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