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