AI
*÷÷g
, LECTURE 1
Definition : d- set F with two
binary operations + and ✗ is a
field if bolt
CF t, ) and (F) {0} e) abelian and the distribution law
groups
0 ×, are
, ,
holds :
latte =
art be , for all and> e c- F- .
The smallest such that
integer p
O
ltt.it#---
p times
is called the characteristic of Tf such then the characteristic
F. no
peeisls ,
of F- is
defined to be zoo .
Yuppose p= a. be where ka ,b<
p .
⇐*
G)
(t)(T
-
Then D=
l+y
=
times latrines
ptimes a
Then 1*7--10 since
acp , so has a
multiplicative inverse .
To deduce (☒☒) =
0 . ☒ since
Acp .
of prime .
Eeamples : Cbhaeactoistic 0 : Ok , ④_[i]= { at bit a. AEUQ
} , R, G
hfbahactetisticp 1Fp= { : on , . . . >
f- if
with arithmetic modulo p
it vector Vooee field Iv -1,0)
Definition abelian
F- is
space a
ghoup
:
an ,
together with a scalar multiplication # ✗ V→ V such that for all and c-IF ,
A) no c- V :
(a) acre + no)= are + a no
(2) later =
are + to
(3) (ab) a = a CARD
(4) i. A = or
Let Vbe vector ooo#
a
space
.
Linear d- set SEV
independence is
linearly independent if whenever an
:
. ..
>
anElf ,
and s, , . . .
, SNES :
, A
,
S ,-1 . . .
+
an sn= 0 a.= . . .
=
are 0 .
Spanning : d- set Ssu is
spanning if foe all a c- U there exists am . ..
. aneF and
S
,,
. . .
, SNES with D= an sa + . . -
+
an Sm .
Basis d- set B C- V is basis
of U
if V3 is
spanning and
linearly independent
:
a .
The site B is the dimension V.
of of
LAI -
Prelims :
body oectot
space wilt finite spanning set has a basis and
the dimension
of such vector
spaces is
well-defined .
}
^
ceeeaonples : (1) V= IF with standard basis { ( 1- 0 , . . .
,
07s . . .
>
Cos . .
> 0,17
(2) V= FED with standard basis {I >✗> ✗ 3 . . .
}
(3) Let V=R☒= { ( am an Az , ) . . . / ai c- R} .
Then :
5- { } where (1) 0 )
en en . . .
ee . 0, . . .
, . . .
,
is
linearly independent but its
W is subset V.
span
a
proper of
Let N' → IR If a) of (2) if )
f :
;
, . . .
buffaloe ce , en t . . .
+
an en= 0
for some MEN, an . . . > anEIR .
}
Gance la
=If
,
en-1 . . .
+ amend 8)
linearly independent set
0
i. e.
aj
=
,
foe any je
☒ .
Let f= then ) c- V (takes natural number to e)
. . .
eoeeg
retell ,
f ¢ Ypan G) =W .
Linear
chaps Yuppose : V and W are oectot
spaces ooee IF .
d- map TV → Wis a
linear transformation Coe just linear
map) if foe all AEF > A > N' c- V ,
Tae + v7 Too) +Tfo')
T( are
=
A) at (a) + T( v7
'
yTca A)
=
-1 .
= a Too)
it
bijective linear
map is called an isomorphism of vector
spaces .
Eeamples : (e) The linear
map
T RED → IRA] :
given by fed to ✗ f G) is an
it isomorphism IRCD its RED
injection defines
to
image
:
an ✗ .
(2) The linear T.WS 112*-7 IRE] en= ( o ) XM
"
given by 0,1 > to
map ,
. . .
> 0, . . .
, defines an
isomorphism .
(3) Let Thom ( vis) be the linear Vtow ac-IF > AEV
set
of maps from . For ,
and 5T C- Thorn (V ,
W ) define :
(at) (a) := a( THD
(1-+5) (a) : = Too) + SAD .
wilt these Flom MW) ooo#
definitions is a vector
space .
Woo assume that ↳Wale finite dimensional .
Gooey linear map TV →W is determined
by
its values on a basis B. foev
(as B. is T:B → W it can be extended
spanning) . Nice roosa ,
given any map
leriealy independent
to linear T:V→W Casts is
a
map .
Let D= {em . . .
> em
} and B' =
{ ein semi}
. . -
be bases foe V and Weesp .
*
Let [ be the matrix with Ci >
,j entry aij
such that :
☐
Tcej) aijei -1
amj 'm
=
. . -
+ e .
vIw
Tcej) ajei aiy.ee?+..-+amj
= + . . . + .
'
V3 B en
ej
e,
ein
. . . . . . .
( ]
Aaj
÷ ; (at Cej)=aCTCejD =
ei
aij
. . - - -
. -
: :
=
acaijei -1 . . .
+
amjem)
eñ amj =
aanjei + . . -
+
aamjem
We call B. the initial basis and B' the final basis .
Walt that :
[aT]B =
a
Cpg [ B) and [1-+5] Jj ftp.tpgcsifz
Furthermore Jeon Ivo , v7 for finite dimensional
-
if Se some vector
space
U roith basis v3
"
then :
,
v5 w § u
[50-1] ☐ =
[☐ [ v3 B
'
B
"
pg
,
gg pg B.
Theorem : The
assignment
T →
CIB is an isomorphism of oectot
spaces from Thom ( Vix
) to the Cmxm
) matrices ooo it It
space of
-
.
*÷÷g
, LECTURE 1
Definition : d- set F with two
binary operations + and ✗ is a
field if bolt
CF t, ) and (F) {0} e) abelian and the distribution law
groups
0 ×, are
, ,
holds :
latte =
art be , for all and> e c- F- .
The smallest such that
integer p
O
ltt.it#---
p times
is called the characteristic of Tf such then the characteristic
F. no
peeisls ,
of F- is
defined to be zoo .
Yuppose p= a. be where ka ,b<
p .
⇐*
G)
(t)(T
-
Then D=
l+y
=
times latrines
ptimes a
Then 1*7--10 since
acp , so has a
multiplicative inverse .
To deduce (☒☒) =
0 . ☒ since
Acp .
of prime .
Eeamples : Cbhaeactoistic 0 : Ok , ④_[i]= { at bit a. AEUQ
} , R, G
hfbahactetisticp 1Fp= { : on , . . . >
f- if
with arithmetic modulo p
it vector Vooee field Iv -1,0)
Definition abelian
F- is
space a
ghoup
:
an ,
together with a scalar multiplication # ✗ V→ V such that for all and c-IF ,
A) no c- V :
(a) acre + no)= are + a no
(2) later =
are + to
(3) (ab) a = a CARD
(4) i. A = or
Let Vbe vector ooo#
a
space
.
Linear d- set SEV
independence is
linearly independent if whenever an
:
. ..
>
anElf ,
and s, , . . .
, SNES :
, A
,
S ,-1 . . .
+
an sn= 0 a.= . . .
=
are 0 .
Spanning : d- set Ssu is
spanning if foe all a c- U there exists am . ..
. aneF and
S
,,
. . .
, SNES with D= an sa + . . -
+
an Sm .
Basis d- set B C- V is basis
of U
if V3 is
spanning and
linearly independent
:
a .
The site B is the dimension V.
of of
LAI -
Prelims :
body oectot
space wilt finite spanning set has a basis and
the dimension
of such vector
spaces is
well-defined .
}
^
ceeeaonples : (1) V= IF with standard basis { ( 1- 0 , . . .
,
07s . . .
>
Cos . .
> 0,17
(2) V= FED with standard basis {I >✗> ✗ 3 . . .
}
(3) Let V=R☒= { ( am an Az , ) . . . / ai c- R} .
Then :
5- { } where (1) 0 )
en en . . .
ee . 0, . . .
, . . .
,
is
linearly independent but its
W is subset V.
span
a
proper of
Let N' → IR If a) of (2) if )
f :
;
, . . .
buffaloe ce , en t . . .
+
an en= 0
for some MEN, an . . . > anEIR .
}
Gance la
=If
,
en-1 . . .
+ amend 8)
linearly independent set
0
i. e.
aj
=
,
foe any je
☒ .
Let f= then ) c- V (takes natural number to e)
. . .
eoeeg
retell ,
f ¢ Ypan G) =W .
Linear
chaps Yuppose : V and W are oectot
spaces ooee IF .
d- map TV → Wis a
linear transformation Coe just linear
map) if foe all AEF > A > N' c- V ,
Tae + v7 Too) +Tfo')
T( are
=
A) at (a) + T( v7
'
yTca A)
=
-1 .
= a Too)
it
bijective linear
map is called an isomorphism of vector
spaces .
Eeamples : (e) The linear
map
T RED → IRA] :
given by fed to ✗ f G) is an
it isomorphism IRCD its RED
injection defines
to
image
:
an ✗ .
(2) The linear T.WS 112*-7 IRE] en= ( o ) XM
"
given by 0,1 > to
map ,
. . .
> 0, . . .
, defines an
isomorphism .
(3) Let Thom ( vis) be the linear Vtow ac-IF > AEV
set
of maps from . For ,
and 5T C- Thorn (V ,
W ) define :
(at) (a) := a( THD
(1-+5) (a) : = Too) + SAD .
wilt these Flom MW) ooo#
definitions is a vector
space .
Woo assume that ↳Wale finite dimensional .
Gooey linear map TV →W is determined
by
its values on a basis B. foev
(as B. is T:B → W it can be extended
spanning) . Nice roosa ,
given any map
leriealy independent
to linear T:V→W Casts is
a
map .
Let D= {em . . .
> em
} and B' =
{ ein semi}
. . -
be bases foe V and Weesp .
*
Let [ be the matrix with Ci >
,j entry aij
such that :
☐
Tcej) aijei -1
amj 'm
=
. . -
+ e .
vIw
Tcej) ajei aiy.ee?+..-+amj
= + . . . + .
'
V3 B en
ej
e,
ein
. . . . . . .
( ]
Aaj
÷ ; (at Cej)=aCTCejD =
ei
aij
. . - - -
. -
: :
=
acaijei -1 . . .
+
amjem)
eñ amj =
aanjei + . . -
+
aamjem
We call B. the initial basis and B' the final basis .
Walt that :
[aT]B =
a
Cpg [ B) and [1-+5] Jj ftp.tpgcsifz
Furthermore Jeon Ivo , v7 for finite dimensional
-
if Se some vector
space
U roith basis v3
"
then :
,
v5 w § u
[50-1] ☐ =
[☐ [ v3 B
'
B
"
pg
,
gg pg B.
Theorem : The
assignment
T →
CIB is an isomorphism of oectot
spaces from Thom ( Vix
) to the Cmxm
) matrices ooo it It
space of
-
.
