It begins by introducing the problem and objectives, then provides theoretical background on linear systems. Next, it defines the variables, operations, and equations of the specific problem. The document then describes how linear systems are applied in civil engineering and the Gaussian elimination method is used to solve the system. Finally, the problem is coded and solved using MATLAB software.
Chaos synchronization in a 6-D hyperchaotic system with self-excited attractorTELKOMNIKA JOURNAL
This paper presented stability application for chaos synchronization using a 6-D hyperchaotic system of different controllers and two tools: Lyapunov stability theory and Linearization methods. Synchronization methods based on nonlinear control strategy is used. The selecting controller's methods have been modified by applying complete synchronization. The Linearization methods can achieve convergence according to the of complete synchronization. Numerical simulations are carried out by using MATLAB to validate the effectiveness of the analytical technique.
This document discusses solving systems of linear equations using matrices and Gaussian elimination. It begins with an introduction to representing systems of linear equations as augmented matrices. It then explains how to use Gaussian elimination or Gauss-Jordan elimination to transform the augmented matrix into row-echelon form or reduced row-echelon form in order to solve the system. Several examples are provided to demonstrate solving systems with 2, 3, and an inconsistent number of variables using this process of transforming the augmented matrix.
This document describes an experimental evaluation of combinatorial preconditioners for solving linear systems. It compares Vaidya's algorithm for constructing combinatorial preconditioners to newer algorithms presented by Spielman, including a low-stretch spanning tree constructor and tree augmentation approach. The algorithms were implemented in Java and experimentally evaluated using a test framework on various matrices. The main results found that the new augmentation algorithm did not consistently outperform Vaidya's algorithm, though it did sometimes have significantly better performance. Using low-stretch trees as a basis for augmentation provided a consistent but modest improvement over Vaidya.
This document provides an overview of quadratic functions and equations. It contains 29 slides that cover topics such as: graphing quadratic functions and determining their properties based on the leading coefficient; writing quadratic functions in vertex form using completing the square; solving applications involving quadratic modeling; and using factoring, taking square roots, completing the square, the quadratic formula, and the discriminant to solve quadratic equations. The learning objectives are also listed which are to understand concepts about quadratic functions and equations and learn techniques for graphing, modeling with, and solving them.
Multiple regression analysis is used to understand the relationship between a dependent variable and multiple independent variables. It allows us to determine the effect of independent variables like age and work experience on a dependent variable like income. The analysis examines metrics like the coefficient of determination (R2), F-test, t-test and assumptions around normality, multicollinearity, homoscedasticity and autocorrelation. Properly conducted, multiple regression can be used to predict the value of a dependent variable based on the values of independent variables.
This presentation is a part of the COP2272C college level course taught at the Florida Polytechnic University located in Lakeland Florida. The purpose of this course is to introduce students to the C++ language and the fundamentals of object orientated programming..
The course is one semester in length and meets for 2 hours twice a week. The Instructor is Dr. Jim Anderson.
This document discusses systems of linear equations in two variables. It explains that a solution to a system is an ordered pair that satisfies both equations. The solution can be found by graphing the equations on a coordinate plane, with the point of intersection giving the solution if there is a single solution. The document provides examples of finding solutions by substitution, addition, and for systems with no or infinite solutions. It also discusses using systems to analyze break-even points for businesses.
In this tutorial, we discuss how to do a regression analysis in Excel. I will teach you how to activate the regression analysis feature, what are the functions and methods we can use to do a regression analysis in Excel and most importantly, how to interpret the regression analysis results. Source: https://tinytutes.com/tutorials/regression-analysis-in-excel/
The document discusses linear programming problems and their graphical solutions. It introduces:
- Graphing linear inequalities in two variables by representing the solution set as a half-plane defined by the inequality. Any point on or below the graph line satisfies the inequality.
- Solving linear programming problems with two unknowns using graphical methods by representing the feasible region as the intersection of half-planes defined by the constraints.
- More advanced algebraic methods, like the simplex method, for solving problems with three or more unknowns.
This document describes a numerical methods course that covers various numerical techniques for solving engineering problems. The course topics include root-finding, solving systems of linear equations, curve fitting, numerical integration and differentiation, and solving ordinary differential equations. It also introduces MATLAB for implementing numerical methods and visualizing data and functions.
9.6 Systems of Inequalities and Linear Programmingsmiller5
This document provides an overview of systems of inequalities and how to graph and solve them. It discusses representing systems of inequalities symbolically and identifying the solution as the overlapping region of the graphed inequalities. Examples are provided of writing systems of inequalities from word problems and using graphs to find the solutions. Linear programming is also introduced as an application of systems of inequalities to optimize an objective function subject to constraints.
Support Vector Machines USING MACHINE LEARNING HOW IT WORKSrajalakshmi5921
This document discusses support vector machines (SVM), a supervised machine learning algorithm used for classification and regression. It explains that SVM finds the optimal boundary, known as a hyperplane, that separates classes with the maximum margin. When data is not linearly separable, kernel functions can transform the data into a higher-dimensional space to make it separable. The document discusses SVM for both linearly separable and non-separable data, kernel functions, hyperparameters, and approaches for multiclass classification like one-vs-one and one-vs-all.
1. The document introduces Ahmed Nobi and provides his contact information and background. It outlines a session on programming fundamentals including variables, data types, operators, selection, and loops.
2. The session will cover variables, data types, operators, and selection. It provides examples of each concept and explains how to name variables, common data types, mathematical and logical operators, and if/else statements.
3. Blocks are explained as using curly braces {} to group lines of code that should be executed together, such as the main block or blocks within if/else statements.
The document summarizes the simplex algorithm for solving linear programs. It describes how the algorithm works by starting with an initial basic feasible solution and iteratively arriving at solutions with greater objective values through pivoting. Pivoting exchanges a non-basic variable for a basic variable to obtain an equivalent linear program with a higher objective value. The algorithm terminates when all coefficients in the objective function are negative, indicating an optimal solution has been found.
A tour of the top 10 algorithms for machine learning newbiesVimal Gupta
The document summarizes the top 10 machine learning algorithms for machine learning newbies. It discusses linear regression, logistic regression, linear discriminant analysis, classification and regression trees, naive bayes, k-nearest neighbors, and learning vector quantization. For each algorithm, it provides a brief overview of the model representation and how predictions are made. The document emphasizes that no single algorithm is best and recommends trying multiple algorithms to find the best one for the given problem and dataset.
Boosting algorithms work by combining multiple weak learners to create a strong learner. Weak learners are models that are only slightly correlated with the true output. Boosting algorithms fit these weak learners in sequence, giving higher weight to observations that previous weak learners misclassified. This process converts weak learners into a strong learner. Common boosting algorithms include AdaBoost, gradient boosting, and XGBoost. Gradient boosting works by gradually minimizing a loss function to fit new models that correlate with the negative gradient of the previous models.
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This chapter discusses regression models, including simple and multiple linear regression. It covers developing regression equations from sample data, measuring the fit of regression models, and assumptions of regression analysis. Key aspects covered include using scatter plots to examine relationships between variables, calculating the slope, intercept, coefficient of determination, and correlation coefficient, and performing hypothesis tests to determine if regression models are statistically significant. The chapter objectives are to help students understand and appropriately apply simple, multiple, and nonlinear regression techniques.
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2. 2Rev.S08
Learning Objectives
Upon completing this module, you should be able to:
1. Evaluate functions of two variables.
2. Apply the method of substitution.
3. Apply the elimination method.
4. Solve system of equations symbolically.
5. Apply graphical and numerical methods to system of
equations.
6. Recognize different types of linear systems.
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3. 3Rev.S08
Learning Objectives (Cont.)
7. Use basic terminology related to inequalities.
8. Use interval notation.
9. Solve linear inequalities symbolically.
10. Solve linear inequalities graphically and numerically.
11. Solve double inequalities.
12. Graph a system of linear inequalities.
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4. 4Rev.S08
System of Equations and Inequalities
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- System of Linear Equations in Two Variables- System of Linear Equations in Two Variables
- Solutions of Linear Inequalities- Solutions of Linear Inequalities
There are two major topics in this module:
5. 5Rev.S08
Do We Really Use Functions of Two
Variables?
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The answer is YES.
Many quantities in everyday life depend on more than
one variable.
Examples
Area of a rectangle requires both width and length.
Heat index is the function of temperature and humidity.
Wind chill is determined by calculating the temperature
and wind speed.
Grade point average is computed using grades and
credit hours.
6. 6Rev.S08
Let’s Take a Look at the
Arithmetic Operations
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The arithmetic operations of addition, subtraction,
multiplication, and division are computed by functions
of two inputs.
The addition function of f can be represented
symbolically by f(x,y) = x + y, where z = f(x,y).
The independent variables are x and y.
The dependent variable is z. The z output depends
on the inputs x and y.
7. 7Rev.S08
Here are Some Examples
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For each function, evaluate the expression and interpret
the result.
a) f(5, –2) where f(x,y) = xy
b) A(6,9), where calculates the area of a
triangle with a base of 6 inches and a height of 9
inches.
Solution
• f(5, –2) = (5)(–2) = –10.
• A(6,9) =
If a triangle has a base of 6 inches and a height of 9
inches, the area of the triangle is 27 square inches.
8. 8Rev.S08
What is a System of Linear Equations?
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A linear equation in two variables can be written in the
form ax + by = k, where a, b, and k are constants, and
a and b are not equal to 0.
A pair of equations is called a system of linear
equations because they involve solving more than one
linear equation at once.
A solution to a system of equations consists of an x-
value and a y-value that satisfy both equations
simultaneously.
The set of all solutions is called the solution set.
9. 9Rev.S08
How to Use the Method of Substitution to
solve a system of two equations?
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10. 10Rev.S08
How to Solve the System Symbolically?
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Solve the system symbolically.
Solution
Step 1: Solve one of the
equations for one of the
variables.
Step 2: Substitute
for y in the second
equation.
11. 11Rev.S08
How to Solve the System Symbolically?
(Cont.)
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Step 3: Substitute x = 1 into the equation
from Step 1. We find that
Check:
The ordered pair is (1, 2) since the solutions check in
both equations.
12. 12Rev.S08
Solve the system.
• Solution
• Solve the second equation for y.
• Substitute 4x + 2 for y in the first equation, solving for
x.
• The equation −4 = −4 is an identity that is always true
and indicates that there are infinitely many solutions.
The two equations are equivalent.
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Example with Infinitely Many Solutions
14. 14Rev.S08
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How to Use Elimination Method to Solve
System of Equations?
Use elimination to solve each system of equations, if
possible. Identify the system as consistent or
inconsistent. If the system is consistent, state whether
the equations are dependent or independent. Support
your results graphically.
a) 3x − y = 7 b) 5x − y = 8 c) x − y = 5
5x + y = 9 −5x + y = −8 x − y = − 2
15. 15Rev.S08
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How to Use Elimination Method to Solve
System of Equations? (Cont.)
Solution
a)
EliminateEliminate yy by addingby adding
the equations.the equations.
Find y by substituting
x = 2 in either equation.
The solution is (2, −1). The system is
consistent and the equations are independent.
16. 16Rev.S08
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How to Use Elimination Method to Solve
System of Equations? (Cont.)
If we add the equations we obtain theIf we add the equations we obtain the
following result.following result.
The equationThe equation 0 = 00 = 0 isis anan
identityidentity that isthat is always truealways true..
The two equations are equivalent.The two equations are equivalent.
There areThere are infinitelyinfinitely
many solutionsmany solutions..
{({(xx,, yy)| 5)| 5xx −− yy = 8}= 8}
b)
17. 17Rev.S08
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How to Use Elimination Method to Solve
System of Equations? (Cont.)
c) If we subtract the second equation fromIf we subtract the second equation from
the first, we obtain the following result.the first, we obtain the following result.
The equationThe equation 0 = 70 = 7 is ais a
contradiction that iscontradiction that is never truenever true..
Therefore there areTherefore there are no solutionsno solutions,,
and theand the system is inconsistentsystem is inconsistent..
18. 18Rev.S08
Let’s Practice Using Elimination
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Solve the system by using elimination.
Solution
Multiply the first equation by 3 and the second equation
by 4. Addition eliminates the y-variable.
Substituting x = 3 in 2x + 3y = 12 results in
2(3) + 3y = 12 or y = 2
The solution is (3, 2).
19. 19Rev.S08
Terminology related to Inequalities
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• InequalitiesInequalities result whenever theresult whenever the equals signequals sign inin
an equation isan equation is replaced withreplaced with any one of theany one of the
symbols:symbols: ≤, ≥, <, >≤, ≥, <, >
• Examples of inequalities include:Examples of inequalities include:
•2x –72x –7 >> x +13x +13
•xx22
≤≤ 15 – 21x15 – 21x
•xy +9 xxy +9 x << 22xx22
•3535 >> 66
20. 20Rev.S08
Linear Inequality in One Variable
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•AA linear inequality in one variablelinear inequality in one variable is an inequality that canis an inequality that can
be written in the formbe written in the form
axax ++ bb >> 0 where a ≠ 0.0 where a ≠ 0.
(The symbol may be replaced by(The symbol may be replaced by ≤, ≥, <, >≤, ≥, <, > ))
•Examples of linear inequalities in one variable:Examples of linear inequalities in one variable:
• 55xx + 4 ≤ 2 + 3+ 4 ≤ 2 + 3xx simplifies to 2simplifies to 2xx + 2 ≤ 0+ 2 ≤ 0
• −−1(1(xx – 3) + 4(2– 3) + 4(2xx + 1) > 5 simplifies to 7+ 1) > 5 simplifies to 7xx + 2 > 0+ 2 > 0
•Examples of inequalities in one variable which areExamples of inequalities in one variable which are notnot
linear:linear:
• xx22
< 1< 1
21. 21Rev.S08
Let’s Look at Interval Notation
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TheThe solutionsolution to ato a linear inequality in one variablelinear inequality in one variable is typicallyis typically anan
interval on the real number lineinterval on the real number line. See examples of interval notation. See examples of interval notation
below.below.
22. 22Rev.S08
Multiplied by a Negative Number
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Note that 3Note that 3 << 5, but if both sides are5, but if both sides are multiplied bymultiplied by −− 11, in, in
order to produce a true statementorder to produce a true statement the > symbol must bethe > symbol must be
usedused..
33 << 55
butbut
−− 33 >> −− 55
So when both sides of an inequality are multiplied (orSo when both sides of an inequality are multiplied (or
divided) by a negative number the direction of thedivided) by a negative number the direction of the
inequality must be reversedinequality must be reversed..
23. 23Rev.S08
How to Solve Linear Inequalities
Symbolically?
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The procedure for solving a linear inequality symbolically is the same asThe procedure for solving a linear inequality symbolically is the same as
the procedure for solving a linear equation,the procedure for solving a linear equation, exceptexcept when both sideswhen both sides
of an inequality are multiplied (or divided) by a negative number theof an inequality are multiplied (or divided) by a negative number the
direction of the inequality is reversed.direction of the inequality is reversed.
Example of Solving aExample of Solving a Example of Solving aExample of Solving a
LinearLinear EquationEquation SymbolicallySymbolically LinearLinear InequalityInequality SymbolicllySymboliclly
SolveSolve −−22xx + 1 =+ 1 = xx −− 22 SolveSolve −−22xx + 1 <+ 1 < xx −− 22
−−22xx −− xx == −−22 −−11 −−22xx −− xx << −−22 −−11
−−33xx == −−33 −−33xx << −−33
xx = 1= 1 xx >> 11
Note that we divided bothNote that we divided both
sides bysides by −−3 so the direction3 so the direction
of the inequality wasof the inequality was
reversed.reversed. In interval notationIn interval notation
the solution set is (1,∞).the solution set is (1,∞).
24. 24Rev.S08
How to Solve a Linear Inequality
Graphically?
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Note that the graphs intersect at the point (8.20, 7.59). The graph ofNote that the graphs intersect at the point (8.20, 7.59). The graph of
yy11 is above the graph of yis above the graph of y22 to the right of the point of intersection orto the right of the point of intersection or
whenwhen xx > 8.20. Thus, in> 8.20. Thus, in interval notation,interval notation, the solution set is (8.20,the solution set is (8.20,
∞)∞)
SolveSolve
25. 25Rev.S08
How to Solve a Linear Inequality
Numerically?
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Note that the inequality above becomesNote that the inequality above becomes yy11 ≥≥ yy22 since we letsince we let yy11 equal the left-equal the left-
hand side andhand side and yy22 equal the right hand side.equal the right hand side.
To write the solution set of the inequality we are looking for the values of x inTo write the solution set of the inequality we are looking for the values of x in
the table for whichthe table for which yy11 is the same or larger thanis the same or larger than yy22.. Note that whenNote that when xx == −−1.3,1.3, yy11
is less thanis less than yy22;; but whenbut when xx == −− 1.4,1.4, yy11 is larger thanis larger than yy22.. By the IntermediateBy the Intermediate
Value Property, there is a value ofValue Property, there is a value of xx betweenbetween −− 1.4 and1.4 and −− 1.3 such that1.3 such that yy11 == yy22..
In order to find an approximation of this value, make a new table in which x isIn order to find an approximation of this value, make a new table in which x is
incremented by .01 (note thatincremented by .01 (note that xx is incremented by .1 in the table to the leftis incremented by .1 in the table to the left
here.)here.)
SolveSolve
26. 26Rev.S08
How to Solve a Linear Inequality
Numerically? (cont.)
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SolveSolve
To write theTo write the solution setsolution set of the inequality we are looking for the valuesof the inequality we are looking for the values
of x in the table for whichof x in the table for which yy11 is the same as or larger thanis the same as or larger than yy22.. Note thatNote that
whenwhen xx is approximatelyis approximately −−1.36,1.36, yy11 equalsequals yy22 and when x is smaller thanand when x is smaller than
−−1.361.36 yy11 is larger thanis larger than yy22 ,, so the solutions can be writtenso the solutions can be written
x ≤x ≤ −−1.361.36 oror ((−−∞,∞, −−1.36] in interval notation1.36] in interval notation..
27. 27Rev.S08
How to Solve Double Inequalities?
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• Example:Example: Suppose the Fahrenheit temperatureSuppose the Fahrenheit temperature xx
miles above the ground level is given bymiles above the ground level is given by
TT((xx) = 88 – 32) = 88 – 32 xx. Determine the altitudes where the. Determine the altitudes where the
air temp is from 30air temp is from 3000
to 40to 4000
..
• We must solve the inequalityWe must solve the inequality
30 <30 < 88 – 3288 – 32 xx < 40< 40
To solve:To solve: Isolate the variable xIsolate the variable x inin the middlethe middle of theof the
three-part inequalitythree-part inequality
28. 28Rev.S08
How to Solve Double Inequalities?
(Cont.)
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Direction reversed –Divided eachDirection reversed –Divided each
side of an inequality by a negativeside of an inequality by a negative
Thus, between 1.5 and 1.8215Thus, between 1.5 and 1.8215
miles above ground level,miles above ground level,
the air temperature isthe air temperature is
between 30 and 40 degreesbetween 30 and 40 degrees
Fahrenheit.Fahrenheit.
29. 29Rev.S08
How to Graph a System of Linear
Inequalities?
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The graph of a linear inequality is a half-plane, which
may include the boundary. The boundary line is included
when the inequality includes a less than or equal to or
greater than or equal to symbol.
To determine which part of the plane to shade, select a
test point.
30. 30Rev.S08
How to Graph a System of Linear
Inequalities? (Cont.)
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Graph the solution set to the inequality x + 4y > 4.
Solution
Graph the line x + 4y = 4 using a dashed line.
Use a test point to determine which half of the plane to
shade.
Test
Point
x + 4y > 4 True or
False?
(4, 2) 4 + 4(2) > 4 True
(0, 0) 0 + 4(0) > 4 False
31. 31Rev.S08
Example
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Solve the system of inequalities
by shading the solution set. Use
the graph to identify one solution.
x + y ≤ 3
2x + y ≥ 4
Solution
Solve each inequality for y.
y ≤ −x + 3 (shade below line)
y ≥ −2x + 4 (shade above line)
The point (4, −2) is a solution.
32. 32Rev.S08
What have we learned?
We have learned to:
1. Evaluate functions of two variables.
2. Apply the method of substitution.
3. Apply the elimination method.
4. Solve system of equations symbolically.
5. Apply graphical and numerical methods to system of
equations.
6. Recognize different types of linear systems.
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Click link to download other modules.
33. 33Rev.S08
What have we learned? (Cont.)
7. Use basic terminology related to inequalities.
8. Use interval notation.
9. Solve linear inequalities symbolically.
10. Solve linear inequalities graphically and numerically.
11. Solve double inequalities.
12. Graph a system of linear inequalities.
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34. 34Rev.S08
Credit
Some of these slides have been adapted/modified in part/whole from the
slides of the following textbook:
• Rockswold, Gary, Precalculus with Modeling and Visualization, 3th
Edition
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