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Lecture 6 Intersection of Hyperplanes and Matrix Inverse

Lecture 6 Intersection of Hyperplanes and Matrix Inverse. Shang-Hua Teng. Expressing Elimination by Matrix Multiplication. Elementary or Elimination Matrix. The elementary or elimination matrix

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Lecture 6 Intersection of Hyperplanes and Matrix Inverse

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  1. Lecture 6Intersection of Hyperplanes and Matrix Inverse Shang-Hua Teng

  2. Expressing Elimination by Matrix Multiplication

  3. 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

  4. Elementary or Elimination Matrix

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

  6. The Product of Elimination Matrices

  7. Elimination by Matrix Multiplication

  8. Linear Systems in Higher Dimensions

  9. Linear Systems in Higher Dimensions

  10. Linear Systems in Higher Dimensions

  11. Booking with Elimination Matrices

  12. Multiplying Elimination Matrices

  13. Inverse Matrices • In 1 dimension

  14. Inverse Matrices • In high dimensions

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

  16. Some Special Matrices and Their Inverses

  17. Inverses in Two Dimensions Proof:

  18. Uniqueness of Inverse Matrices

  19. Inverse and Linear System

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

  21. 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

  22. One More Property Proof So

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

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

  25. 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

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

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

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

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

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