FP2

Question 1

Division Algorithm

Answer 1

take a,b
find a/b floor that is q
r=a-bq
a=bq+r

Question 2

Euclidean algorithm finds

Answer 2

HCF/GCD

Question 3

How to euclidean algorithm

Answer 3

take number and apply division algorithm
take second number and apply DA with remainder
when r=0, the previous remainder is the GCD

Question 4

What is Bezouts identity

Answer 4

Unique integer solution to gcd=ax+by

Question 5

How to solve ax+by

Answer 5

Make sure equal to GCD of a,b
use euclids algorithm
back substitution

Question 6

how to write a=b modm

Answer 6

a=b + km

Question 7

4 mod rules

Answer 7

if a=b and c=d modm
a+/-c = b+/-d
ac=bd
ka=kb
a^n=b^n

Question 8

Trick for solving modular stuff

Answer 8

make equal to power of 1

Question 9

4 divisibility test

Answer 9

last 2 digits divisible by 4

Question 10

If you want to divide in mod arithmetic

Answer 10

must divide modulus by gcd of m and divisor

Question 11

What do you do to solve ax=b modm if gcd(a,m) does not divide b

Answer 11

find multiplicative inverse
note gcd(a,m) = 1

Question 12

how to find multiplicative inverse

Answer 12

use bezouts identity with a and m to find an expression of the form ap +mq = 1
multiply by p as ap = 1 mod m

Question 13

fermats little theorem

Answer 13

a^p = a modp if p is prime

Question 14

4 axioms of a group

Answer 14

closure
identity
inverse
associativity

Question 15

Find closure

Answer 15

cayley table

Question 16

find identity

Answer 16

e in group st ae=ea=a

check both

Question 17

find inverse

Answer 17

ab=ba=e

Question 18

associativity

Answer 18

a(bc)=(ab)c

Question 19

what a symmetric group

Answer 19

group of all possible permutations on an object

Question 20

what notation is useful for group permutations

Answer 20

two row notation

Question 21

What is a cyclic group

Answer 21

a group where every element is repeated by applying the operation to the generator

Question 22

what is a dihedral group

Answer 22

group of symmetries of a n sided regular polygon

Question 23

What is the order of a group

Answer 23

number of distinct elements

Question 24

what is the order of an element

Answer 24

smallest k st a^k=identity

Question 25

condition for a cyclic group

Answer 25

|a| | |G|

Question 26

what is the order of the identity

Answer 26

1

Question 27

what is difference between a proper subgroup and a subgroup

Answer 27

a subgroup can be the whole group

Question 28

How to find subgroups

Answer 28

cayley table will only show elements from that subgroup

Question 29

notation for element that generates a subgroup

Answer 29

pointy brackets

Question 30

What is Lagrange's theorem

Answer 30

if H is a subgroup of G | |H| | |G|

Question 31

How to show group is isophorphic

Answer 31

set up one to one function | show group operation is conserved

Question 32

isomorphic symbol

Answer 32

= with ~ on top

Question 33

solve homogenous 1st order

Answer 33

back substiution

Question 34

non homogenous first order a=1

Answer 34

un = u(0) + sum of g(r) from 1 to n

Question 35

non homogenous first order

Answer 35

Find CF of homogenous choose appropriate PS sum use inital condition

Question 36

2nd order

Answer 36

use coefficents to form AE (assume solution is Ar^n) | use PS

Question 37

What do you have to do in reccurance PS

Answer 37

make sure you havent already got a term

Question 38

how to find eigenvalue

Answer 38

det (A-lamda I )

Question 39

how to find eigenvector

Answer 39

sub in lambda and find relationship between a and x

Question 40

what is a normalised eigenvector

Answer 40

unit eigenvector

Question 41

what if not relationship between x and y with eigenvalue

Answer 41

eigen vectors are 1,0 and 0,1

Question 42

How to find diagonal form

Answer 42

Find eigens of A combine to make P find P-1 D= P-1AP

Question 43

when can you use orthogonal diagonalisation

Answer 43

when A=A^t

Question 44

what is true in orthogonal diagonalisation

Answer 44

D=P^TAP

Question 45

What is cayley hamilton theorem

Answer 45

every square matrix satisfies its own characteristic equation

Question 46

what is a characteristic equation

Answer 46

this is the equation to find eigenvalue

Question 47

Operation of transformation composition

Answer 47

p . q means q then p

Question 48

major arc

Answer 48

theta less than pi/2

Question 49

minor arc

Answer 49

theta more than pi/2