Algebra

hE/sZ^/  E /KE >  ^'K > ^dZK & h>d   'ZKEKD/ z 'ZK/Eh^dZ/ ^

>'Z / >

'KDdZ/

E >1d/ W

D dD d/ /  Y

W

E  s





>   > W

/  

' ,

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1 D  /   W , s

^   d     ^

s

d

>

'  

^

'  / /     W   

 >

>

>

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      d s



3

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D

,

D W

: > ys//  Z   &   ys/ W

>

>

& > ^D 



 



D > >



>  d  t

' Z

 ,

  ^   , , '





4

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 , ' d

Z d > ' W

'

,

,

 ,  W  , ^ ^ ^  ,  > D t

5

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/  h h & 



D 

&

>



 ^ ^ &
+ :V ×V → V (u , v ) → u + v

s s s
⋅ : F ×V →V (α , v) → αv

& s

 >
∀ u , v ∈V ⇒ u + v ∈V

> >

s

s

6

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∀u , v, w ∈ V : u + (v + w) = (u + v) + w

 ∃ Ov ∈V / ∀v ∈ V , v + Ov = Ov + v  ∀v ∈ V , ∃ − v ∈ V / v + (−v) = (−v) + v = Ov 
∀u , v : u − v = u + ( −v )

>
∀u , v ∈ V : u + v = v + u



&
∀ α ∈ F ∧ ∀ v ∈ V ⇒ αv ∈ V

∀α , β ∈ F ∧ ∀v ∈ V : α ( β v ) = (αβ )v



s

∀α ∈ F ∧ ∀v, u ∈ V : α (v + u ) = αv + αu
 &
∀α , β ∈ F ∧ ∀v ∈ V : (α + β )v = αv + β v

>
1∈ F , ∀ v ∈V : 1.v = v

^ 

s

(V ,+, F ,.)
F=R

 s^ F =C

s

7

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 /   ^ V = IR 2 u = ( x, y )



IR 2 = {( x, y ) / x ∈ IR ∧ y ∈ IR}



u+v v

u = ( x, y ) v = ( x ′, y ′)
u

&

αv

u + v = ( x + x ′, y + y ′)

v

α ∈ IR

&

 ^ u = ( x, y )
v = ( x′, y′) ∈ IR 2 u + v = ( x + x′, y +y′)
x + x′ ∈ R
y + y ′ ∈ IR

> ^ W
u = ( x, y ) v = ( x′, y′)

u + v ∈ IR 2

w = ( x′′, y′′) ∈ IR 2

(u + v ) + w = u + (v + w)

8

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(u + v ) + w = [( x, y ) + ( x′, y′)] + ( x′′, y′′) = ( x + x′, y + y′) + ( x′′, y′′) = ( x + x′ + x′′, y + y′ + y′′) u +(v + w) = ( x, y ) + [( x′, y′) + ( x′′, y′′)] = ( x, y ) + ( x′ + x′′, y′ + y′′) = ( x + x′ + x′′, y + y′ + y′′)

s
(u + v ) + w = u + (v + w)

 ^
u = ( x, y ) ∈ IR 2

IR 2

0 v = (0,0)
∀u ∈ IR 2 , u + Ov = Ov + u

u + Ov = ( x, y ) + (0,0) = ( x + 0, y + 0) = ( x, y ) Ov + u = (0,0) + ( x, y ) = (0 + x,0 + y ) = ( x, y ) > ^ 
u = ( x, y )
∀u ∈ IR 2 , u + Ov = Ov + u

− u = ( −x,− y )

u + (−u ) = (−u ) + u = 0 v
u + ( −u ) = ( x, y ) + ( − x,− y ) = ( x − x, y − y ) = (0,0) ( −u ) + u = ( − x,− y ) + ( x, y ) = ( − x + x,− y + y ) = (0,0)

^ W

u = ( x′, y′)

v = ( x′, y′) ∈ IR 2

u + v = ( x, y ) + ( x′, y′) = ( x + x′, y + y′) v + u = ( x′, y′) + ( x, y ) = ( x′ + x, y′ + y ) = ( x + x′, y + y′)

/Z



IR 2

IR 2

α ∈ IR

u = ( x, y ) ∈ IR 2αu = (αx, αy )

αx ∈ IR ∧ αy ∈ IR

9

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 >

αu = (αx, αy ) ∈ IR 2
IR 2

^ W

α , β ∈ IR

u = ( x, y ) ∈ IR 2 (αβ )u = α ( β u )

(αβ )u = (αβ )( x, y ) = (αβx,αβy )

α ( βu ) = α (βx, βy ) = (αβx,αβy )
^ W
α ∈ IR
u, v ∈ IR 2

α (u + v ) = αu +...