( 2019 )

MATH 2022 Week 12

Worksheet

MATH 2022 Week 12 Worksheet

QY True or false :

Ca) 27 is a cyclic group under addition.

T F

C b ) Zz is a field under addition and

multiplication . T f

Cc) 274 is a cyclic group under addition.

T F-

(d) Be,is a field under addition and

multiplication . T f

Ce ) 22×224 is a cyclic group under addition.

T F

(f) 222×213 is a field under addition and

multiplication . T F

(g) 222×21 ,is a cyclic group under addition

.

T F

Q4 Solve the following systemset 3 y t 4 Z t 2W = O

2x ty t 3 Z t 45 = I

and write down the number of solutions

(a) over 22g :

(b) over If :

Q3/ Let

ro -- si:S::] .

to -

- C :S::]where O C- IR

. Thew

Ro, Ron = RO

,t Oz , TO

,To

,

= RO,

- Oz

Ro,

To,

= To,

to, , To

,RO

,

= TO,

- On

Simplify the following expressions :

(a) Ty, Rita Teth =

(b) Rit Titan Rita Tp =

Cc) TIL Rt, Tate

,

=

Cd ) Ritz TIG Rut,

=

Only Consider the group G of symmetriesof a regular n -

gon , generated by a

rotation a and a reflection p .

Thena

"

=p! I

, xp =p a-'

, pa-

- I'

p .

Lety = ptabp

' I' p' Ip .

(a) Simplify 8 to become a power of a

or a power of a composed with p :

8 =

(b) Simplify 8 further whew

Ci ) n =3 : 8 =

Cii ) n -- 4 : 8 =

iii ) n -

- f : 8 =

Civ ) n-

- 6 : 8 =

Cv ) n = 7 : 8 =

QT Find the rank and nullity of

m -. c :::::3

and a basis for Mt, working over Is :

Qty Let B = { Choi,

Coil ) } be the standard

basis for 225,

and let

C = { C 1,21,

12,4 },

D= { C 3,41,

C 2,3 ) } .

(a) write down

[ id ) Is =,

[ id ]DB =

(b) Use matrix inversion to finalB

Cid]? -

- Lid]=)D

(c) Now find

[ id ] I = ( id ) ! ( id ) ! =

Cid ] ? = Lid) :C id )? =

(d) Hence write down

[ C 1,21 ],

= , [ Chill ],

=

[ C 3,47 ),

=,

[ 12,37),

=

QF Let V= Lea,

e- a >,

a subspace of differentiable functions .

LetB = { en,

e- n },

C ={ sinha,

cosh a }" t

D ; v -7 V,

f tf'

fffv

(a) findC the differential operator) .

(D) BB =,

LD]:-[id ] I = , lid ] ! -

-

Cb) Verify that the following expression

simplifies to (D) I :

lie :c Dj:L id :-.

Qq Let L : IR'

→ IR-

where

Lcaey) = C 4h try ,- a ty )

.

(a) Write down the matrix with respect to

the standard basis B = { Clio ),

Co ,I ) } :

Me - [ L ] BB =

(b) Find the eigenvalues of Me and corresponding

eigenvectors :

(c) find P and diagonal D such that

M,

= PDP- l

:

p =

,D=

C'd) Find a basis C of R"

such that 4)'

=Dc e

and [ id)B

=P :

c =