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BAI CM20144 Applications I: Mathematics for Applications Mark Wood cspmaw@cs.bath.ac.uk http://www.cs.bath.ac.uk/~cspmaw

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BAI Fields Solving Equations in Z p Matrix Inversion Test 3 Todays Tutorial

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BAI Definitions

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BAI Algebra Domain (eg. R, Q, Z p ) plus operations (eg. +, -, x) Definitions

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BAI Algebra Domain (eg. R, Q, Z p ) plus operations (eg. +, -, x) Group Algebra with associativity, identity and inverse Definitions

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BAI Algebra Domain (eg. R, Q, Z p ) plus operations (eg. +, -, x) Group Algebra with associativity, identity and inverse Ring Domain with + and x Addition creates an abelian group (commutative) Mult n has associativity, identity and distributivity Definitions

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BAI Algebra Domain (eg. R, Q, Z p ) plus operations (eg. +, -, x) Group Algebra with associativity, identity and inverse Ring Domain with + and x Addition creates an abelian group (commutative) Mult n has associativity, identity and distributivity Field Ring where multiplication also has inverse Examples: R, Q, C, Z p, (not Z 4 ) Definitions

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BAI Solving Equations in Z p

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BAI Write down multiplication table Find multiplicative inverses Solving Equations in Z p

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BAI Write down multiplication table Find multiplicative inverses Write down augmented matrix Solving Equations in Z p

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BAI Write down multiplication table Find multiplicative inverses Write down augmented matrix Solve using Gauss-Jordan Get rid of negatives Must use field operations: modulo arithmetic Stay positive Solving Equations in Z p

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BAI 2x 1 - x 2 = 3 x 1 + 4x 2 + x 3 = 2 -x 1 + 2x 3 = -7 Example in Z 5

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BAI x 1 + 2x 3 – x 4 = 1 -x 1 + x 2 - x 3 + x 4 = 1 -2x 3 + 2x 4 = 1 Example in Z 3

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BAI Matrix Inversion

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BAI Write down matrix to be inverted Matrix Inversion

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BAI Write down matrix to be inverted Append appropriate identity matrix Matrix Inversion

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BAI Write down matrix to be inverted Append appropriate identity matrix Find reduced echelon form using G-J Matrix Inversion

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BAI Write down matrix to be inverted Append appropriate identity matrix Find reduced echelon form using G-J Look for form identity : inverse If exists, so does inverse It not, then inverse does not exist Matrix Inversion

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BAI 2 0 4 -1 3 1 0 1 2 Example: Find Inverse in Q

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BAI Can express system of equations as AX = B A is the matrix of coefficients Matrix Inversion: Application

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BAI Can express system of equations as AX = B A is the matrix of coefficients Find A -1 using G-J matrix inversion Matrix Inversion: Application

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BAI Can express system of equations as AX = B A is the matrix of coefficients Find A -1 using G-J matrix inversion Solve by re-arranging: X = A -1 B Matrix Inversion: Application

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