MATH 2022 Linear and Abstract Algebra

LECTURE 18 Thursday04/04/2019-

Working towards a general theory of-

vector spaces d linear transformations-

- vectorspaces are ubiquitous in mathematics

- generalised abstract properties of the

Cartesian my - plane- convert problems to manipulations of

coordinates-

is I-

lineartnusformations-a.ci?IITeeteIa+ersO

- converting R filtering mathematicalinformation

Cartesian products-

first define the Cartesiawproduet of sets

A, ,

An,

- - .

,Aw

# )A

,x Aux - - . x Aw

|= { Cai

,are

,. . .

,an ) ) a

,E A

,,

areas,

. . .

,an f An }

7-cakedawn-tu.pl#

If Ai,

. . .

,Aw have operations of the same type

then the Cartesian product inherits these

operations coordinatewise#

i

Eg. If Ai

,. . .

,Aw have t then define t

also for A, X . . . X Aw by

Cai,

. . .

,an ) t C bi

,. . .

,bn ) = ( qtb , ,

. rn

,autbn )

coIiut i . .

⇒-

IR x IR = { Cr, y ) I a

, y f IR }

is the usual Cartesian plane ,or my - plane .

÷devised by Descartes

ayi-thettth.cat

b - . - - .

÷. Ca

,b ) Ca ,

b ) t Cc,

d ) = Cate,

btd )

a. in . :÷::i¥

Eg . Rex 222 = { Ca,

b ) I a,

b f 212 }

= { C 0,07,

Coil ),

Chol,

4,1 ) }and

,for example ,

( lil ) t Chi ) = ( Itt,

It ) = ( o,

o ),

( 0,1 ) t ( I,

o) = ( otl,

Ito ) = ( 1,1 ) .

Eg. 222×22 ,

= { ca,

b ) I at 222,

be 22, }

= { Co,ol

,Coal

,6,21

,Clio )

,Cl

,I )

,4,2 ) }

and,

for example ,

C 1,1 ) t Cl,

I ) = Citi,

Iti ) = ( o, 2)

,

Clint ( oil) = Cito,

Ith ) = C I,

o ),

( 1,17 t ( 1,0 ) = ( Itt,

Ito ) = ( o, I ) .

If A,

= An = . . . = Aw then we call Aix . . . x An

a Cartesian .

Most commonly,

A,

= . . . = An = F

where F is a field,

and we write

f"

= f x . . . xf = { Ca , ,. .

; an ) I a, ,

. . .

, an tf },

which has coordinate wise addition,

multiplication and

scalar multiplication-

:

1-( a, ,

- - -

,an ) t cbi

,- . .

,but = ( ai tb

, ,. - .

, ant bn )-

④

Cai ,- - yaw ) C by . . .

, but = ( a ,b , ,. . .

,Anbu )(

live.n.ai-cxan.n.la#-C** )

we also

P-tgfo.g.org#w:+Ltrivialarithmeti&

observe the similarity of et I and Ctx ) to addition

and scalar multiplication of row vectors :

-/ Ca ,. . . an] t [ b

,. . . bn ] = [ a ,tb ,

. - . antbn ] ft )-

|XCa,._.an]=[Xa,...Xan# ( * * )-

so we in fact identify ( think of as equal)( ai ,

. . .

,an ) I [ a

,. . r an ] .

Thus,

for n 31,

we may identify

F-"

= { C a,

. . . an ] I a, ,

. . ganef }.

Notice analogues of ⇐ ) and ⇐* ) for column vectors :

lH%⇐ ,

-

lY= ⇐-

Putv = { [!!) I a

, ,. . .

,a

-tf } .

Thew transposition ( taking the transpose) is a

bijectionbetween Fw and V :

Ca, ,

. - .

,an ) = [ a

,- - - an ] 1-7 [ a

,. . . an]T

iwhich respects addition ④ and scalar multiplication #*) .

-F" and V are isomorphic vector spaces

-

-

C see later for general definitions ) .

-Warning : some authors identity n - tupleswithcolumuvec.tw

"

( at ,- - -

,an ) = [ a

,. . - an ]

and use transposition to move formally betweenwww.ewum#

Linear transformations (special case )-

:

- general case for abstract vector spaces later-

LetL .

. fm → f~

be a function where F is a field and m,

u > o.

Call L a

lineartransformah.io#ifLCEtw)--LCE)tLCEI./Lcxe--xuo)for all I , I t Fm and I E f

.amin÷eI÷÷

-Exercise : L '

. f"

→ f"

is a linear transformation

i÷÷÷÷::÷:÷:÷::eayup.e.ewegw.ea.c.mwu.g.in

-

Notation ( to decongest expressions) : We writeLCny.rynnlforlfcny.r.ie

Example : L : IRL - s IR"

where

Lcr, y) = ( 2kt 3g ,

- set 5g).

egg- LC 1,27 = (247+36)

,- It 5cal ) = ( 8

,9)

,

43,

- 2) = ( 43 ) +3C - 2),

-3+544 ) = ( o

,-13 )

,

Leo,

o ) = ( Nol tko ),

- o +56 ) ) = ( 0,

0 ) .

We claim that L is a linear transformation.

÷:÷bi÷:::::::

Example : L : IR'

-5 IR"

where

Lcr, y) = ( 2kt 3g ,

- set 5g).

We claim that L is a linear transformation,

Let I , EE IR"

and X EIR,

so

I = C my ) and I = Cn'

, y' )

for some my ,a

'

, y 't IR .

Thew LCE t ur ) = L ( C a g) the ', y

' ))~

= LC Cute'

, ytg' ) )

= Lcntn'

, yty' )

Thew LCE t I ) = L ( C my ) the ', y

' ))= L ( Catoe

'

, yty' ) )

= Lcntn'

, yty' )

= ( until tzcyty ' ),

- Cntn ' ) t Hy ty' I)

= ( Latin'

t 3 y +3 y'

,

- n - n'

+ 5g t Tgi )= ( 2nt3y t 2n

'

+3g'

,- n t 5y - n 't 5g

' )= ( antsy ,

- at 5g ) t C 2nlt3.gl,

- n' try' )

= Lcn , y) t Lcn'

. y' )

=

Lettuce)

.

✓ PI

Also LC X I ) = L ( X Cry) )= L ( C xn

, Xy ) )= LC du

, Ky )

= ( 2 Chu ) t 3 Cky),

- C xn ) t stay))= ( x ( antsy )

,X C - set 5g ) )

= X ( entry ,- at 5g)

:: ÷÷:÷÷This completes the verification that L is a linear

transformation . µ

Putm = fY? ] .

Observe that

man -- Eis'll ;I= I

so that,

in this example ,

-

Lcn .gl -- cm

, y' ) if Mf ;] -

- fry! ]-and we say that the matrix M reprtg L

.

Exampled : Let O C- IR and Lo : IR'

→ IR'

whereLo ( a, y) = ( a cos O -

y Sino,

se since t y cos O ) .

÷÷÷÷÷÷÷÷÷÷±T

a

,•

hockey ).

.

i.' IF

.

.

..

- • cry )

>n

Geometrically ,

it is clear that L -- Lo preserves

addition R scalar multiplication :

=THE )

LCIITUI ) LCXE)•

n

KITE ) w

79Itd

, .

'

7A of

,

'

'

-

.. LCE) '' I

. UI ) r-

.r .

'

7r

'

- 7

LCE)'

-

..

,

- o •I

'

••N

ire . LCITII -- 411+41 ) and Lay ) -

- XUE )

so that L is a linear transformation.

PutRo -

-

[ ?!:I ].

a e-

ropy . :-

:X ;]

. m .

I :c::ILoca.y7-cm.gl/iffRofyJ--fa#

-i. e . Lo is represented by the matrix

'

Ro.

Exampled C not a linear transformation) :

Let T : IR-

→ IR"

where

T ( key ) = ( retro, y t yo )

for some fixed point C no, yo ) C- R ?

I::÷i÷iii÷They) = ( retro

, ytyo )T A

Taco ) = ( no. yo)

•

yo - - - - - . . - . • #/ i •

'

, ! Cay )• I 2 se

Coco) No

If Cro, yo ) = C o

,o ) then They ) = C my)

,

the ideutitymapping of IR'

,which is a

linear transformation Ctrivially) .

If C no, yo ) f- Co

,0 ) then

T ( o ( o,

o ) ) = Tco,

0 ) = ( no, yo ) f- C 0,0 )

yet o T Coco) = o ( no, yo ) = C 0,0 )

,

so that T does not preserve scalar multiplication,

so that T is not a linear transformationµ

If Cro, yo ) = C o

,o ) then The ,y ) = C n , y) ,

the ideutitymapping of IR'

,which is a

linear transformation Ctrivially) .

:in:c:ii.it:c . . .eu . ÷÷÷yet o T Coco) = o ( no, yo ) = C 0,0 )

,

so that T does not preserve scalar multiplication

so that T is not a linear transformationµ µ

÷÷÷::::÷:¥:¥÷::÷i÷i÷÷