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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 =
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