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The document discusses the LU factorization method for solving systems of linear equations. It provides an example of applying the Gauss elimination method to a system of 4 equations with 4 unknowns. This results in an upper triangular system that can be easily solved with back substitution. The multipliers used in the row operations are stored in the lower triangular matrix L, while the upper triangular matrix U contains the coefficients from Gauss elimination. The product of L and U yields the original coefficient matrix A, representing the LU factorization of A.

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IB Maths SL Matrices

A matrix is an ordered set of numbers listed in rectangular form. Matrix A is described as a 2x3 matrix, with 2 rows and 3 columns. B is described as a row matrix and C as a column matrix. Special matrices include the 3x3 zero matrix and the 3x3 identity matrix. Matrices can be added if they are the same order and multiplied by scalars. To multiply matrices, rows of the first are multiplied with columns of the second. The result of multiplying matrices is given a special name. The determinant of a matrix is a single number calculated from its elements, and determines properties like invertibility. For a 3x3 matrix to have an inverse, its determinant cannot be zero, making it non

Distance of a point from a line

This presentation continues with my series of videos on Straight Lines, coordinate geometry.
Here, we learn how to calculate distance of a point from a line and also distance between 2 parallel lines.
This is useful for grade 11 math students. Problems are explained in a simple and easy way.

Solve systemsbygraphing

This document discusses solving systems of equations by graphing. It defines a system of equations as two or more equations using the same variables, where the solution satisfies all equations. There are three possibilities when graphing systems: intersecting lines, where the intersection point is the solution; parallel lines, where there is no solution; and coinciding lines, where there are infinitely many solutions. It provides examples of each type of system and guides the student through solving sample systems by graphing and checking solutions.

Systems of linear equations; matrices

This document discusses Gauss-Jordan elimination for solving systems of linear equations. It begins by introducing the three possible cases for solutions: unique solution, no solution, or infinite solutions. It then provides an example of using Gauss-Jordan elimination to solve a 3x3 system. The steps involve transforming the augmented matrix into reduced row echelon form and then reading the solution variables from the final matrix. Applications of solving systems from word problems are also discussed.

9.6 Systems of Inequalities and Linear Programming

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.

Matrices and determinants

The document provides definitions and concepts related to matrices and determinants. It begins with definitions of matrices, operations on matrices like transpose and trace. It then discusses row echelon form, elementary row operations, and using matrices to represent systems of linear equations. The document will cover topics like inverse matrices, matrix rank and nullity, polynomials of matrices, properties of determinants, minors and cofactors, and Cramer's rule.

ppt of VCLA

This ppt will be useful to useful all the student who want to study about the matrix's basic knowledge.

Diagonalization of Matrices

The document discusses diagonalization of matrices. Diagonalization involves finding an invertible matrix P such that P-1AP is a diagonal matrix. The procedure for diagonalizing a matrix A includes: (1) finding the eigenvalues of A, (2) finding linearly independent eigenvectors corresponding to the eigenvalues, (3) forming the matrix P with the eigenvectors as columns, and (4) showing that P-1AP is a diagonal matrix with the eigenvalues along the diagonal. An example demonstrates finding the eigenvectors of a 2x2 matrix A and constructing the matrix P to diagonalize A.

Integrated Math 2 Section 8-5

This document provides an overview of matrices and determinants. It begins with essential questions about finding the determinant of a 2x2 matrix and using determinants to solve systems of equations. It then defines key terms like square matrix and provides examples of calculating the determinant of a 2x2 matrix. The document explains Cramer's Rule for solving systems of equations using determinants and provides a worked example of applying Cramer's Rule to solve a system of two equations with two unknowns. It concludes by assigning related homework problems.

Systems of linear equations and augmented matrices

This document discusses solving systems of linear equations using augmented matrices. It begins by defining a matrix and augmented matrix. An augmented matrix stores the coefficients and constants of a linear system. Row operations can be performed on the augmented matrix to put it in reduced row echelon form and solve the system. Several examples show how to set up and row reduce augmented matrices to solve systems. The document concludes by identifying the three possible final forms an augmented matrix can take, indicating the number of solutions to the corresponding system.

9.1 Systems of Linear Equations

This document discusses systems of linear equations and methods for solving them. It defines a linear system as a set of equations where all variables have an exponent of 1. There are three possibilities for a system: 1) a single solution, 2) no solution (inconsistent), or 3) infinitely many solutions. Four methods are presented for solving systems: substitution, elimination, graphing, and matrices. Examples are provided to illustrate substitution and elimination. The document also discusses how to determine if a system is inconsistent or has infinitely many solutions based on the outcome of solving the system.

presentation on matrix

This document defines and provides examples of different types of matrices:
- Matrices are arrangements of elements in rows and columns represented by symbols.
- Types include row matrices, column matrices, square matrices, null matrices, identity matrices, diagonal matrices, scalar matrices, triangular matrices, transpose matrices, symmetric matrices, skew matrices, equal matrices, and algebraic matrices.
- Algebraic matrix operations include addition, subtraction, and multiplication where the matrices must be of the same order.

Dmitrii Tihonkih - The Iterative Closest Points Algorithm and Affine Transfo...

This document describes modifications made to the iterative closest point (ICP) algorithm. The authors propose a new matching procedure that uses the angles between line segments connecting points to find initial correspondences. They also formulate the ICP variational problem for an arbitrary affine transformation. A computer simulation applies the standard ICP approach and the authors' algorithm to two point sets related by a known transformation, finding the latter estimates the transformation more accurately.

Matrices and System of Linear Equations ppt

The document discusses matrices and systems of linear equations. It defines matrices and different types of matrices including square, diagonal, scalar, identity, zero, negative, upper triangular, lower triangular, and transpose matrices. It also covers properties of matrix operations and examples of finding the transpose of matrices. The document then discusses row echelon form (REF) and reduced row echelon form (RREF) as well as the different types of solutions that systems of linear equations can have.

Bba i-bm-u-2- matrix -

The document discusses various types of matrices:
- Row and column matrices are matrices with only one row or column respectively.
- A square matrix has the same number of rows and columns.
- A diagonal matrix has non-zero elements only along its main diagonal.
- An identity matrix has ones along its main diagonal and zeros elsewhere.
- A scalar matrix has all elements along its main diagonal multiplied by a scalar.
- A null matrix has all elements equal to zero.
The document also discusses properties such as the transpose of a matrix, symmetric matrices, and how to add, subtract and multiply matrices.

system of non-linear equation (linear algebra & vector calculus)

1) The document discusses row echelon form, reduced row echelon form, and augmented matrices. It also covers systems of nonlinear equations and methods for solving them.
2) Examples are provided of using Gaussian elimination and Gaussian-Jordan elimination methods to solve a system of nonlinear equations for angles α, β, and γ.
3) The solution involves transforming the augmented matrix into reduced row echelon form, from which the values α = π/2, β = π, and γ = 0 can be determined.

Matrices 1

This document provides an overview of matrices and basic matrix operations. It discusses what matrices are, how to perform operations like addition, multiplication, and taking the transpose. It also covers special types of matrices like diagonal, triangular, and identity matrices. It explains how to calculate the determinant of a 2x2 matrix and find the inverse of a 2x2 matrix using the determinant. The goal is for the reader to understand matrices, common operations, and how to calculate the determinant and inverse of a 2x2 matrix after reviewing this material.

Introduction to Matrices

A matrix is a rectangular array of numbers arranged in rows and columns. The dimensions of a matrix are written as the number of rows x the number of columns. Each individual entry in the matrix is named by its position, using the matrix name and row and column numbers. Matrices can represent systems of equations or points in a plane. Operations on matrices include addition, multiplication by scalars, and dilation of points represented by matrices.

IB Maths SL Matrices

IB Maths SL Matrices

Distance of a point from a line

Distance of a point from a line

Solve systemsbygraphing

Solve systemsbygraphing

Systems of linear equations; matrices

Systems of linear equations; matrices

9.6 Systems of Inequalities and Linear Programming

9.6 Systems of Inequalities and Linear Programming

Matrices and determinants

Matrices and determinants

ppt of VCLA

ppt of VCLA

Diagonalization of Matrices

Diagonalization of Matrices

Integrated Math 2 Section 8-5

Integrated Math 2 Section 8-5

Systems of linear equations and augmented matrices

Systems of linear equations and augmented matrices

9.1 Systems of Linear Equations

9.1 Systems of Linear Equations

presentation on matrix

presentation on matrix

Dmitrii Tihonkih - The Iterative Closest Points Algorithm and Affine Transfo...

Dmitrii Tihonkih - The Iterative Closest Points Algorithm and Affine Transfo...

Matrices and System of Linear Equations ppt

Matrices and System of Linear Equations ppt

Bba i-bm-u-2- matrix -

Bba i-bm-u-2- matrix -

system of non-linear equation (linear algebra & vector calculus)

system of non-linear equation (linear algebra & vector calculus)

Matrices 1

Matrices 1

Matrices

Matrices

Introduction to Matrices

Introduction to Matrices

Iterative methods for the solution

The document discusses iterative methods for solving systems of linear equations, specifically the Jacobi and Gauss-Seidel methods. The Jacobi method updates each unknown using the previous values, while Gauss-Seidel uses the most recent values calculated in the current iteration. Both methods are demonstrated through examples. The computational cost of each iteration for Jacobi is 2n^2 FLOPs, where n is the number of equations. The total FLOPs increases linearly with the number of iterations.

Direct methods

The document discusses the LU factorization method for solving systems of linear equations. It provides an example of applying the Gauss elimination method to a system of 4 equations with 4 unknowns. This results in an upper triangular system that can be easily solved with back substitution. The multipliers used in the row operations are stored in the lower triangular matrix L, while the upper triangular matrix U contains the coefficients from Gauss elimination. The product of L and U yields the original coefficient matrix A, representing the LU factorization of A.

Iterative methods for the solution

Oscar Eduardo Mendivelso Orozco studies Petroleum Engineering Numerical Methods. He is a student studying petroleum engineering with a focus on numerical methods. His studies involve the application of numerical techniques to solve problems in petroleum engineering.

Iteration

Iterative structures, also known as loops, repeat sections of code and are used for tasks like calculating multiple values, computing iterative results, printing tables of data, and processing large amounts of input or array data. The three types of loops in C++ are the while loop, do-while loop, and for loop, each with different test conditions to control the loop execution. Loops can also be nested within each other to perform multiple iterations or to loop through multi-dimensional data structures.

Iterative methods for the solution

The document discusses iterative methods for solving systems of linear equations, specifically the Jacobi and Gauss-Seidel methods. The Jacobi method updates each unknown using the previous values, while Gauss-Seidel uses the most recent values calculated in the current iteration. Both methods are demonstrated through examples. The computational cost of each iteration for Jacobi is 2n^2 FLOPs, where n is the number of equations. The total FLOPs for m iterations is 2mn^2.

NUMERICAL METHODS -Iterative methods(indirect method)

The document discusses two iterative methods for solving systems of linear equations: Gauss-Jacobi and Gauss-Seidel. Gauss-Jacobi solves each equation separately using the most recent approximations for the other variables. Gauss-Seidel updates each variable with the most recent values available. The document provides an example applying both methods to solve a system of three equations. Gauss-Seidel converges faster, requiring fewer iterations than Gauss-Jacobi to achieve the same accuracy. Both methods are useful alternatives to direct methods like Gaussian elimination when round-off errors are a concern.

Succession “Losers”: What Happens to Executives Passed Over for the CEO Job?

Succession “Losers”: What Happens to Executives Passed Over for the CEO Job? Stanford GSB Corporate Governance Research Initiative

This document summarizes a study of CEO succession events among the largest 100 U.S. corporations between 2005-2015. The study analyzed executives who were passed over for the CEO role ("succession losers") and their subsequent careers. It found that 74% of passed over executives left their companies, with 30% eventually becoming CEOs elsewhere. However, companies led by succession losers saw average stock price declines of 13% over 3 years, compared to gains for companies whose CEO selections remained unchanged. The findings suggest that boards generally identify the most qualified CEO candidates, though differences between internal and external hires complicate comparisons.Iterative methods for the solution

Iterative methods for the solution

Direct methods

Direct methods

Iterative methods for the solution

Iterative methods for the solution

Iteration

Iteration

Iterative methods for the solution

Iterative methods for the solution

NUMERICAL METHODS -Iterative methods(indirect method)

NUMERICAL METHODS -Iterative methods(indirect method)

Succession “Losers”: What Happens to Executives Passed Over for the CEO Job?

Succession “Losers”: What Happens to Executives Passed Over for the CEO Job?

Linear equations

This document provides information about solving systems of linear equations through various methods such as graphing, substitution, and elimination. It defines what a linear system is and explains the concepts of consistent and inconsistent systems. Graphing is discussed as a way to find the point where two lines intersect. The substitution and elimination methods are described step-by-step with examples shown of using each method to solve sample systems of equations. Additional topics covered include slope, matrix notation, and an example of using a matrix to perform a Hill cipher encryption on a short plaintext message.

Some methods for small systems of equations solutions

This document discusses several methods for solving systems of linear equations:
- The graphical method involves drawing the lines defined by the equations on a graph and finding their point of intersection.
- Cramer's rule provides an expression to find the solution using determinants of the coefficient matrix and matrices obtained by replacing columns.
- Matrix inverse involves finding the inverse of the coefficient matrix and multiplying it by the constants vector.
- Gauss elimination is a two step method involving eliminating variables in the forward step and back substitution to find the solution.
- LU decomposition writes the matrix as the product of a lower and upper triangular matrix to solve the system.

System of equations

This document discusses different methods for solving systems of linear equations, including Cramer's rule, elimination methods, and Gaussian elimination. Cramer's rule uses determinants to find the values of variables by dividing the determinant of the coefficients by the primary determinant. Elimination methods remove one unknown using row operations like addition and subtraction. Gaussian elimination transforms the coefficient matrix into triangular form using row operations, then back substitution can find the unique solution.

System of equations

This document discusses different methods for solving systems of linear equations, including Cramer's rule, elimination methods, and Gaussian elimination. Cramer's rule uses determinants to find the values of variables by dividing the determinant of the coefficients by the primary determinant. Elimination methods remove one unknown using multiplication and addition of equations. Gaussian elimination transforms the coefficient matrix into row echelon form through elementary row operations to solve the system.

System of equations

This document discusses different methods for solving systems of linear equations, including Cramer's rule, elimination methods, Gaussian elimination, and Gauss-Jordan elimination. It provides an example of using Cramer's rule to solve a 2x2 system of equations, resulting in solutions of x=2 and y=1. It also gives a step-by-step example of using Gaussian elimination to determine that a given 2x2 system has no solution.

System of equations

This document discusses different methods for solving systems of linear equations, including Cramer's rule, elimination methods, and Gaussian elimination. Cramer's rule uses determinants to find the values of variables by dividing the determinant of the coefficients by the primary determinant. Elimination methods remove one unknown using multiplication and addition of equations. Gaussian elimination transforms the coefficient matrix into triangular form using row operations, then back substitution can find the unique solution.

System of equations

This document discusses different methods for solving systems of linear equations, including Cramer's rule, elimination methods, and Gaussian elimination. Cramer's rule uses determinants to find the values of variables by dividing the determinant of the coefficients by the primary determinant. Elimination methods remove one unknown using addition or subtraction of equations. Gaussian elimination transforms the coefficient matrix into row echelon form through elementary row operations to solve the system. The document provides examples of applying each method to solve sample systems of linear equations.

Gauss Jordan

This document discusses the Gauss-Jordan elimination method for solving systems of linear equations. It provides biographical information on Gauss and Jordan, who developed the method. It then explains the Gauss-Jordan elimination process, provides examples of solving systems of equations using the method, and discusses applications to mathematical modeling.

Roots of polynomials

The document discusses methods for finding the roots of polynomial equations, including Muller's method and Bairstow's method. Muller's method uses three points to derive the coefficients of a parabola and find an approximated root. Bairstow's method involves synthetically dividing a polynomial by a quadratic factor to find values of r and s that make the coefficients b1 and b0 equal to zero, through an iterative process. It provides an example of applying Bairstow's method to find the roots of a 5th order polynomial.

Chapter 4: Linear Algebraic Equations

1. This document discusses methods for solving linear algebraic equations and operations involving matrices. It covers topics such as matrix definitions, types of matrices, matrix operations, representing equations in matrix form, and methods for solving systems of linear equations including graphical methods, determinants, Cramer's rule, elimination, Gauss-Jordan, LU decomposition, and calculating the matrix inverse.
2. Key matrix operations include addition, multiplication, and rules for inverting a matrix. Methods for solving systems of equations include graphical techniques, determinants, Cramer's rule, elimination, Gauss, Gauss-Jordan, and LU decomposition.
3. LU decomposition involves writing a matrix as the product of a lower and upper triangular matrix, which can

Solution of equations for methods iterativos

This document discusses iterative methods for solving systems of equations, including Jacobi, Gauss-Seidel, and Gauss-Seidel relaxation methods. Iterative methods progressively calculate approximations to the solution, unlike direct methods which require completing the full process to obtain the answer. The Jacobi method can solve simple square systems of equations in an iterative fashion. Gauss-Seidel is also an iterative technique that sequentially solves for each unknown using previous approximations. Gauss-Seidel relaxation is similar but incorporates a relaxation parameter. Examples demonstrate applying these methods to solve systems of equations.

Solution of equations for methods iterativos

This document discusses iterative methods for solving systems of equations, including Jacobi, Gauss-Seidel, and Gauss-Seidel relaxation methods. Iterative methods progressively calculate approximations to the solution, unlike direct methods which require completing the full process to obtain the answer. The Jacobi method is used to solve simple square systems of equations. Gauss-Seidel is an iterative technique that solves systems of linear equations by computing updated solutions sequentially using forward substitution. Gauss-Seidel relaxation is similar but incorporates a relaxation parameter. Examples demonstrate applying these methods over multiple iterations to solve systems.

Solution of equations for methods iterativos

This document discusses iterative methods for solving systems of equations. It describes the Jacobi method, which solves systems by iteratively updating solutions. It also describes Gauss-Seidel method, which improves on Jacobi by using previous updated solutions in the current iteration. Both methods are used to progressively calculate better approximations to the solution until reaching an acceptable level of accuracy.

Chapter 3: Linear Systems and Matrices - Part 1/Slides

The document provides information about linear systems and matrices. It begins by defining linear and non-linear equations. It then discusses systems of linear equations, their graphical and geometric interpretations, and the three possible solutions: no solution, a unique solution, or infinitely many solutions. The document also covers matrix notation for representing linear systems, elementary row operations for transforming systems, and determining whether a system has a solution and whether that solution is unique.

Gauss elimination

Gaussian elimination is a method for solving systems of linear equations consisting of two steps:
1) Forward elimination transforms the coefficient matrix into an upper triangular matrix by eliminating variables from lower-numbered equations. This is done by subtracting appropriate multiples of pivot equations from other equations.
2) Back substitution solves the upper triangular system of equations by substituting the solution of higher-numbered equations into lower ones and solving for each variable sequentially, starting from the last equation.

The geometry of three planes in space

This document discusses using systems of linear equations and matrices to represent and find the intersection of planes in three-dimensional space. It provides examples of using the inverse matrix method and reduced row echelon form (RREF) method to solve systems of 2 and 3 planes. The RREF method can find lines of intersection even when planes do not intersect at a single point, and can reveal when planes share a common line of intersection.

Linear Equations

- A linear system includes two or more equations with two or more variables. When two equations are used to model a problem, it is called a linear system.
- Common methods to solve linear systems include graphing the equations to find their intersection point, substitution where one variable is solved for in one equation and substituted into the other, and elimination where equations are combined by multiplication to eliminate a variable.
- The Hill cipher is a method to encrypt plaintext messages by performing matrix multiplication on the message represented as numbers with an encryption key matrix.

Iterativos methods

This document discusses iterative methods for solving systems of equations, including the Jacobi and Gauss-Seidel methods. The Jacobi method solves systems of equations by iteratively updating the solution variables. The Gauss-Seidel method similarly iteratively solves systems but updates the variables in a specific sequential order for increased convergence. Examples are provided of applying both methods through multiple iterations to arrive at solutions. Relaxation is also introduced as a variation of Gauss-Seidel.

Iterativos Methods

This document discusses iterative methods for solving systems of equations, including the Jacobi and Gauss-Seidel methods. The Jacobi method solves systems of equations by iteratively updating the estimates of the unknown variables. The Gauss-Seidel method similarly iteratively solves systems but updates the estimates sequentially from left to right. Examples applying both methods to solve systems are provided.

System of linear equations 2 eso

This document discusses systems of linear equations and methods for solving them. It defines a system of linear equations as two equations with two unknowns where the solution is a pair of numbers that satisfies both equations. Three methods for solving systems are presented: substitution, where one equation is used to solve for one unknown and substitutes it into the other; matching, where one unknown is solved for in both equations and set equal; and elimination, where the equations are manipulated to eliminate one unknown. The document also defines the three possible types of solutions to a system: a consistent independent system with a single solution, a consistent dependent system with infinite solutions, and an inconsistent system with no solution.

Linear equations

Linear equations

Some methods for small systems of equations solutions

Some methods for small systems of equations solutions

System of equations

System of equations

System of equations

System of equations

System of equations

System of equations

System of equations

System of equations

System of equations

System of equations

Gauss Jordan

Gauss Jordan

Roots of polynomials

Roots of polynomials

Chapter 4: Linear Algebraic Equations

Chapter 4: Linear Algebraic Equations

Solution of equations for methods iterativos

Solution of equations for methods iterativos

Solution of equations for methods iterativos

Solution of equations for methods iterativos

Solution of equations for methods iterativos

Solution of equations for methods iterativos

Chapter 3: Linear Systems and Matrices - Part 1/Slides

Chapter 3: Linear Systems and Matrices - Part 1/Slides

Gauss elimination

Gauss elimination

The geometry of three planes in space

The geometry of three planes in space

Linear Equations

Linear Equations

Iterativos methods

Iterativos methods

Iterativos Methods

Iterativos Methods

System of linear equations 2 eso

System of linear equations 2 eso

- 1. OSCAR EDUARDO MENDIVELSO OROZCO I study Petroleum Engineering Numerical Methods
- 2. INTRODUCTION The problem to be solved in this chapter is that of a system of m linear equations with n unknowns in a matrix defined by: Where A ∈ IR (M, n) and b ∈ IR m are data and x ∈ IR n is the vector incognita. Can be explicitly written as: (1) Usually assume that m = n and that the system has only solution, ie det A =0, or, rg A = n.
- 3. LU FACTORIZATION METHOD The easiest way to explain the LU method is illustrating the basic Gauss method through an example, as is the case of the given matrix and then applying the procedure to a system of four equations with four unknowns:
- 4. In the first step, we multiply the first equation by 12 / 6 = 2 and subtract the second, then multiply the first equation by 3 / 6 = 1 / 2 and subtract the third and finally multiply the first equation by -6 / 6 =- 1 and subtract the fourth. The numbers 2, ½ and -1 are the multipliers of the first step in the process of elimination. Number 6 is the pivotal element of this first step and the first row, that remains unchanged is called the pivot row. The system now looks like this:
- 5. In the next step of the process, the second row is used as pivot row and -4 as the new pivot element we apply the process: multiply the second row by - 12/-4 = 3 and the remainder of the third and then multiply the second row 2 / (-4) = - 1/2 and subtract the fourth. The multipliers are in this case 3 and -1/2 and the system of equations reduces to:
- 6. The last step is to multiply the third equation by 4 / 2 = 2 and subtract the fourth. The resulting system turns out to be: The resulting system is upper triangular and equivalent to the original system (the solutions of both systems overlap.) However, this system is easily solvable by applying the backward substitution algorithm. The solution of the system of equations turns out to be:
- 7. If we put the multiplier used to transform the system into a unit lower triangular matrix (L) occupying each position of helping to produce zero, we obtain the following matrix: Moreover, the upper triangular matrix (U) formed by the coefficients resulting after applying the Gauss algorithm (2) is:
- 8. These two matrices give us the LU factorization of the initial matrix of coefficients, A, expressed by equation (1):