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.
Gauss jordan and Guass elimination methodMeet Nayak
This ppt is based on engineering maths.
the topis is Gauss jordan and gauss elimination method.
This ppt having one example of both method and having algorithm.
The document discusses vector spaces and related linear algebra concepts. It defines vector spaces and lists the axioms that must be satisfied. Examples of vector spaces include the set of all pairs of real numbers and the space of 2x2 symmetric matrices. The document also discusses subspaces, linear combinations, span, basis, dimension, row space, column space, null space, rank, nullity, and change of basis. It provides examples and explanations of these fundamental linear algebra topics.
The velocity of a vector function is the absolute value of its tangent vector. The speed of a vector function is the length of its velocity vector, and the arc length (distance traveled) is the integral of speed.
This chapter covers two-dimensional (2D) plots in MATLAB. It discusses various plot commands like plot, fplot, and errorbar that can be used to create basic and specialized 2D plots. It also describes how to format plots by adding labels, titles, legends, text annotations and grids. Plots with logarithmic and polar axes as well as histograms are demonstrated. The chapter shows how to create multiple plots on the same figure using subplots or in separate figure windows.
The document discusses functions with more than one independent variable and how to represent them graphically. It introduces the concept of level curves, which are lines connecting points where the function z has a constant value. Level curves are similar to contour lines on a topographic map and can be drawn to sketch the graph of a multi-variable function. The document provides an example of a two-variable function and shows its level curves. It also discusses how to view such graphs using a graphing calculator.
The document discusses methods for solving systems of linear equations in two variables, including graphing the lines defined by each equation to find their point of intersection, the elimination method of adding multiples of equations to remove a variable, and the substitution method of solving one equation for one variable and substituting it into the other equation. It provides examples of applying these different solution methods to solve systems of linear equations and determining if a solution exists.
Gauss Elimination & Gauss Jordan Methods in Numerical & Statistical MethodsJanki Shah
The document discusses methods for solving systems of linear equations. It introduces Gauss elimination and Gauss Jordan methods. Gauss elimination transforms the augmented matrix of the system into row echelon form through elementary row operations, then back-substitutes to solve for the variables. Gauss Jordan additionally transforms the matrix to reduced row echelon form to read solutions directly from the matrix. An example demonstrates applying each method to solve a system of equations.
Gauss jordan and Guass elimination methodMeet Nayak
This ppt is based on engineering maths.
the topis is Gauss jordan and gauss elimination method.
This ppt having one example of both method and having algorithm.
The document discusses vector spaces and related linear algebra concepts. It defines vector spaces and lists the axioms that must be satisfied. Examples of vector spaces include the set of all pairs of real numbers and the space of 2x2 symmetric matrices. The document also discusses subspaces, linear combinations, span, basis, dimension, row space, column space, null space, rank, nullity, and change of basis. It provides examples and explanations of these fundamental linear algebra topics.
The velocity of a vector function is the absolute value of its tangent vector. The speed of a vector function is the length of its velocity vector, and the arc length (distance traveled) is the integral of speed.
This chapter covers two-dimensional (2D) plots in MATLAB. It discusses various plot commands like plot, fplot, and errorbar that can be used to create basic and specialized 2D plots. It also describes how to format plots by adding labels, titles, legends, text annotations and grids. Plots with logarithmic and polar axes as well as histograms are demonstrated. The chapter shows how to create multiple plots on the same figure using subplots or in separate figure windows.
The document discusses functions with more than one independent variable and how to represent them graphically. It introduces the concept of level curves, which are lines connecting points where the function z has a constant value. Level curves are similar to contour lines on a topographic map and can be drawn to sketch the graph of a multi-variable function. The document provides an example of a two-variable function and shows its level curves. It also discusses how to view such graphs using a graphing calculator.
The document discusses methods for solving systems of linear equations in two variables, including graphing the lines defined by each equation to find their point of intersection, the elimination method of adding multiples of equations to remove a variable, and the substitution method of solving one equation for one variable and substituting it into the other equation. It provides examples of applying these different solution methods to solve systems of linear equations and determining if a solution exists.
Gauss Elimination & Gauss Jordan Methods in Numerical & Statistical MethodsJanki Shah
The document discusses methods for solving systems of linear equations. It introduces Gauss elimination and Gauss Jordan methods. Gauss elimination transforms the augmented matrix of the system into row echelon form through elementary row operations, then back-substitutes to solve for the variables. Gauss Jordan additionally transforms the matrix to reduced row echelon form to read solutions directly from the matrix. An example demonstrates applying each method to solve a system of equations.
First order linear differential equationNofal Umair
1. A differential equation relates an unknown function and its derivatives, and can be ordinary (involving one variable) or partial (involving partial derivatives).
2. Linear differential equations have dependent variables and derivatives that are of degree one, and coefficients that do not depend on the dependent variable.
3. Common methods for solving first-order linear differential equations include separation of variables, homogeneous equations, and exact equations.
1. Quiz 4 will cover sections 3.3, 5.1, and 5.2 and will be on Thursday, February 18.
2. To find the nth power of a matrix A that has been diagonalized as A = PDP-1, one raises the diagonal elements of D to the nth power to obtain Dn, leaving P and P-1 unchanged.
3. A matrix is diagonalizable if and only if it has n linearly independent eigenvectors, allowing it to be written as A = PDP-1, where the columns of P are the eigenvectors and the diagonal elements of D are the corresponding eigenvalues.
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 describes the cofactor method for finding the inverse of a matrix A. It defines the cofactor Cij as the signed determinant of the matrix made by removing row i and column j from A. The inverse is then given by the transpose of the matrix of cofactors divided by the determinant of A. An example calculates the inverse of the 2x2 matrix A = [a, c; b, d] using this method.
This is your introduction to domain, range, and functions. You will learn more about domain, range, functions, relations, x-values, and y-values. There are definitions and explanations of each concepts. There are questions to help quiz yourself. Test your abilities. Enjoy.
This document contains one mark questions on matrices. It defines key matrix terms like scalar matrix, identity matrix, diagonal matrix. It also contains questions on finding the order and elements of a given matrix, the number of possible matrices with given conditions, constructing matrices based on given elements and properties. Questions also involve matrix operations like addition, multiplication and solving equations involving matrices.
1) Graphing quadratic equations in vertex form involves identifying the vertex (a, h) and the axis of symmetry (x = h) from the equation.
2) The document provides examples of quadratic equations in vertex form and describes the transformations between the graphs and the standard form equations.
3) Key aspects like the vertex, axis of symmetry, direction of opening, and roots are identified and the transformations including stretches, shifts, reflections and compressions are described.
The document discusses the elimination method for solving systems of equations. It provides examples of using addition or multiplication to eliminate a variable and obtain a solution. Key steps include adding or multiplying the equations appropriately, substituting values back to solve for the remaining variable, and obtaining the solution point (x,y).
This document provides a lesson on matrix inverses and solving systems of equations using inverse matrices. It begins with examples of determining whether two matrices are inverses of each other and finding the inverse of a given matrix. It then explains how to use inverse matrices to solve systems of equations by writing the system as a matrix equation and multiplying both sides by the inverse. Examples are provided to demonstrate solving systems using inverse matrices and decoding encoded messages using a given encoding matrix.
Please go through the slides. It is very interesting way to learn this chapter for 2020-21.If you like this PPT please put a thanks message in my number 9826371828.
This document provides information about vector spaces and subspaces. It defines a vector space as a set of objects called vectors that can be added together and multiplied by scalars, subject to certain rules. A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. The null space of a matrix is the set of solutions to the homogeneous equation Ax=0 and is a subspace. The column space of a matrix is the set of all linear combinations of its columns and is also a subspace. Examples are provided to illustrate these concepts.
Writing and Graphing slope intercept formguestd1dc2e
The document discusses graphing and writing equations in slope-intercept form (y=mx+b). It explains that the slope (m) is the coefficient and the y-intercept (b) is the constant. It provides examples of graphing lines by plotting the y-intercept and counting up/right by the slope. It also demonstrates writing equations given the slope and/or a point, including finding missing values like the y-intercept.
The document defines a subspace as a non-empty subset W of a vector space V that is itself a vector space under the operations defined on V. It notes that every vector space has at least two subspaces: itself and the zero subspace containing only the zero vector. To prove that W is a subspace of V, we only need to verify that W is closed under the vector space operations. Examples are provided to illustrate this, such as showing that the set W={(x,0,0)| x in R} is a subspace of R3 by verifying it is closed under vector addition and scalar multiplication.
This document provides instructions and examples for solving various types of logarithmic and exponential equations. It explains how to change between logarithmic and exponential forms, use the change of base formula, make the bases the same, use properties of logarithms and exponents, and solve compound interest problems using the appropriate formulas. Examples are provided for each type of problem to demonstrate the steps to take.
The document discusses inverse matrices. It begins by explaining that the inverse of a nonzero number a is its reciprocal 1/a. It then states that the inverse of a matrix A, written as A-1, is a matrix such that AA-1 = A-1A = I, where I is the identity matrix. However, not all matrices have inverses. There are two types of matrices regarding inverses: invertible matrices that have inverses, and noninvertible matrices that do not. The document provides an example of finding the inverse of a matrix and discusses when a matrix may not have an inverse.
This document discusses linear independence, basis, and dimension in linear algebra. It defines linear independence as vectors being linearly independent if the only solution that produces the zero vector is the trivial solution with all coefficients equal to zero. A basis is defined as a set of linearly independent vectors that span the vector space. The dimension of a vector space is the number of vectors in any basis of that space. The dimensions of the four fundamental subspaces (row space, column space, nullspace, and left nullspace) of a matrix are defined in terms of the rank of the matrix.
The document discusses solving literal equations by isolating variables. It defines literal equations as equations with more than one variable. The rules for solving literal equations are: 1) simplify each side if needed, 2) move the variable being solved for to one side using the opposite operation, 3) isolate the variable being solved for by applying the opposite operation to each side. Examples are provided of solving for different variables in equations and formulas. Practice problems are given at the end to solve for specific variables.
6.2 special cases system of linear equationsmath260
The document discusses special cases of systems of linear equations, including inconsistent/contradictory systems that have no solution, dependent systems that have infinitely many solutions, and the process of putting a system's augmented matrix into row-reduced echelon form (rref-form) to identify which type it is. It provides examples of an inconsistent system with equations x+y=2 and x+y=3, and a dependent system with equations x+y=2 and 2x+2y=4.
The document presents information on partial differentiation including:
- Partial differentiation involves a function with more than one independent variable and partial derivatives.
- Notation for partial derivatives is presented.
- Methods for computing first and higher order partial derivatives are explained with examples.
- The concepts of homogeneous functions and the chain rule for partial differentiation are defined.
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.
1 Part 2 Systems of Equations Which Do Not Have A Uni.docxeugeniadean34240
1
Part 2: Systems of Equations Which Do Not Have A Unique
Solution
On the previous pages we learned how to solve systems of equations using Gaussian
elimination. In each of the examples and exercises of part 1(except for exercise 1 parts d and e)
the systems of equations had a unique solution. That is, a single value for each of the variables.
In example 3 we found the solution to be 7 23 3, . This means that the graphs of the two lines in
example 3 intersect at this unique point. In 2-space, the xy-plane, we have the geometric bonus
of being able to draw a picture of the solutions to a system of two equations two unknowns.
Clearly, if we were asked to draw the graphs of two lines in the xy-plane we have 3 basic
choices/cases:
1. Draw the two lines so they intersect. This point of intersection can only happen once for
a given pair of lines. That is, the two lines intersect in a unique point. There is a unique
common solution to the system of equations. Discussed in part 1.
2. Draw the two lines so that one is on "top of" the other. In this case there are an infinite
number of common points, an infinite number of solutions to the given system. Discussed
in part 2.
3. Draw two parallel lines. In this case there are no points common to both lines. There is
no solution to the system of equations that describe the lines. Discussed in part 2.
The 3 cases above apply to any system of equations.
Theorem 1. For any system of m equations with n unknowns (m < n) one of the following cases
applies:
1. There is a unique solution to the system.
2. There is an infinite number of solutions to the system.
3. There are no solutions to the system.
Again, in this section of the notes we will illustrate cases 2 and 3. To solve systems of
equations where these cases apply we use the matrix procedure developed previously.
Example 6. Solve the system
x + 2y = 1
2x + 4y = 2
2
It is probably already clear to the reader that the second equation is really the first in
disguise. (Simply divide both sides of the second equation by 2 to obtain the first). So if we
were to draw the graph of both we would obtain the same line, hence have an infinite number of
points common to both lines, an infinite number of solutions. However it would be helpful in
solving other systems where the solutions may not be so apparent to do the problem
algebraically, using matrices. The matrix of the system with its simplification follows. Recall,
we try to express the matrix
1 2 1
2 4 2
in the form 1
2
1 0
0 1
b
b
from which we can read off the
solution. However after one step we note that
1 2 1
2 4 2
1 22 R R
1 2 1
0 0 0
. It should be clear to the reader that no matter what further
elementary row operations we perform on the matrix
1 2 1
0 0 0
we cannot change it to the form
we hoped for, namel.
First order linear differential equationNofal Umair
1. A differential equation relates an unknown function and its derivatives, and can be ordinary (involving one variable) or partial (involving partial derivatives).
2. Linear differential equations have dependent variables and derivatives that are of degree one, and coefficients that do not depend on the dependent variable.
3. Common methods for solving first-order linear differential equations include separation of variables, homogeneous equations, and exact equations.
1. Quiz 4 will cover sections 3.3, 5.1, and 5.2 and will be on Thursday, February 18.
2. To find the nth power of a matrix A that has been diagonalized as A = PDP-1, one raises the diagonal elements of D to the nth power to obtain Dn, leaving P and P-1 unchanged.
3. A matrix is diagonalizable if and only if it has n linearly independent eigenvectors, allowing it to be written as A = PDP-1, where the columns of P are the eigenvectors and the diagonal elements of D are the corresponding eigenvalues.
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 describes the cofactor method for finding the inverse of a matrix A. It defines the cofactor Cij as the signed determinant of the matrix made by removing row i and column j from A. The inverse is then given by the transpose of the matrix of cofactors divided by the determinant of A. An example calculates the inverse of the 2x2 matrix A = [a, c; b, d] using this method.
This is your introduction to domain, range, and functions. You will learn more about domain, range, functions, relations, x-values, and y-values. There are definitions and explanations of each concepts. There are questions to help quiz yourself. Test your abilities. Enjoy.
This document contains one mark questions on matrices. It defines key matrix terms like scalar matrix, identity matrix, diagonal matrix. It also contains questions on finding the order and elements of a given matrix, the number of possible matrices with given conditions, constructing matrices based on given elements and properties. Questions also involve matrix operations like addition, multiplication and solving equations involving matrices.
1) Graphing quadratic equations in vertex form involves identifying the vertex (a, h) and the axis of symmetry (x = h) from the equation.
2) The document provides examples of quadratic equations in vertex form and describes the transformations between the graphs and the standard form equations.
3) Key aspects like the vertex, axis of symmetry, direction of opening, and roots are identified and the transformations including stretches, shifts, reflections and compressions are described.
The document discusses the elimination method for solving systems of equations. It provides examples of using addition or multiplication to eliminate a variable and obtain a solution. Key steps include adding or multiplying the equations appropriately, substituting values back to solve for the remaining variable, and obtaining the solution point (x,y).
This document provides a lesson on matrix inverses and solving systems of equations using inverse matrices. It begins with examples of determining whether two matrices are inverses of each other and finding the inverse of a given matrix. It then explains how to use inverse matrices to solve systems of equations by writing the system as a matrix equation and multiplying both sides by the inverse. Examples are provided to demonstrate solving systems using inverse matrices and decoding encoded messages using a given encoding matrix.
Please go through the slides. It is very interesting way to learn this chapter for 2020-21.If you like this PPT please put a thanks message in my number 9826371828.
This document provides information about vector spaces and subspaces. It defines a vector space as a set of objects called vectors that can be added together and multiplied by scalars, subject to certain rules. A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. The null space of a matrix is the set of solutions to the homogeneous equation Ax=0 and is a subspace. The column space of a matrix is the set of all linear combinations of its columns and is also a subspace. Examples are provided to illustrate these concepts.
Writing and Graphing slope intercept formguestd1dc2e
The document discusses graphing and writing equations in slope-intercept form (y=mx+b). It explains that the slope (m) is the coefficient and the y-intercept (b) is the constant. It provides examples of graphing lines by plotting the y-intercept and counting up/right by the slope. It also demonstrates writing equations given the slope and/or a point, including finding missing values like the y-intercept.
The document defines a subspace as a non-empty subset W of a vector space V that is itself a vector space under the operations defined on V. It notes that every vector space has at least two subspaces: itself and the zero subspace containing only the zero vector. To prove that W is a subspace of V, we only need to verify that W is closed under the vector space operations. Examples are provided to illustrate this, such as showing that the set W={(x,0,0)| x in R} is a subspace of R3 by verifying it is closed under vector addition and scalar multiplication.
This document provides instructions and examples for solving various types of logarithmic and exponential equations. It explains how to change between logarithmic and exponential forms, use the change of base formula, make the bases the same, use properties of logarithms and exponents, and solve compound interest problems using the appropriate formulas. Examples are provided for each type of problem to demonstrate the steps to take.
The document discusses inverse matrices. It begins by explaining that the inverse of a nonzero number a is its reciprocal 1/a. It then states that the inverse of a matrix A, written as A-1, is a matrix such that AA-1 = A-1A = I, where I is the identity matrix. However, not all matrices have inverses. There are two types of matrices regarding inverses: invertible matrices that have inverses, and noninvertible matrices that do not. The document provides an example of finding the inverse of a matrix and discusses when a matrix may not have an inverse.
This document discusses linear independence, basis, and dimension in linear algebra. It defines linear independence as vectors being linearly independent if the only solution that produces the zero vector is the trivial solution with all coefficients equal to zero. A basis is defined as a set of linearly independent vectors that span the vector space. The dimension of a vector space is the number of vectors in any basis of that space. The dimensions of the four fundamental subspaces (row space, column space, nullspace, and left nullspace) of a matrix are defined in terms of the rank of the matrix.
The document discusses solving literal equations by isolating variables. It defines literal equations as equations with more than one variable. The rules for solving literal equations are: 1) simplify each side if needed, 2) move the variable being solved for to one side using the opposite operation, 3) isolate the variable being solved for by applying the opposite operation to each side. Examples are provided of solving for different variables in equations and formulas. Practice problems are given at the end to solve for specific variables.
6.2 special cases system of linear equationsmath260
The document discusses special cases of systems of linear equations, including inconsistent/contradictory systems that have no solution, dependent systems that have infinitely many solutions, and the process of putting a system's augmented matrix into row-reduced echelon form (rref-form) to identify which type it is. It provides examples of an inconsistent system with equations x+y=2 and x+y=3, and a dependent system with equations x+y=2 and 2x+2y=4.
The document presents information on partial differentiation including:
- Partial differentiation involves a function with more than one independent variable and partial derivatives.
- Notation for partial derivatives is presented.
- Methods for computing first and higher order partial derivatives are explained with examples.
- The concepts of homogeneous functions and the chain rule for partial differentiation are defined.
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.
1 Part 2 Systems of Equations Which Do Not Have A Uni.docxeugeniadean34240
1
Part 2: Systems of Equations Which Do Not Have A Unique
Solution
On the previous pages we learned how to solve systems of equations using Gaussian
elimination. In each of the examples and exercises of part 1(except for exercise 1 parts d and e)
the systems of equations had a unique solution. That is, a single value for each of the variables.
In example 3 we found the solution to be 7 23 3, . This means that the graphs of the two lines in
example 3 intersect at this unique point. In 2-space, the xy-plane, we have the geometric bonus
of being able to draw a picture of the solutions to a system of two equations two unknowns.
Clearly, if we were asked to draw the graphs of two lines in the xy-plane we have 3 basic
choices/cases:
1. Draw the two lines so they intersect. This point of intersection can only happen once for
a given pair of lines. That is, the two lines intersect in a unique point. There is a unique
common solution to the system of equations. Discussed in part 1.
2. Draw the two lines so that one is on "top of" the other. In this case there are an infinite
number of common points, an infinite number of solutions to the given system. Discussed
in part 2.
3. Draw two parallel lines. In this case there are no points common to both lines. There is
no solution to the system of equations that describe the lines. Discussed in part 2.
The 3 cases above apply to any system of equations.
Theorem 1. For any system of m equations with n unknowns (m < n) one of the following cases
applies:
1. There is a unique solution to the system.
2. There is an infinite number of solutions to the system.
3. There are no solutions to the system.
Again, in this section of the notes we will illustrate cases 2 and 3. To solve systems of
equations where these cases apply we use the matrix procedure developed previously.
Example 6. Solve the system
x + 2y = 1
2x + 4y = 2
2
It is probably already clear to the reader that the second equation is really the first in
disguise. (Simply divide both sides of the second equation by 2 to obtain the first). So if we
were to draw the graph of both we would obtain the same line, hence have an infinite number of
points common to both lines, an infinite number of solutions. However it would be helpful in
solving other systems where the solutions may not be so apparent to do the problem
algebraically, using matrices. The matrix of the system with its simplification follows. Recall,
we try to express the matrix
1 2 1
2 4 2
in the form 1
2
1 0
0 1
b
b
from which we can read off the
solution. However after one step we note that
1 2 1
2 4 2
1 22 R R
1 2 1
0 0 0
. It should be clear to the reader that no matter what further
elementary row operations we perform on the matrix
1 2 1
0 0 0
we cannot change it to the form
we hoped for, namel.
1. Thomas' algorithm, also called the tridiagonal matrix algorithm (TDMA), is used to solve systems of equations with a tridiagonal matrix structure.
2. It works by applying Gaussian elimination to convert the system of equations into an upper triangular system that can then be solved using backward substitution.
3. An example problem demonstrating the algorithm is worked through, showing the conversion to upper triangular form and solution via backward substitution.
This document provides an overview of matrices and determinants from a college algebra textbook. It begins with a chapter overview stating that a matrix is a rectangular array of numbers used to organize information. It then discusses representing a linear system of equations as an augmented matrix, which contains the same information as the system in a simpler form. Elementary row operations that can be performed on the augmented matrix are introduced. Gaussian elimination, a method for solving linear systems using the row-echelon form of the augmented matrix and back-substitution, is also covered at a high level.
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.
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.
Chapter 3: Linear Systems and Matrices - Part 1/SlidesChaimae Baroudi
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.
This document discusses methods for solving systems of linear equations, including Gauss elimination and iterative Gauss methods like the Gauss-Jacobi and Gauss-Seidel methods. It provides explanations of row-echelon form and reduced row-echelon form. Examples are given to demonstrate using elementary row operations to solve systems of linear equations and the Gauss-Jacobi iterative method.
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.
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.
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.
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.
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.
The document discusses linear systems of equations and their solutions. It begins by defining key terms like echelon form, reduced row echelon form, and the rank of a matrix. It then explains how to use Cramer's rule and Gaussian elimination to determine if a system has a unique solution, infinite solutions, or no solution. Specifically, it shows that if the determinant of the coefficient matrix is non-zero and none of the Di values are zero, then the system has a unique solution according to Cramer's rule. It also provides examples of solving homogeneous and non-homogeneous systems.
This presentation will be very helpful to learn about system of linear equations, and solving the system.It includes common terms related with the lesson and using of Cramer's rule.
Please download the PPT first and then navigate through slide with mouse clicks.
The document discusses linear systems of equations and their properties. It defines a linear system as a collection of linear equations involving the same variables. A linear system can have either no solution, exactly one solution, or infinitely many solutions. A system is considered consistent if it has one solution or infinitely many solutions, and inconsistent if it has no solution. The document provides examples of consistent and inconsistent systems, and shows how to use row reduction to determine if a system is consistent or inconsistent.
Math lecture 6 (System of Linear Equations)Osama Zahid
The document provides information on solving systems of linear equations and quadratic equations. It discusses multiple methods for solving each type of equation, including:
- Solving systems of linear equations by elimination and substitution.
- Using Cramer's Rule to solve a single variable in a system without solving the whole system.
- Solving quadratic equations directly, with the quadratic formula, by factorizing, and by completing the square.
- It gives examples of each method and explains the steps involved.
This document introduces matrices through the theory of simultaneous linear equations. It discusses how matrices can represent systems of linear equations and how elementary row operations can be used to solve such systems. Specifically, it shows that elementary row operations preserve equivalence between systems of linear equations. It then provides examples of using row operations to determine that a system has no solution and to solve a system with an infinite number of solutions.
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2. The Gauss-Jordan Method
J. Gauss:
1777-1855
M. Jordan:
1838-1922
GAUSS / JORDAN (G / J) is a device to solve
systems of (linear) equations.
3. Carl Friedrich Gauss
1777-1855
At the age of seven, Carl
Friedrich Gauss started
elementary school, and his
potential was noticed almost
immediately. His teacher,
Büttner, and his assistant,
Martin Bartels, were amazed
when Gauss summed the
integers from 1 to 100
instantly by spotting that the
sum was 50 pairs of numbers,
each pair summing to 101.
4. DEFINITION
The elimination process applied to obtain the
row reduced echelon form of the augmented
matrix is called the Gauss-Jordan elimination.
The rows consisting of all zeros comes
below all non-zero rows;
the leading terms appear from left to right in
successive rows
5. The Gauss-Jordan Method
The Gauss-Jordan elimination method is a
technique for solving systems of linear
equations of any size.
The operations of the Gauss-Jordan method
are
1. Interchange any two equations.
2. Replace an equation by a nonzero constant
multiple of itself.
3. Replace an equation by the sum of that
equation and a constant multiple of any
other equation.
6. Gauss-Jordan Elimination
Any linear system must have exactly one solution, no
solution, or an infinite number of solutions.
Previously we considered the 2 ´ 2 case, in which the term
consistent is used to describe a system with a unique
solution, inconsistent is used to describe a system with no
solution, and dependent is used for a system with an
infinite number of solutions. In this section we will
consider larger systems with more variables and more
equations, but the same three terms are used to describe
them.
7. Matrix Representations of Consistent,
Inconsistent and Dependent Systems
The following matrix representations of three linear
equations in three unknowns illustrate the three different
cases:
Case I: consistent From this matrix
representation, you can
determine that
ö
x = 3
y = 4
z = 5 ÷ ÷ ÷
ø
æ
ç ç ç è
3
4
5
1 0 0
0 1 0
0 0 1
8. Matrix Representations
(continued)
Case 2: inconsistent From the second row of
the matrix, we find that
0x + 0y +0z =6
or
0 = 6,
an impossible equation.
From this, we conclude
that there are no solutions
to the linear system.
ö
÷ ÷ ÷
ø
æ
ç ç ç
è
4
6
0
1 2 3
0 0 0
0 0 0
9. Matrix Representations
(continued)
Case 3: dependent When there are fewer non-zero
rows of a system than
ö
there are variables, there
will be infinitely many
solutions, and therefore
the system is dependent. ÷ ÷ ÷
ø
æ
ç ç ç
è
4
0
0
1 2 3
0 0 0
0 0 0
10. Reduced Row Echelon Form
A matrix is said to be in reduced row echelon form or,
more simply, in reduced form, if
• Each row consisting entirely of zeros is below any row
having at least one non-zero element.
• The leftmost nonzero element in each row is 1.
• All other elements in the column containing the
leftmost 1 of a given row are zeros.
• The leftmost 1 in any row is to the right of the leftmost
1 in the row above.
12. Solving a System
Using Gauss-Jordan Elimination
Example: Solve
x + y – z = –2
2x – y + z = 5
–x + 2y + 2z = 1
13. Solving a System
Using Gauss-Jordan Elimination
Example: Solve
x + y – z = –2
2x – y + z = 5
–x + 2y + 2z = 1
Solution:
We begin by writing the system as an augmented matrix
14. Example
(continued)
We already have a 1 in the
diagonal position of first column.
Now we want 0’s below the 1.
The first 0 can be obtained by
multiplying row 1 by –2 and
adding the results to row 2:
Row 1 is unchanged
(–2) times Row 1 is added to
Row 2
Row 3 is unchanged
é 1 1 - 1 - 2
ù
ê 2 - 1 1 5
úÞ ê ú
êë- 1 2 2 1
úû
15. Example
(continued)
The second 0 can be obtained
by adding row 1 to row 3:
Row 1 is unchanged
Row 2 is unchanged
Row 1 is added to Row 3
16. Example
(continued)
Moving to the second
column, we want a 1 in the
diagonal position (where
there was a –3). We get this
by dividing every element in
row 2 by -3:
Row 1 is unchanged
Row 2 is divided by –3
Row 3 is unchanged
17. Example
(continued)
To obtain a 0 below the 1 ,
we multiply row 2 by –3 and
add it to the third row:
Row 1 is unchanged
Row 2 is unchanged
(–3) times row 2 is added
to row 3
18. Example
(continued)
To obtain a 1 in the third
position of the third row, we
divide that row by 4. Rows 1
and 2 do not change.
19. Example
(continued)
We can now work upwards to get
zeros in the third column, above
the 1 in the third row.
Add R3 to R2 and replace R2
with that sum
Add R3 to R1 and replace R1
with the sum.
Row 3 will not be changed.
All that remains to obtain
reduced row echelon form is to
eliminate the 1 in the first row, 2nd
position.
ö
÷ ÷ ÷
ø
æ
ç ç ç
è
0
-
2
1
1 1 0
0 1 0
0 0 1
20. Example
(continued)
To get a zero in the first row
and second position, we
multiply row 2 by –1 and add
the result to row 1 and
replace row 1 by that result.
Rows 2 and 3 remain
unaffected.
ö
÷ ÷ ÷
ø
æ
ç ç ç
è
0
-
2
1
1 1 0
0 1 0
0 0 1
ö
÷ ÷ ÷
ø
æ
ç ç ç
è
1
-
2
1
1 0 0
0 1 0
0 0 1
21. Final Result
We can now “read” our
solution from this last matrix.
We have
x = 1,
y = –1
z = 2.
Written as an ordered triple,
we have (1, –1, 2). This is a
consistent system with a
unique solution.
ö
÷ ÷ ÷
ø
æ
ç ç ç
è
1
-
2
1
1 0 0
0 1 0
0 0 1
22. Example 2
Example: Solve the system
3x – 4y + 4z = 7
x – y – 2z = 2
2x – 3y + 6z = 5
23. Example 2
Example: Solve the system
3x – 4y + 4z = 7
x – y – 2z = 2
2x – 3y + 6z = 5
Solution:
Begin by representing the
system as an augmented matrix: 3 4 4 7
æ - ö
ç ç 1 - 1 - 2 2
¸ ¸
çè 2 - 3 6 5
ø¸
24. Example 2
(continued)
Since the first number in
the second row is a 1, we
interchange rows 1 and 2
and leave row 3
unchanged:
æ 3 - 4 4 7
ö
ç ¸ ç 1 - 1 - 2 2
¸
çè 2 - 3 6 5
ø¸
2
æ 1 - 1 2
ö
ç ¸
ç 3 - 4 4 7
¸
ç è 2
- 3 6 5
¸ ø
-
25. Example 2
(continued)
In this step, we will get
zeros in the entries beneath
the 1 in the first column:
Multiply row 1 by –3 , add
to row 2 and replace row 2:
–3R1 + R2 → R2.
Multiply row 1 by –2, add
to row 3 and replace row 3:
–2R1+R3 → R3.
æ 1 - 1 2
ö
ç ¸
ç 3 - 4 4 7
¸
ç ¸ è - 3 6 5
ø
2
æ 1 - 1 2
ö
ç ¸
ç 0 - 1 10 1
¸
ç è 0
- 1 10 1
¸ ø
-
2
2
-
26. 26
Final Result
To get a zero in the third
row, second entry we
multiply row 2 by –1 and add
the result to R3 and replace
R3 by that sum: Notice this
operations “wipes out” row 3
so row 3 consists entirely of
zeros.
Any time you have fewer
non-zero rows than variables
you will have a dependent
system.
-
2
æ 1 - 1 2
ö
ç ¸
ç 0 - 1 10 1
¸
ç è 0
- 1 10 1
¸ ø
æ 1 - 1 - 2 2
ö
ç ç 0 - 1 10 1
¸ ¸
çè 0 0 0 0
ø¸
27. Representation of a Solution of a
Dependent System
æ - - ö
ç - ¸ ç ¸
çè ø¸
We can interpret the
second row of this matrix
as
–y + 10z = 1, or
10z – 1 = y
So, if we let z = t
(arbitrary real number,)
then in terms of t,
y = 10t – 1.
Next we can express the variable x
in terms of t as follows: From the
first row of the matrix, we have x –
y -2z = 2.
If z = t and y = 10t – 1, we have
x – (10t – 1) – 2t = 2 or x = 12t + 1.
Our general solution can now be
expressed in terms of t:
(12t + 1,10t – 1, t),
where t is an arbitrary real number.
1 1 2 2
0 1 10 1
0 0 0 0
28. Procedure for
Gauss-Jordan Elimination
Step 1. Choose the leftmost nonzero column and use appropriate
row operations to get a 1 at the top.
Step 2. Use multiples of the row containing the 1 from step 1 to
get zeros in all remaining places in the column containing this 1.
Step 3. Repeat step 1 with the submatrix formed by (mentally)
deleting the row used in step and all rows above this row.
Step 4. Repeat step 2 with the entire matrix, including the rows
deleted mentally. Continue this process until the entire matrix is
in reduced form.
Note: If at any point in this process we obtain a row with all
zeros to the left of the vertical line and a nonzero number to the
right, we can stop because we will have a contradiction.
29. Applications
Systems of linear equations provide an excellent opportunity to
discuss mathematical modeling. The process of using mathematics
to solve real-world problems can be broken down into three steps:
Step 1. Construct a mathematical model whose solution will
provide information about the real-world problem.
Real-world
problem
3. Interpret 1. Construct
Mathematical
Model
Mathematical
Solution
2. Solve
30. Applications
(continued)
Step 2. Solve the mathematical model.
Step 3. Interpret the solution to the mathematical model in
terms of the original real-world problem.
Example: Purchasing. A company that rents small moving
trucks wants to purchase 25 trucks with a combined capacity
of 28,000 cubic feet. Three different types of trucks are
available: a 10-foot truck with a capacity of 350 cubic feet, a
14-foot truck with a capacity of 700 cubic feet, and a 24-foot
truck with a capacity of 1,400 cubic feet. How many of each
type of truck should the company purchase?
31. Solution
Step 1. The question in this example indicates that the relevant
variables are the number of each type of truck.
x = number of 10-foot trucks
y = number of 14-foot trucks
z = number of 24-foot trucks
We form the mathematical model:
x + y + z = 25 (Total number of trucks)
350x + 700y + 1,400z = 28,000 (Total capacity)
32. Solution
(continued)
Step 2. Now we form the augmented coefficient matrix of
the system and solve by using Gauss-Jordan elimination:
1 1 1 25
350 700 1, 400 28,000
é ù
ê ú
ë û
(1/350)R2 R2
1 1 1 25
1 2 4 80
é ù
ê ú
ë û
–R1 + R2 R2
1 1 1 25
0 1 3 55
é ù
ê ú
ë û
–R2 + R1 R1
é 1 0 - 2 - 30
ù
ê ú
ë 0 1 3 55
û
Matrix is in reduced form.
x – 2z = –30 or x = 2z – 30,
y + 3z = 55 or y = –3z + 55.
33. Solution
(continued)
Let z = t. Then for t any real number
x = 2t – 30
y = –3t + 55
z = t
is a solution to our mathematical model.
Step 3. We must interpret this solution in terms of the original
problem. Since the variables x, y, and z represent numbers of trucks,
they must be nonnegative. And since we can’t purchase a fractional
number of trucks, each must be a nonnegative whole number.
34. Solution
(continued)
Since t = z, it follows that t must also
be a nonnegative whole number. The
first and second equations in the
model place additional restrictions on
the values t can assume:
x = 2t – 30 > 0 implies that t > 15
y = –3t + 55 > 0 implies that t < 55/3
Thus the only possible values of t that
will produce meaningful solutions to
the original problem are 15, 16, 17,
and 18. A table is a convenient way to
display these solutions.
10-ft
truck
14-ft.
truck
24-ft
truck
t x y z
15 0 10 15
16 2 7 16
17 4 4 17
18 6 1 18