Download presentation

Presentation is loading. Please wait.

Published byRyan Newman Modified over 3 years ago

1
BAI CM20144 Applications I: Mathematics for Applications Mark Wood cspmaw@cs.bath.ac.uk http://www.cs.bath.ac.uk/~cspmaw

2
BAI Fields Solving Equations in Z p Matrix Inversion Test 3 Todays Tutorial

3
BAI Definitions

4
BAI Algebra Domain (eg. R, Q, Z p ) plus operations (eg. +, -, x) Definitions

5
BAI Algebra Domain (eg. R, Q, Z p ) plus operations (eg. +, -, x) Group Algebra with associativity, identity and inverse Definitions

6
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

7
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

8
BAI Solving Equations in Z p

9
BAI Write down multiplication table Find multiplicative inverses Solving Equations in Z p

10
BAI Write down multiplication table Find multiplicative inverses Write down augmented matrix Solving Equations in Z p

11
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

12
BAI 2x 1 - x 2 = 3 x 1 + 4x 2 + x 3 = 2 -x 1 + 2x 3 = -7 Example in Z 5

13
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

14
BAI Matrix Inversion

15
BAI Write down matrix to be inverted Matrix Inversion

16
BAI Write down matrix to be inverted Append appropriate identity matrix Matrix Inversion

17
BAI Write down matrix to be inverted Append appropriate identity matrix Find reduced echelon form using G-J Matrix Inversion

18
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

19
BAI 2 0 4 -1 3 1 0 1 2 Example: Find Inverse in Q

20
BAI Can express system of equations as AX = B A is the matrix of coefficients Matrix Inversion: Application

21
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

22
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

Similar presentations

OK

Chapter 4 – Finite Fields. Introduction will now introduce finite fields of increasing importance in cryptography –AES, Elliptic Curve, IDEA, Public Key.

Chapter 4 – Finite Fields. Introduction will now introduce finite fields of increasing importance in cryptography –AES, Elliptic Curve, IDEA, Public Key.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on motivation in psychology Ppt on online hotel booking Ppt on geometry of the universe Ppt on you can win if you want Ppt on different types of dance forms list Ppt on google cloud storage Mp ppt online counselling 2012 Ppt on english chapters of class 10 Download ppt on area of parallelogram Ppt on power grid failure east