34
ry Tu (re,','/ :" *T V * ,,ratir*e w*'d'a 'Wf'*"'-* '"* @; V^V ----=' V ( ^Y, ^t)€ "f@ "Y 4/r,4" , ffi-.A"; *4"'*;l &'^ ,' ff','t* /*^ uhn' u;'(' *{'; ' Nfu '{kn "fr^"'d edh /< .p-*/*fl* ( rc=fr 'n't a) f tr n^'#'Y"* t \/ V nln$e Y ') V /,Fek / ,OoL(.rt *Vnnl,* @ /J ,AtuWr4 '+ .l2tt o **rl o/r:^ V oz'';a"*t O ,fr lY 1r4, <^ /'tni O!o^v..-+v V,*) = eO# Ja"*^*^ *a- (V,@,O ) *at' /K- F*' ,o'rr*il 4 n) ,d'; (V,@) -*t' f-/ *i *.f 'r:"n{ol4',,*'rt lAr) (u/rf )o ri-= (xe"'@(fO*) Ut xePo*) =(*o96(*o'*) Vt) ?pO,i'= xo(7c^r) i,i'l,p nl-- ',v / \/ K v

Spatii vectoriale

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Page 1: Spatii vectoriale

ry Tu (re,','/ :" *TV * ,,ratir*e w*'d'a 'Wf'*"'-*

'"*

@; V^V ----=' V

( ^Y, ^t)€ "f@ "Y

4/r,4", ffi-.A"; *4"'*;l &'^

,' ff','t* /*^ uhn' u;'(' *{';

' Nfu '{kn "fr^"'d edh /<

.p-*/*fl* ( rc=fr 'n't a) f

tr n^'#'Y"*

t \/V nln$e Y')

V /,Fek/

,OoL(.rt *Vnnl,* @/J

,AtuWr4 '+ .l2tt o

**rl o/r:^ V oz'';a"*t O

,fr lY 1r4, <^ /'tni

O!o^v..-+vV,*)

= eO#

Ja"*^*^ *a- (V,@,O ) *at' /K- F*',o'rr*il 4 n) ,d'; (V,@) -*t' f-/*i *.f 'r:"n{ol4',,*'rt

lAr) (u/rf )o ri-= (xe"'@(fO*)

Ut xePo*) =(*o96(*o'*)Vt) ?pO,i'= xo(7c^r)

i,i'l,p nl-- ',v /

\/

K

v

Page 2: Spatii vectoriale

-7 -

t^*d*---r* \/ ; K=i{.^lJ

qta)/Ke

(

)

-h, *f,Rt 7Gr'?'' -=(j^)

u/*,:F,r)Qz (*7)rQ),Tffi')

,l; &/r,o

l^=C) ;z:'R'r/1

O,R'^t--4tR-nV'il) F--'-> -qJi " (l^) 'l=ff^)

O ' R- n 7R* ------*' i?o

(4 , x) t---':' dQ x

0;t ^

ny'ur*r' *t

,4-/* ' (n*, o ,o) 4,t, f* d-""i ,*'/

#r**./,*lv ', (R'", @) ff ,//;r.^ ( *,14 | )

fro'f;*, 'nA 7*h,,

.*u/',1- '

(4, (o r/)e x = For@fc, , v *f *R, Vae R\.

&; ,?

xQ-=(Tr)

Page 3: Spatii vectoriale

^. ?L

7Q//(

f) ?./K

\/R,

^'(

f::,;..)wtir)'(u..) w=["o (!) 'b'[)]

= ec-)@fu")

&({,-u,^ ( * y)o z= Fe'lgfcx ) v x'7.

f e") ao(.6d) =(ax)Ot'il , $ x*Rt v

u,"J) \'' ,.*J, )

' (*'J)

w xe R,*,J r,*{ ,,(^) , l=ff^)

,/Jo.*", *Q (,@A)ffi xo (f.:O):(

: [u^,,,*A,\

l*^']o [-!

,\ = (ro-)

\u*. . *J",)

=

(*", -^)'(*^J =

?,uW; c# -1o 3W)=(o^)6F"J )

il/"&J*fLl,kl/!

i'.ff

. y'o/n'o6 t'*l <

,y*1; ,*A*rl ^o-1

,

' '/ ?t!€ /

v *Va

n4 ,&***nt*rA (*r) f U,

a,*) f" K= R

cn (v,e,o ) *,t,

{vi *O7.=-4,J

\/ - /v *L

-taAQrl,t

,/*(rg(.)

@7rJ-wC r*

*-rdl

"@ry =V

Page 4: Spatii vectoriale

*4-

M,J

, (v,@ ) 'fl"r oa'/'o'-'

&o$*;'(4, -(r,,l

(t, ?'rr. *r')@fc) r

VxrVtUo,fet?'

;, z€V, *,/'R xrb o

,4/*,^" g.PQ) ."T^-''/-- *n' ^f ': **@ o

ila eJ J"4o' @

(,44",tJ

P=J

&t" efol

= (to")@fo.)'?n*.w; ;

ry)oz= (*e!*fc')tr;:f i^

(4 uo(^@J) *t*)@(/%/)r

v *'n/ € v) {r"tR

w ^'il* v' xeR ..-^r^.-,1fr*.Y xe F%/)=<o(V )- !il)*= *f,*:

:**@y*:eo*)@(/% , _.v{xaR-0S,* *bU@{) =(*o*)@(A{' ,v^'H Y{o

l*, eP*";o(fo*) , v 7e'4**'f eR

1;t ;6;:= -r: Fr": xo?P) r *'(rt

Page 5: Spatii vectoriale

*f-

&* (TF-- *rfO,) , \/ x+ V, d *,/"?

(4, 4ppx = a , v 7 € V

ryTY treV

,r{-lu"*^' LO*.*o= t ) &* t97:7'{xeV'/R

0#,;-*., *o#-( ., (4e ,o) = f* ''u/*'J *"1'/\

o (tr^'IP) '+ = *du^anna

o

rR = *'**ll'^u

,)u/ft""/ ,*il ' "'*l'

@ tK^L^i' I / n ufnJ ' 7*/ '^J =

= { drr qX*"-+'+,rn " 'a'"*K '1< i< ^lJ

'o, /K[*J,w

V q*R,

+ "^)

.'/'

,*/^-^{n',ti/ ,f*de

@ t(r. I /, r -- /K, / = fu{' J, r s' K'L;#""c(

Page 6: Spatii vectoriale

J*,'K x

u'rJ)

'8'f*^

:/KxK(,1)

e:@,o):rt'/a,,"u

t--i JAq : .r -----= /</oV*flu,=/c4+63):*x*r

T*---\ K

L,

L----' *C/: T ---/'<

Pfl(,)= u/,"t,d"rn-f4 *.#''^'{

$"zq- ( &'-e^k'twte ;2-/r-o-n **d'*ir* /--r flU

/ /

J.+L/

ei TY (V,o,') '(*n ^-f* ".e'/";^/ *;5-tq'*-4'J'

S E V ^* -'*ldrtn€ *4 "'"#';

" 5 ,,J.m) P'&:- 4 ("J,) /*; f* {* "\/\/

dr, - - ,4^ G K ,4'? "o'ry= ert +' - '/ d^'f^ /'&,,^*n 'eF nj- *&(/

, 5 A.r,L, Q.{;J gra /u,./,,,-*-r-/1,,

-,--r,1,n e l{ ,a ^!. 4,'ry*. -o 1d;= AV ,o/'o*' {=. ''d

<^ = orc-

Page 7: Spatii vectoriale

*7 -

" 5 "A rvl &d dn; 5 s 4;f +4"('c'(/

$<'uemw,t*zt&d-t K-",f */'-/' "*/-Y./ V t'e "il:^ V "= I S I ''

/Kun/t S = rt"g

,F{w,rct^' ') '&aa V=l A' J ' */***' t#2V = a

' ,, ,. <- e -/ ,/^ -- ')/*,*,

/sr/= lit I

llg,/,.,,t ,\) /K

V * ru,t,r^^&>ur( rrrvoArrtrr *6 "M'frt '/;A "-**

t' 4^ Pot-

Ti xa

J 4:7r, ' $n=v- e R 'a 7 z= 4(,7-'-a 4"e*'

' V &,, - )*'R' 'aI

=a'ffj 17;H+. '/v^(ir)

,,n,

d,:* k ={rLR

-^JY* N Y t---/r 'l '/

/)txo-'V&

/I) fut' &i" Rn (/*n *****i I M' /,n! '* ,r'r" rs(t .Jo'b=1",='(:) ,rz"H ) ''(^'(il J 'tu

M ,4,#'* *8 t *^4 "d4=rr,4 i vxnre\ J 9,--{r.,nR'ai''

.YR*.' x= (:)

=[il:H+ ft): *,t,+ .'' f a-rem-

x: ()

.p,f X/- . -

{-fr*-

#'

/1

/J= /J.

vD 'B -,a

={r=CrR

4e, -r --f 4€a = d6n--A

Page 8: Spatii vectoriale

t) B^x" f* KDJ

/*' il,:^ lk^fxJ = (t r I/K

<rt b= | /, x, Xt, - ,^*J )

Page 9: Spatii vectoriale

_a

,%-(i)l

/r'/a&at e

^o

4*e '%.

// s/ (-

. t/f2o.t-clo'ta

1"4-(i) ,#,'(!)

3

/ fi{r*y,il?@ ,

.a i 14h+AraL-t

x^=c u )

4r# r* 4e4 = On-U

€xe,t'u'l', -* ^ *-/, ,co A - \;=:-f I

4rytn,ecx, ,,fux* ; Rt,/

tr/d* #^'- *o HaIR3= 3-rr-,

I R

A',r*, lnl= 3 &*'' /'A^"*/,atto/a; in /3 */'} o" "'1^ ''

'

t n/, ,-(/ u,,4,tbe n?

>fl d, = 4l ::

{-7**

7Y 4,4.,{uG R o i 44*^& 1,'4r={z=Q

r'/*- 4, t i {,t};,"<"4r= CR, <::") - (i)- -, (il * 4t'(l) tf,) 4,1 <rt 4t -o

qz)1*,4 {:=DI *r: o

r (-;;j.') h)*/d*

-//*/?O ad+nt 4o

mrA/u'ut

,J/,rr*,r"'*/; It;a )

Page 10: Spatii vectoriale

*/ct .*

fiffi,-,"^ -rf-l 4= 4,-1,, , /t*^ 6 /"*ent* "'l**o

ll ary uy*nu*( .','t'5 = l*,, -,*, J tR' fu*u/ rq e f^

1:D n--"/ y A=/l ] .'/ li ri--{/)/{na"' 4v V= K-,tu/r- ,*nJP;t( ^

,,/h /"H* / ,/r'V p*i# b/J

,oi *L'=4fr.- r*f^

4,. - r{^ tr.rz @ dr-eur/i ,?* ;

A@+' i; n "f ,uek/tual,J ,,n't'*,,/,;

f*ru n= lr=(l)

,&'= (!), % =(,,')i,

/il,rh, 'h,/*r ,* o/lr*,'',,,CI;**-=

rrf. 4 4 t 4n); * i3 CIr5 (-, I fl =ls/

4= lt,. .,,

_lr.1>U.n '( ll-rvf C/

/r

y uo,fuz, t*=(f a

4, 4rr4t * R 'a' i

- [t+42 ft)

*4,1)

p (:;::)Lq ({) =(!)r$).&) c2

1v-) )*, r ltt/3= 5

1 {,+ x5= fI

| /t'3

*#* *"a,7*/*( e /'4;'*{(

^/uX, "-''* ) ,

Page 11: Spatii vectoriale

e)/l t t

l, o /

Ito,

(

)

=, ,'=(: !; /)

i: )g) -u) 0^o^/

*ud'"6**/*& -/-' n* 'i/oz" 4

- [)'

({) =(,"fj/6trd" L

lltmm z,F

n,&;^, r;*

L,n L, - L.

Lr* L'-l'

&) "(: :7t*,"H;

d,=-ft e(z=*3) 4>=3

!r^d ,&n."a #"A^ )

/ {,\ ^-t 1{\ ILf**' } e\"=r) l;:\= 4'lt I '(,*,^alu."tou"6^-L,

|.;) l,/ , ^,n. Ln ,_Jr*/o_*a,&l^;.* 4-n *4' '"&/a &u/'1 {

i"*UrsUU

oO

JO

o

Page 12: Spatii vectoriale

J"/"'*t&"Lbct

T- V = k-,f4,/,on A V' ',*"- /,c

,/,nP B A "hz" E.t

*lz*

,fuZu' .ua&u f* ^n/"i*/

nxduio'/-, /3-|(,,--,,eJ : 4/=J{,,. ,f-\J*&n *r*'* *'of;*, J &rt// .& /*

'?*J*, -r,/., *' -.**C'*f,4 /*uh''&'

'lt" A* 4/ ^; N .c0, /*rc- 4'

Page 13: Spatii vectoriale

-t3-

d;- ^ /wu ( **fir*o t4 /r*r, ,& A ,fuxo

'T*./-;, ; *lA;; *a,tb.di6 u*/*,&/

,;,Arc I'

,= {7')

,€s=(i)J * {u,*- *'l/, fl,/,=(4 frri\

/, ,/",/"

an(c'o

/,= drgtlrtz*{t(t

fr='f,",*fr(,*fs(,fr= fie,"fr€z* f,...

,("t t*,orvrl, *,lA)' *-"6"*/ri' 'u'"d ^"'l* 'sawcdw 7

,Wr/ "! 't** Z , od "; to"/'> &,, &, , 6, G 2 a4-'

'r= o, r,n e, €, ' er(.. <q2 (::\= *, lt') * 4r,01 )\^,./ (. J

:g:) ^

(;)

)

t,/-/0 -5 -tz\

4'-11- T 4 I16 -!- ol

l-' -6 I1,, z ,rl

G 6 a/

a5

4/

1-L

,'3

[4rfr-z

4^

^ee&/\r

x_L

7-(^

T3T,

4,\ -/,Q,2 l-- 4*, I I

t:-f (

&

P o ooru CI

a owtooalooo/

B0A

( cAalA-_30L

@-tA/u+

AF\/J

H-tr CI

a moc {n

-/o

6

IL

-5I

5-

-tLi1

l2_

r,il

6

-3

.- | zA, -, e, + 3'e-t= ,: r= I ::) ,(; i i l

I uo,ni;,',#; = *.: 'r' ( ;: l'{; i o )

A / /'-bcdheula,*, n f,

,n,t6r/a %^ /,'/^

c *6

tAoz

L.* zl*zL,L,4* L, - sL.Lr4. L.* zLu

Qe-zL,-;Q

Ls"- r; l" t s L.

Page 14: Spatii vectoriale

/arl,e.p,

U4h-en

_/4 _-5 -/L\ r,i : ll:.€ &llt7 7/ \'3

)

') ^/4/

,ldL(t

/2

+,

'47

(,, rceo-trt/*,a-fl1 Z, /./Jnt

,{"AP /5 814,,4 I

(*) =r-t')

. on'06^o& A" /= (!) ; /*o a s'.^/

(f;),,(!) Wl

=E)/

, ow,o/n*,{-/, A /.= (:') ^ A* t *-l

(il) 2,' [i) P I*)=E)

--r\4lel6l?t3e

6

flj =

/1,=t ld,

I

\r.

pr,r,

r'

*3LT

-/

*/av-66lL6

f,C^** /' t*w & A '/*rc 4 A /ozo d/ "'ffi'

tq8,9,

,*63T0

6l6

Page 15: Spatii vectoriale

--/. /rtec/,gurt'{"e

w fr V -- ,{- ry*t* ,,,'"u6;^{ * t/ g V B ,**{**&&o*e *wv'o/a-rt\/Jr"

il ,4,n, ,*/wof,i^ ,*reY"/ -/ /---T'--

V &*F

,a#*--"' A I V

il e V * *nl*'"o#'r o'w'/^-

xr*,rt,y* U V x'n.K/u tvn/rrU

'U

" V*,yn(/ z+4 e Uu0* $ n"" U, V'<* /K " {Y e U

fu- Tu lr= /t(y *&";''1 .a

tt ; /.4.,,,prt*JJZ, Vr*U +,rk ,+ucf.a,r,ct

*if*

/<.a4GJ

,I{r"r*I;

-*-:r*,) I O, J': V ,xo',.f *'tff '''/t"&;o'{'" "^e /.' V

4fi";a1v,*f*"* hut/^, .../t/ *re; u + v \: *hr^(l =,6/L,oV , &*' u* v,

"/ LAJ (M ,,

rJ ?k -s={ rr,- ,&tJE V /

lsr{v6)Yi*=4'ry/ ' /tr%/ +'l )*o*in ''

U,OU.= J nl-aV I $€-U

r

fui (,/,01/r= lar| ,*ft,,,o;

/r( op&"oZ* c,"t #, €) U,

V= K- f *"/-, "'( :i U,, U, t*$*

g "t*-ro )

( ,t,J**4 ")

4zef,J* \/

k) n)-e il-l Z V.J

I/n\J

mr/*ofo,{

Page 16: Spatii vectoriale

- /(; _

' T' V *' *F* ''j",t'*''"/ fi;t f -

^^d U2; ) 4- * q, //, *,/fk: ,/k* u

,&^ ,(u,. u, ) : d:r/ u,) r il*)*( 1) - Mhurq rtu')K\

4

i^ 4"*t/,*,/o,t , .t,;,, (rl,@ur) = il'L,utl, ' "den(tt. )l.'k

Orv,, l-

,a) Jo ttt\ -j*b/ Jc />(

lrl-,A)

frhr[. '!"

*l , '72J,yMtrvn trwl , Md ,/ur-r*"T;*'^ .'

n*-f\ e a, v u,f * R' v''t o a4U -rJl lK (=-)

{y

7y *,f . 7? , *,J € U ''1'*X"> c,0 *, rfr/ e (/

A,e- 7G U '>,? -=(i:)i Yrnrzr'T^=o

t-u -P J=$i) Y l''J"J'=*

.+/-,* n,,rJ- " &)rf (f) =ff'#.)

Page 17: Spatii vectoriale

-/7 *

?*rt* ffi *--ru € U rt'A*'V ^o- uoo-'L)'* 't-

( {^,,,f y,) 4 ( x+1P7, )' ( <ou'fJ, ) : o\ / /'J" /\/,4rM^, (o r, .

f,J, ) * ( oo,,f-/, ) . (^,- fr/, ) :

= * 0y:9'f ft:f'il =. r-)

t/'/1c 'l #<r?j,?r,*M; .a *oru " (1 ' 'il* '

@d^ , {"{rk a &u>,*'** /"*a) ;

LJ = J ,- () ,'fP^ ) Y,+x,+ *,=oJ =

l*= E) Ie

l'= (T') /'",*uorRJ :

? f=f ;:). ( i,') 1 ""4t € nJ =

€1 7= ., (J',) . ,t 0') / .'';72 n R J =

)l= -rr._tt l=

J

etn U 'il, ^#'YflM/' (j),li )'t?t*rffi; 6 U <

]) r{*,^ (/=

/*, l(:),0)JJrf, ,& ,cern#r*lv ,/-"

,-''3//(

"

. f!,')

,3) ,=lz=zzH*Y, F)1r,"*4

.,1. q k-,/* (J. i*' "ttu*"n fu'e') ol'^ U

"9 tf ,un, [r,)

,f,')* */*ffi )

Page 18: Spatii vectoriale

^,/, I

4 {il=(f) '-i;:": (u:,*')- H (?

(8-

((;),IDJ/-t l,-er^ { (;l-hv

., '7=

eV\f l+dlc/

(,/,

tG)=l(

"al

*- tJ, , - t J :

)17'r€"jr'fl ) 1J'*

A"trilil*

ilel"[;rar* ,t/'ocn

I

{,,&, (=R

^ ,1. i,

,,r**_ -,(;')r ",(i)= {i)(?-) I -d'-1'= 3

I r?-;;61,-t F;;=q

o/*' d; (/: Lv7

e7f2 lk),(i)lIH'HJ

4 d,-lg) :4 -1J

?

-2a.t+Z=41UJ

+Ur

); X

u,lx*(#

,*-& ,'

@U, our,

U, " J

IJ,

"[,

,J:-:\ tzJ

,) ,,

,il

(

Wo "4'r,t€mk'r}t-nt /rH-,; rfuK \1 &L

J/

rc)

l''v'

Jtr)

II

-1

-[

d,

**R-J"

Page 19: Spatii vectoriale

y'4,1 u,(),4r,= W "l f u,",1* "r;h/ *V ], /

U Uu h ,a*, a{o ccI

hY {rr{, e R

*t3*

il,

"{*e^*'t*^e f*: -e , R | = l'a'JI, (/, 4 7/?\

4=-4=o

ed^, ,&''uu

*' ,ilr- U, =l\,fe'R

A, o. /"; '.

,ai 1 f * 4, 4'$ )'/o;/^>* q

= ff) e ., (i).. (i)"/{)

--4=l#,,,y,J o/,',8, -

J,,v, = (;) , 4,= (il J

^3f(/-fueato

o'/'t t="'4n

"4

/,t =ftU j ^./'; u,-lB.f =I-

u.

ft)

,&$#

U,=1 E)n uI/; E'

: {,'*, * {, &>

/"^1", U, *t'

4>.(i) _ln "^1,

' J^r- =(i)

J

' 1t"-(i )l?ew P

il. *fl' t'-o bnxt f 'il"-a'd' r

f* /n U'

t,{\(J.'i* t'v?^ /'t,. L/' Y n}-sN' J

TY qIe U,0U.

-= (a) , *,J,n rK

) L/, - e(z\ --

-I lx,=c,I L--4l.----:_---)

I d', --OlI 2---

Page 20: Spatii vectoriale

-La *

U^, U, FD "-tJrZ=-c(o

;. u-

L',? | 7-27" 2 =Q

) ^='/z *?

I t:=J*Q ',1 =o

*=J-Z-O -?-? /?f=/:\l"/

o(/, = ((!) I +n /*,ou)= o

flU,

-7-P

u,

&; (Ur u,) = il%7/?'

= u_n ,

-5

aFe U,Ot).

"g* nle 4

f\-t t-=C,?p 1

\ z= Z

I y-*o(/

?re.H; En

, '&zo- ti d;en v'^tt"nf ,/rA (/ * t/,J/'

+ d;- (,R

c

r', /

- fu(u,nrt,R.

u,

-a/.,,(u,tu, )= 3,R' ,-.3

L/, n U, -1 //<

-f**"

&^^7?3 = 34sqe .rpora>-'cuo ,o

A, - U^ "*&/*' R3

/{ArA z"' %;; -frffuo4,^o d;pnv:.,..-)

Page 21: Spatii vectoriale

?-c* a pl/on^r;.,a ;: ,f-#* u'+ u'

,ae'do'-' 'q''/.* 8: E, Ub^ ,*d f; 2:9 ''o.

(j, u #,4

Y^ ooz,-,1 o*d*

Ty 6= 4uB.=1h,6*v'&a^a), .'-*AU*

R

,J,,

A=

l\

:){32

*4*"

' | (:)=

%Jlz -/Ir c

I e--J'j3,

*,,*.-(i)j ,B-'l*=(iI

t/

& rt+w+'1,14 cn l4r#',#r3*NP /#U,n0-

,*Un*3,;

W, ,&.e*,r^n q n dz = /%rJ y (j,n (1,=R},

rtfu*-W"rrwu,Q 4t- 4t + Uz ,'c.u 4, € (1, ?^

*, " (i,

"'ffi t'*u'6'

/'etfu. , nseo)tui e, e U, i //ze (/, a 2 n)-//r7,ttz-

N,e U, -,t) ,//t : (#) tt n*tryr*n'a ='n'{;'u-(t['),rf^l=J

ilze (J, 'vt. il,; lf) cr4 a G R

[,e

Page 22: Spatii vectoriale

€ (/,* U-

* LL-

,gr*' rc$i"f"-", ,n/,{rL e R ,,4 ,n

(:) =?r') rH M f'y-'::=,'.

u,= 7:)

,Y,=6

l!lol>lz

rn

tl

t\,,ll.}-'

*T2

3<'t'=(i) ' *'= (j)

I 2'TlE-\u

2-

I :5--'rP UE='&= *

?u*r-Ha,co t:)-l, /

nul

@v.

u, =

#,-/*r;4^ ,l* 6 ,i */*,en <;tu^e

f* [/, , tr, U, fi d., U, n il"

lrdfu------=-

, Bazq $, ,/;enu'une ,errn/t'u

t/,

,,f"w., 8r.l &,, #r, *, I "J P- I#^A;* B, ^ '1';

4 nt; * 4 e).,;,= # ) 4n ., (n'r t, ( | o 4, ({)- I il/

)Zd,t4*{/t=o.&tD \ *,/2d,t4ds=o

I

| 3'/.2 t343'=c

Page 23: Spatii vectoriale

Y/lJwn c'o

truL*V c,o

*23 *

/i rd, ,,^t / ,r-6 o&r*' k

A lc , fi^A= lilo

,$/,dq 4={r.{= c ,.e{.'a'Isl? 7 l=o -dD 4,\, *',ilu| t I 43 sl

E,=l t'L

A

-_v-ActT A:C )

u /2. tl'f \rts'lt zl 4 I(.o:'3 If f f /V ljL C.

,/+, ehd'*.lJ

^t; u, =l& I'7R-

" 8o.6 .,4d"- | o'h,

A: ltILf

vy,

$ot

' 8"6

(j, fiu, =

,#

)

) ^r / qte U, y n'*e

:/j

ot

d-z

lLahQ

c\^b, {r}z u}3

,hz. r* u, ",H t: 1"v,, *+rJ f '/',,{.lt l*,

'1t, J ,/**; /-,,/* (y', /,/,J

h .dr'*^h'o.ne C-J- A.

,\,,Fr, -\J "i-t/=r J,/, |"t,d',d.J o''1'/3lAt#.

( i -71' t\= . =,, lw, '&,J n / ;

l-t --o t J'f f A

tl)*en h/04( J

Page 24: Spatii vectoriale

TY r*e A,0t/,

U, *{1 a} =

LIL :) 4f =

^u-Vfl lU r-

V,JrLl- e

/ , ^1..Cgm/a mA'kP

J

4t-

d* -* dt

*u-(/,

llzn-n

'&L"'a,"a d!

t4Fe

,^6'

'ittcn\80c" nFe Uo

'vp il-v

qt e u-

tff+ lrlli =2

-o *=f,u, tf 2.d,

J4,^"-^ .l L*' + &' - f ' -f L

| 4r /.2 4' tr''F'

i^t**. :,.:=,ff"' *.*;,; ;: <,- !o$^ ?r)(r'-'r- )4 q',' 12 A 4 ae*a o-"** /'

J

,r,) n 2 ?r, = f,"f- /' 6

*ft,

,,#--)/4,r1 /

ft;+

il:I

lr'

?-p 4--*

Y^&*""

'vb r' 'r" * Ert

tr, = trr.

=r,#)-p 4f=

g!4t'**,n

Page 25: Spatii vectoriale

r *T^4 od, = L

* (1,

= z+2 I;3

J#,^*- **h@. tt1;, ! _ u,+ rt,=.R3

(/, n [1=

;;*",R'-3 t Ktm^r:ufo,,frttgL_a B, u B* -

| *",,o, ,6,,*- J

&t/r"rh:-* '*,flJ tru,l"u" '

u-l' ' f '-t\ {4r'2nr4/*:U ? +t ;) ry li { il,. * *u'r 1

tt 1Y, q 1$z-

Ww /*,t* U, * U, *-t' l*Y" f' '"h' J ' fu o'l-';-'t'. ('/'=

5

/

Page 26: Spatii vectoriale

-L6 *

,rtt,*fu ,/,-r'ottt/r

'4(u',u)= *'/r*1' L. ,ol; vY--- ' il; v,

/l"ilu.

(-, /(<+

V' V "+ ,8r., ,$ e V

d n *4

a'--t

V *,fz*\/V <,fr e//<

/

€^*ql;' : il- n ,,r'ufi'u /"a-'^*,1 ur'("*/; -/'luJ"o "#

,^) /" 7R3 ' i?u , /(+) - ('&") /

'q I '?'---,* , /G)= ff)-) { rc--,P',/(t)=f:)

d) | rP'---7P- , /(i)- U;)

Page 27: Spatii vectoriale

*27 -

2R'R,,

)):\\_t-I

/*ltz d z4 l

(.:ru). t(,1 ).,

: ./+),yfrul

/,J

./r,*a-* (=, J ,4 e -{*,.(R)

/Xztra-\

( ^'!:,,)

I r+ n'f

I 'fJ,lfu- r,/"

/ff.)

f 7,tzZ, \. ( 'f )", ^,-J, I

/-p -/F

7

-4/fi.M '

-Q.t

L

Page 28: Spatii vectoriale

(, ,Lf*k"J "4 ad'a'''ato eF(r aV'A"o,/'\--r' , // n / | ,,2

/ Q.t^J -->> q.f^J , {n t/rr el-t1'5ffi a,Lu/v -/.+'ana

/J

(a / @.trJ^0^kJ\-/"/ x {fe4r[^1/(?)= lrr4e ,

n\-J

fr,"fr,) p a,,('n,hfi1 / r

,)A 4*1,'o /

/t*u.o

/r^) = /r roX'n s/x1

_./-e/)A

i*4 ,4,^9.-vaLA

6

J

,A- "n

,:J

l*'a,*

/ v--'

i v ---^v7v' -.*,r'o*o 4,'oC^jd-

/

Page 29: Spatii vectoriale

*29^

vt y*!,, -/*w;4,tr /4't/ = (* " V; /r*l =tr,J I V

J o,* t/ "a"r /HAJ<v,

/

/

,'r't..yece ,*a-

un.-*,ttq"e+&^ln

J

vJ

(.a' ru/'/ao_l.*? I-7 - tl'

"A,d */a *tf 'A''rt /'/-"/-//K// il^^ Y^/

ffiay,,rn #"* *n 7' 1s,-,tt

,tJ.rn N,U

TJdo;|.\ 4. / R'- R"LrJ ":-__7

"{uaa-

#/*'*' d; v F

,K

D*JJo /r( Ail)lr c-o

"/,^) / _afr, f*!oL) /i ^r affi2'uzt""e

#ld^*--*T*

^) { y[*/r '/'""h*a

/ A ='a*/x* (A-/)t\P'ql

-4,:*^*;"

/*ze " d;enf,|t'azr* ,r*r/"" '?-/ ,O

h-f

c2 {* /::)rf (;:)) =.il;).FlH

v *f eR, v (t,) (t:) * R'

*

Wf*fu-f *ffi %/=fi*v"^/ ,0_

tWut .' J"r.J

?"*tt i"A'A' r/ JI

I+(x)' x €

J'e / v-=V/ ^a&*4,'/" , .r' -1d.{,&/), VlyY*,/)

Page 30: Spatii vectoriale

7* (t ),(f:) * R', ;;': R

s4*,- q-/t) .yt)=/( :;6;: )ffi/,5 ku, +fo.) + (o/,"fr4X+ (&o,.f i")- (*t'f t;)X\'

i (t ko,f*.) r k/, "ft,r)X' =

= o (, + { x * (o, */, )x'* ('o,*/, )x^) /

-f' [o,, /,1 * 1o, -/, ) xi ( *,' /' ) '') =

')/o') -f/(!:) q /=/*('/';b';= // (+

i,) tr"/ = l(f).fl {G). coJrr J :

= I G) e R'/ 'ar /x'{o-/)x? 1"'/)*'- o'f

(;) "zJ = lou'l

0,

P* n / {xt (n-/)z=, &'4^3 ='c

. n=/:of*ll, A=C. -z-'1 /(_

I A;Ol*| .a- /= c' ,nt/= c

--/ r,i )'' '*'7 = I

,drr, /(-/ =

Kl

Page 31: Spatii vectoriale

*3t-

&; ,6me",,a 6d'*'e'' h''dt''u'1'

tt ^z J' L,' ,D ^/' Y- q .:-4 ?- fl* r/o^.7-,f*,hK ':; *rU'Q/'-*/'LOJ^/ .>-2 z= 0* t%J,{

.r'r, .tL ?^/ = Lir n,

'J-/

f,/*e; Y-/ ? , 3/, \

Y-/ = I /1 :) /ac R"l = la'k'p-t)4i1a'/)a"/t'i

: lb- o,Y'r t^x3)* (/x-rx't /x') /^ ' / *R J =

* l.a (,*x\ *i. /ftt?-u') / ''/ * RJ

'r z A! x--^" ^r'= 1,/+^-----)#*--[,d ln*?a

@ I",AJ ^ J f* Y-l

rf*/,;u lt A t u l, ' 1?.7A vu *f - o

*?,pQ =c -at'.2 x'(r+x'*rrt)"f (x-x'/"J - c

.d // f P o (*-f ))"r ('*.f) X'= o

I7, ilJo"/''jd*il; {6 A^/ Y^/ *4 l\ q Ih ,t-oJ'-y= z'r/^

Page 32: Spatii vectoriale

ft)/ /" (R) --,,.{,/* c, ,J

)

-t-d /

-32 -

$"2.;J, : fi,p u*,,t ,n uf,,**J------=-i

n / a- / \ fza,t 31' ct trrlf (" /)=( ",ol

@, /(,,r,i =l G j)

tt,a /) rrt,=

){"

//

4

t r4 I I=lc iln'{-(

J;^ tfir**-e 4*u fu/ y J-"/

*fi,o) //(: i)=lJ:110?) I lza*

3/* c* 21 lo"'f)-l:ll" '/

[ "/ -' d

I fLa+z/r,o,2J'=c /

'?) I ( lo ,** zJ:c jI L:"'*=:

L4 *l+*Lt0

a

o

oC

oo

ZL/

-l

4X

E3tzotf{Loonlooco

m3TAo000

IrJr'Jt

. ,1 _rRl = l- (1,

Page 33: Spatii vectoriale

*33 -

at ,{*n P* {-f .*t, | (! )Jr,/:t,; il;', ik

.r*{"lt!:f) lt G i),ai (: il=(r',f:)l

- I IV, X) \" I I a,{ u,/ aR ar, ]'or u^t***".r,/: { I- 1[i. "J;

) / d &'lb/ 4 u v-t/\ b) I:/:q;=J' i

: I lJ' U' \ I aA*'''/ 6t "'rt '*f**f:1[J. J, I I

&4*oor1* : U^A 'e',n ,u*/*/ '*''7u"'*'-( /ti* {*l,ffixr*e,u'/sr -/'4-' * /,,- ,{+'ri **tef ** *ry-nb,'

q,',nl..u /;-e ( 14 <: /, + 1" )

' AU*' ^'d-,"*'j €) *e{n il- = -{.

v, Y*,/'|ff:{i,) /r,,J-,1,*nI =

= lu, (J':)i l'C :).1" U : ) /;"1''7"a)

r,,^ /J

Sr; ,fiPrn-t" dn'- *nt-t'z/'rttt' 44dh '"

/,-^ 4-tal= &L^'(-'f - MU|A- / <v) 4: / r /'9y- /R

aj ill"J^l= -\il/

Page 34: Spatii vectoriale

,/*t" f,,,7* 7-" {/cz I (: :J ,[/:l , {: i)]0#,,^r'-