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CHAPTER 4:  Direct Methods for Solving Linear Equations Systems Erika Villarreal
Here comes your footer     Page
Here comes your footer     Page  3 planes intersect at a point part 2 of three illustrating  secret sharing  this version has added emphasis of the point created in  LightWave  by  stib
SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer     Page  1.  Gauss Elimination For simplicity, we assume that the coefficient matrix  A in Eq. (2.0.1) is a nonsingular 3 ×3 matrix with  M = N = 3. Then we can write the equation as
SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer     Page  1.  Gauss Elimination
SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer     Page  1.  Gauss Elimination
SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer     Page  1.  Gauss Elimination
SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer     Page  2.   Gauss-Jordan Elimination We can use  this technique to determine if the system has a unique solution, infinite solutions, or no  solution. Echelon Form and Reduced Echelon Form: 1. Echelon Form A matrix is in echelon form if it has leading ones on the main diagonal and zeros below the leading ones. Here are some examples of matrices that are in  echelon form. 2. Reduced Echelon Form A matrix is in reduced echelon form if it has leading ones on the main diagonal and zeros above and below the leading ones. Here are some examples of matrices that are in reduced echelon form.
SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer     Page  2.   Gauss-Jordan Elimination
SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer     Page  2.   Gauss-Jordan Elimination Gaussian Elimination: Gauss-Jordan Elimination: Gaussian Elimination puts a matrix in echelon form. Example: Solve the system by using Gaussian Elimination. 1. Put the matrix in augmented matrix form. 2. Use row operations to put the matrix in echelon form 3.Write the equations form the echelon form matrix and solve the equations. Gauss-Jordan Elimination puts a matrix in reduced echelon form.Example: Solve the system by using Gauss-Jordan Elimination. 1. Put the matrix in augmented matrix form. 2. Use row operations to put the matrix in echelon form
SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer     Page  3.  LU DECOMPOSITION It shows how an original matrix is decomposed into two triangular matrices, an upper and lower. LU decomposition involves only operations on the coefficient matrix [A], providing an efficient means to calculate the inverse matrix or solving systems of linear algebra. The first step is to break down or transform [A] [L] and [U], ie to obtain the lower triangular matrix [L] and the upper triangular matrix [U].
SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer     Page  3.  LU DECOMPOSITION STEPS TO FINDING THE UPPER TRIANGULAR MATRIX (MATRIX [U]) That is: - + Factor * pivot position to change
SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer     Page  3.  LU DECOMPOSITION STEPS TO FINDING THE LOWER TRIANGULAR MATRIX (MATRIX [L]) To find the lower triangular matrix seeks to zero values above each pivot, as well as become an every pivot. It uses the same concept of "factor" described above and are located all the "factors" below the diagonal as appropriate for each. Schematically they look for the following:    Originally it was: Because [A] = [T] [U] to find [L] and [U] from [A] not alter the equation and have the following: Therefore, if Ax = b, then Lux = b, so that Ax = LUX = b.
SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer     Page  3.  LU DECOMPOSITION STEPS TO SOLVE A SYSTEM OF EQUATIONS BY THE LU DECOMPOSITION METHOD
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Chapter 4

  • 1. CHAPTER 4: Direct Methods for Solving Linear Equations Systems Erika Villarreal
  • 2. Here comes your footer  Page
  • 3. Here comes your footer  Page 3 planes intersect at a point part 2 of three illustrating  secret sharing  this version has added emphasis of the point created in  LightWave  by  stib
  • 4. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 1. Gauss Elimination For simplicity, we assume that the coefficient matrix A in Eq. (2.0.1) is a nonsingular 3 ×3 matrix with M = N = 3. Then we can write the equation as
  • 5. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 1. Gauss Elimination
  • 6. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 1. Gauss Elimination
  • 7. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 1. Gauss Elimination
  • 8. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 2. Gauss-Jordan Elimination We can use this technique to determine if the system has a unique solution, infinite solutions, or no solution. Echelon Form and Reduced Echelon Form: 1. Echelon Form A matrix is in echelon form if it has leading ones on the main diagonal and zeros below the leading ones. Here are some examples of matrices that are in echelon form. 2. Reduced Echelon Form A matrix is in reduced echelon form if it has leading ones on the main diagonal and zeros above and below the leading ones. Here are some examples of matrices that are in reduced echelon form.
  • 9. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 2. Gauss-Jordan Elimination
  • 10. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 2. Gauss-Jordan Elimination Gaussian Elimination: Gauss-Jordan Elimination: Gaussian Elimination puts a matrix in echelon form. Example: Solve the system by using Gaussian Elimination. 1. Put the matrix in augmented matrix form. 2. Use row operations to put the matrix in echelon form 3.Write the equations form the echelon form matrix and solve the equations. Gauss-Jordan Elimination puts a matrix in reduced echelon form.Example: Solve the system by using Gauss-Jordan Elimination. 1. Put the matrix in augmented matrix form. 2. Use row operations to put the matrix in echelon form
  • 11. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 3. LU DECOMPOSITION It shows how an original matrix is decomposed into two triangular matrices, an upper and lower. LU decomposition involves only operations on the coefficient matrix [A], providing an efficient means to calculate the inverse matrix or solving systems of linear algebra. The first step is to break down or transform [A] [L] and [U], ie to obtain the lower triangular matrix [L] and the upper triangular matrix [U].
  • 12. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 3. LU DECOMPOSITION STEPS TO FINDING THE UPPER TRIANGULAR MATRIX (MATRIX [U]) That is: - + Factor * pivot position to change
  • 13. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 3. LU DECOMPOSITION STEPS TO FINDING THE LOWER TRIANGULAR MATRIX (MATRIX [L]) To find the lower triangular matrix seeks to zero values above each pivot, as well as become an every pivot. It uses the same concept of "factor" described above and are located all the "factors" below the diagonal as appropriate for each. Schematically they look for the following:    Originally it was: Because [A] = [T] [U] to find [L] and [U] from [A] not alter the equation and have the following: Therefore, if Ax = b, then Lux = b, so that Ax = LUX = b.
  • 14. SOLVING A SYSTEM OF LINEAR EQUATIONS Here comes your footer  Page 3. LU DECOMPOSITION STEPS TO SOLVE A SYSTEM OF EQUATIONS BY THE LU DECOMPOSITION METHOD
  • 15.