lecture7

# lecture7 - Lecture 7 Intersection of Hyperplanes and Matrix...

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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
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 10 7 3 2 8 3 9 4 2 2 4 2 = + - - = - + = - + z y x z y x z y x
Gaussian Elimination in 3D Using the second pivot to eliminate y from the third equation 12 5 4 2 2 4 2 = + = + = - + z y z y z y x

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Gaussian Elimination in 3D Using the second pivot to eliminate y from the third equation 8 4 4 2 2 4 2 = = + = - + z z y z y x
Now We Have a Triangular System From the last equation, we have 8 4 4 2 2 4 2 = = + = - + z z y z y x

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Backward Substitution And substitute z to the first two equations 2 4 2 2 4 2 = = + = - + z z y z y x
Backward Substitution We can solve y 2 4 2 2 4 4 2 = = + = - + z y y x

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Backward Substitution Substitute to the first equation 2 2 2 4 4 2 = = = - + z y y x
Backward Substitution We can solve the first equation 2 2 2 4 8 2 = = = - + z y x

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Backward Substitution We can solve the first equation 2 2 1 = = - = z y x
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 10 7 3 2 8 3 9 4 2 2 4 2 = + - - = - + = - + z y x z y x z y x Ax = b [ ] - - - - = 10 7 3 2 8 3 9 4 2 2 4 2 b A [ A b ]
Pivot 1: The elimination of column 1 - 1 2 - - - - 10 7 3 2 8 3 9 4 2 2 4 2 - - - 10 7 3 2 4 1 1 0 2 2 4 2 - 12 5 1 0 4 1 1 0 2 2 4 2

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Pivot 2: The elimination of column 2 - 1 - 12 5 1 0 4 1 1 0 2 2 4 2 - 8 4 0 0 4 1 1 0 2 2 4 2 Upper triangular matrix
Backward Substitution 1: from the last column to the first - 8 4 0 0 4 1 1 0 2 2 4 2 Upper triangular matrix - 2 1 0 0 4 1 1 0 2 2 4 2 - 2 1 0 0 2 0 1 0 2 2 4 2 2 1 0 0 2 0 1 0 6 0 4 2 - 2 1 0 0 2 0 1 0 2 0 0 2

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