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Lecture 7 Intersection of Hyperplanes and Matrix Inverse

Lecture 7 Intersection of Hyperplanes and Matrix Inverse. Shang-Hua Teng. Elimination Methods for 2 by 2 Linear Systems. 2 by 2 linear system can be solved by eliminating the first variable from the second equation by subtracting a proper multiple of the first equation and then

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Lecture 7 Intersection of Hyperplanes and Matrix Inverse

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  1. Lecture 7Intersection of Hyperplanes and Matrix Inverse Shang-Hua Teng

  2. Elimination Methods for 2 by 2 Linear Systems • 2 by 2 linear system can be solved by eliminating the first variable from the second equation by subtracting a proper multiple of the first equation and then • by backward substitution • Sometime, we need to switch the order of the first and the second equation • Sometime we may not be able to complete the elimination

  3. Singular Systems versus Non-Singular Systems • A singular system has no solution or infinitely many solution • Row Picture: two line are parallel or the same • Column Picture: Two column vectors are co-linear • A non-singular system has a unique solution • Row Picture: two non-parallel lines • Column Picture: two non-colinear column vectors

  4. Gaussian Elimination in 3D • Using the first pivot to eliminate x from the next two equations

  5. Gaussian Elimination in 3D • Using the second pivot to eliminate y from the third equation

  6. Gaussian Elimination in 3D • Using the second pivot to eliminate y from the third equation

  7. Now We Have a Triangular System • From the last equation, we have

  8. Backward Substitution • And substitute z to the first two equations

  9. Backward Substitution • We can solve y

  10. Backward Substitution • Substitute to the first equation

  11. Backward Substitution • We can solve the first equation

  12. Backward Substitution • We can solve the first equation

  13. Generalization • How to generalize to higher dimensions? • What is the complexity of the algorithm? • Answer: Express Elimination with Matrices

  14. Step 1Build Augmented Matrix Ax = b [A b]

  15. Pivot 1: The elimination of column 1

  16. Pivot 2: The elimination of column 2 Upper triangular matrix

  17. Backward Substitution 1: from the last column to the first Upper triangular matrix

  18. Expressing Elimination by Matrix Multiplication

  19. Elementary or Elimination Matrix • The elementary or elimination matrix That subtracts a multiple l of row j from row i can be obtained from the identity matrix I by adding (-l) in the i,j position

  20. Elementary or Elimination Matrix

  21. Pivot 1: The elimination of column 1 Elimination matrix

  22. The Product of Elimination Matrices

  23. Elimination by Matrix Multiplication

  24. Linear Systems in Higher Dimensions

  25. Linear Systems in Higher Dimensions

  26. Linear Systems in Higher Dimensions

  27. Booking with Elimination Matrices

  28. Multiplying Elimination Matrices

  29. Inverse Matrices • In 1 dimension

  30. Inverse Matrices • In high dimensions

  31. Inverse Matrices • In 1 dimension • In higher dimensions

  32. Some Special Matrices and Their Inverses

  33. Inverses in Two Dimensions Proof:

  34. Uniqueness of Inverse Matrices

  35. Inverse and Linear System

  36. Inverse and Linear System • Therefore, the inverse of A exists if and only if elimination produces n non-zero pivots (row exchanges allowed)

  37. Inverse, Singular Matrix and Degeneracy Suppose there is a nonzero vector x such that Ax = 0[column vectors of A co-linear] then A cannot have an inverse Contradiction: So if A is invertible, then Ax =0 can only have the zero solution x=0

  38. One More Property Proof So

  39. Gauss-Jordan Elimination for Computing A-1 • 1D • 2D

  40. Gauss-Jordan Elimination for Computing A-1 • 3D

  41. Gauss-Jordan Elimination for Computing A-1 • 3D: Solving three linear equations defined by A simultaneously • n dimensions: Solving n linear equations defined by A simultaneously

  42. Example:Gauss-Jordan Elimination for Computing A-1 • Make a Big Augmented Matrix

  43. Example:Gauss-Jordan Elimination for Computing A-1

  44. Example:Gauss-Jordan Elimination for Computing A-1

  45. Example:Gauss-Jordan Elimination for Computing A-1

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