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

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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

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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

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

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

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

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

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Backward Substitution And substitute z to the first two equations

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Backward Substitution We can solve y

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Backward Substitution Substitute to the first equation

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Backward Substitution We can solve the first equation

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Backward Substitution We can solve the first equation

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

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

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Pivot 1: The elimination of column 1

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Pivot 2: The elimination of column 2 Upper triangular matrix

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

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Expressing Elimination by Matrix Multiplication

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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

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Elementary or Elimination Matrix

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Pivot 1: The elimination of column 1 Elimination matrix

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The Product of Elimination Matrices

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Elimination by Matrix Multiplication

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Linear Systems in Higher Dimensions

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Booking with Elimination Matrices

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Multiplying Elimination Matrices

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Inverse Matrices In 1 dimension

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Inverse Matrices In high dimensions

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Inverse Matrices In 1 dimension In higher dimensions

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Some Special Matrices and Their Inverses

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Inverses in Two Dimensions Proof:

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Uniqueness of Inverse Matrices

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Inverse and Linear System

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Therefore, the inverse of A exists if and only if elimination produces n non-zero pivots (row exchanges allowed)

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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

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One More Property Proof So

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Gauss-Jordan Elimination for Computing A -1 1D 2D

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Gauss-Jordan Elimination for Computing A -1 3D

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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

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

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Example:Gauss-Jordan Elimination for Computing A -1

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