EV ,¥,x...Recap: Definition: A vector space over F is a set V equipped w/ two operations: (addition) t: VXV → V, cx¥,y¥) titty ' e V (Samalplication).: Fxv → V, C,¥,x i →

Lecture 2: Subspaces

Recap :

Definition : A vector space over Fis a set V equipped w/

two operations :

( addition ) t : VXV→ V

,

cx¥,y¥) titty

'

e V

( Samal plication) .: Fxv → V ,

C ,¥,x i →at EV

Satisfying 8 properties :

( VSI ) : I -15'

= 5'

+ I'

tix ,J' EV

( Vsz ) = ( I' ty ) -1 I = It C g 't 't ) VI. I , 't EVti::3: einen ,( VS 5) = IF

= I'

VI EVa

• (l vs61 : I,¥b¥I' = acbxt I V-a.be 't , EV( VS7) ' -

of City)= at tag' ta EF , VI. 5

'

c- V

%S8 ) : ( at b) I = af + BI

'

ta,

b EF,

KIEV

Remark : an element in F is called scalar .' ' a l v ✓ is called Vector .

Proposition : Let V be a vector spaceover F

. Then :

(a) The element I'

in CVS3 ) is unique , called Zeiter .

( b ) thx'

EV,

the element I'

in Cuse ) is unique , called

the additive inverse of I ( Denote as - I )(c) It I = I 't I ⇒ I =D

'

( Cancellation law )

( d ) O = I' VI E V .

A

F

Ce) C - a) I =- Caf ) = at - I

'

) fact , f I' e V

Cf ) a J'

= I'

Va EF.

A

F

Proofi ca ) . If J'

and J' '

are to elements satisfying ( VS3) .a A

✓ v

Then : J = I'

to' '

=,

⇒ 8=8'

J ' = f '

to'

= I to'

( Ust )( b ) . Given I

'

EV. Suppose we

have I, I

' ' EV satisfying

( VS 4).

Then :→

I + g'

=I = I +5

' '

ash 0 → ,

Then : I = I to = It city'

' ) = C I) ty' '

-- y

c-

(c) I + I = I 't I

⇒ ( I 't E) x C- I ) =t I

'

) t C - I )

⇒ It if= jeezEI⇒ I =5

'

C V8 )by Ccl

(d) of = Coto ) I'

= O # to # ⇒ of = 5'

Il

of to'

( ate - at )aCe ) J = Ca . a) I

'

= at + L - a) I'

⇒ C- atI = - Cat )( byd)

Other part - - leave as exercise

Cf ) af = a Cotto ' )

Istaotao

'

11

atto'

⇒ af = J'

(by la )

SubspaceDefinition : A subset W of a vector space

U over a field F is

called a subspace of V ifW is a vector space over

F

Under the same addition and scalar multiplicationinherited from V

.

Proposition .- Let V be a vector spaceover F - A subset WCV

is a subspace iff the following3 conditions holds -

( a , Ju E W( b ) Tty

' EW,

tix, Jew ( closed

under addition )

( c ) AI EW,

faff,

I'

EW ( closedunder scalar

multiplication )

Proof . . ( ⇒ ) If we V is a subspace ,then Cb ) and C c ) must hold

because W is a vector space .

W has an zero element ('

.

'

W is a vector space )-1

Ow

Then : Jw tow = Jw in V ."

( can,aquatic ) Jwt Jv'

⇒'

Iw

-

- Juin

W

( ⇐ ) If ca ) - Cc ) hold , then additionand scalar multiplication

are well - defined on W ( by (b) and Ces ) and ( V S3 )

follows from Lal .

( VS I ) , CVS 2 ) , ( vs5)- ( VS8) hold for V , so they

hold

for W as well .

Remain to checklust ) .

cVLet I EW . Then , we

have - I'

EV.

F EWUr

But - I = thx'

E W

( by ca ),

'

. W is a vector space overF under the same addition and

scalar multiplication .

Examples : . For any vector space V ,{ I } CV ; ✓ CV C trivial subspaces )

"

W. For V= Mnxnlf )

,

( I2)'

w , = { diagonal matrices } CV

+ Wz= { A EM nxn CF ) ? detCA7=o } CV NOT subspace

( def C A + B ) # def CAI t def CBI )

( L Ews- - { A e Mmm CF ) - - tr CA ) - o } CVn

11

(3 ai ) €aii

• For V = PLF ) ( aotaixt aux 't . . . t aux" )

Pn CF ) :# { f EPLF ) = deg (f) En } is a subspace

W Et { f EPLF ) = degcf ) = n }

- Consider V = Fn = { c X , ,xz , . . . ,Xn ) : Xj E F for j - - 1.2 . . . ,n )

Consider linear system :

f:::c:::::÷

.

'

I ⇐ axe . e

Ami Xi t Amzxz t- - .

t Amn In =

bmgives a subset , the solution set SCV .Is S a subspace ? ?NO if l be , bar . . ,km) toYes iff 5=8 . ( Null space )

Theorem : Any intersection of subspaces ofa vector space V is

a subspace of V .

Proof : Let { Wi } ; , I be acollection of subspaces of V .

Set W : It n wi.

CV.

i E I

' i Tv e Wi for tie ? c'

.

Tv E W.

For any I' twandy' EW , we have I c- Wi , if c- Wi forth'

.

Then - . I' t I'

c- Wi for Hiei ⇒ ityew

For Few , a c- F , wehave at c- Wi fu allies .

i . at c- W .

i. W is a subspace .

Question: W , = Subspace ; Wz= subspace

He

Win Wz is subspace

Is Wiuwa a subspace ?? No in general !

Linear combination and Span

Definition : Let V be a vector space over F and SCV a

non - empty subset .

• We say a vector Jeff is a linearcombination of vectors of S

if I iii. Tiz , . . , Tines and a , .az , . . , anC- F sit .

I = ai Ii , taeuzt . . . t Antin .

Renate I is usually called a linearcombination of

iii. Tiz , . . , In and

a , .az , . . , anthe coefficients of the linear combination .

• The span of S,

denoted as Span ( S I , is the set of alllinear combination of Vectors of S a

Span ( S )Eef { a , I

'

, tazuzt . . . + anutn = Aj C- F , I JES for J -4.2 . . . , n ,NEIN }

Remark : By convention , span ( &? Et{ I }empty

set

• I E Span I Itx'

, I- x

' I*

×

Example :. Fn = span { I , , Ez , - - ,

In } where Ij = Co , o , . . . , ¥, o - . o )• PLF ) = span { I , X , x

2

,.

, #= } j- th

• Mnxn LF ) = span ( S ) ,

S = { Ei ; -- ( IitIf i = is i.jen }. Given iii.

f ka!:) iii. =/ :÷.

) , . . . , a' n=(?÷:)xiiitx. ' " in

#= U ,Then :

Iie span Chini . , . . . .in } ) iff-

. god:. IiIaa:÷IIfa'III. =u.( V ,;Vz , - . , Un )An , X , t Anz Xzt . - - tannin

= Uh

is consistent. ( has Sol )