!" UR #%$ &'$)(+*, -)./ /43657.98:.9/<;.=?>@/4.BA:.CEDFHGHIFH=HJK>@FH3LCNM=O>@FHP I/4Q=R=EDLC M;I |M | = n 021 n0 n = 0S ;QT/4;I'=H/<LC IU;'V9FW57.98:.9/<;.= -)./X=)FH= FH=E> Z / I 0 0 0 Y n > 0 M = M ∪ {w} |M | = n − 10 /_I ;Q`>HD;; [ DLCKM=E>@FHP I/4Q=O;\Q3DFE;U;.9IVFH=HJ].9^@/4;I/4.93I 0 0 0 Y S C ⊆ B |C | = |M | = n − 1 .9/4=H.aA:D;/<;b\Q= /<;I D=H=ed<f;U;Ig;/4LC Dd<;ihj/4=H.D3PQ 5E/Z=EDI/4Q= 0Ec Y B 0 = (B \ C 0 ) ∪ M 0 V w w= X u∈B\C 0 λu u + X λv v v∈M 0 P ;ULCH3./Z57.9= f3U.9=kDd4dZ. /ZI JdZ./4LC ;QmlnQd4JIU. 01 λ Y u ∈ B \ C0 0S 1 · w + Ig./Z= 0 (−λv )v = 0 /<>@.3U;GH3UFELCpVFH3qd4/4=Hu.D3.=erq=ED5ECEf=HJ/ZJP./_Ig\Q= d<;Qt.u^@/<;IU/Z.3v∈M M 0Es u ∈ B \ C/_I 0 Yo1 /_I DNw /<;Ip>HD=E= [ DLCvhj. 0 λu 6= 0 0 .9/4=H.zAbD;Yk (B \ {u}) ∪ {w} = (B \ C) ∪ M Y Y Y 0 x)y /4;:\Q= C = C 0 ∪ {u} V0 / D=p\.93U.9/4=)5ED3IbDFELC TW {}|@~jZ7 , p( $)*(+7 M;I ./Z=?.9=E>HdZ/<LC.93UV9.FHJIU.93zd4/4=H.D3.3gDF ;QT57.;I.9C Im.>@.BAbD;/4;'\Q= DFE;%>@.3 YeS V V Jd4.9/<LCH.= =HVDCHd\Q=.P IQ3U.9= /4.;.]DCHdCH.9/4I''k9H+!?\Q= FH=E>8i/43U> /ZI s 0jc Y V 57.9V.9/<LCH=H.9I 0 dim V dim {0} = 0 0 #%$ &'$)(+*, -)/4=E> FE=E> A:D;.=k\Q= ;QalnQd4JIDFE;X-@DIVqw >HD;; FE=E> /4=E>@. D = 0 x HS S Y Y B B 0 FH=E> VS |B 0 | ≤ |B| >@/4.ziQd4d4.9=B\Q= F P . H C 3 I S |B| ≤ |B 0 | 0 B B0 Y ¡pE¢!7£+£ | ¢ ¤4 , M=%.>@. .9=O>@dZ/<LC@¥>@/ .=E;/ZQ=EDd4.9=p.9P IQ33DF P!D=H= D='.>@.¦dZ/4=H.D3iFH=ED5HCEf=EJ/4J. Y Y Y V Y 5H/<;iV9FB.9/4=H.93gAbD;/<;b\Q= .93U8:.9/ZI.93U= .=HJ. M ⊆V V 0 j4¨}4© EN #%$ &'$)(+*, M;I [email protected]/4=H.]AbD;/<;a\Q= >ED=H=WJ/Z5@I.;z./Z=E.9=W.9P IUQ3 /4. M V S /4;Ia>HDCH.3aDFELCdZ/4=H.D3FE=ED5HxCOf∈=HJV/ZJ \ span(M ) 0§c .93 .93U8:.9/ZI.93I. .=HJ. M ; I / M0 = M ∪ {x} 0 M 0 YkFEY ;8 =HQ@LCKP./Z=H.tAbD;/<;¦\Q= >HD=H=K8i/Z.>@.93UCHQd4.6>HD; 3UJF .9= I /ZI D=O;ID!II 0 VS s Y Y M MS 0 Dk=EDLC`-@D!IUVzw P./Z=H. .=HJ.¦d4/Z=E.D3iFE=ED5HCOf=HJ/ZJ.93b.9P IQ3.=p/4= .CH3qDd4; c 0 x V Y dim V .9P IUQ3U.9=B.9= ICEDd_IU.9=eP!D=H= ;/4=E>B8i/Z3i/4= -HLCH3/ZIIU.9=pln.93I/4J dim V − |M | S 0 ª (+2$ | ¢«$¬t22( £+®2¯2°O$) -)./Z.= F =E> E V98:.9/¤d4/4=H.D3.¦gfF .¦±H57.93q.9/4=H. 3UG7.93 C FH=E> ;/4=E>eD57.9d<;LCE. Y Y³²´ V W F 0Hc'0 0 V /_I-@P«DW µm3UFHGHG7.9= /_Ig./Z=E.93iV9FO;fIV9d4/<LCH.9=T¶GO.3UDI/4Q=t·t>H.93 FEd_IU/ZGHd4/4P«DI/4Q= d4D3.= Y Y 0 5 5H/4d4>@FE=HJ H CH./ZIz¹ 7º!» l+DdZd<;:ln±H3qDdZd4. FH=O> J/4dZI¼ ¸ /4=H. s S f :V →W λ∈F u, v ∈ V ½ 0 f (λu) = λf (u) ¾ >@/4.a.93;IU.z-)F .a/Z= >H/Z.aV8X./_IU.m/4= y@0 f (u + v) = f (u) + f (v) ¾ YkY VS W0 M=E;57.;Q=E>@.3. FO;;b8:.9J.9=¿ ½À DFELC J.9dZI.= Y 0 f (0) = 0 f (x + y) = f (x ) + f ( y ) f ( λ x) = λ f ( x ) f (x ) f (x ) f (y) α x UR _"9 a λx x+ y y α 6 : "m " "< "W " « U!" U U@U }" ! α ¤u !"#aX$&%' ( !")Uj " f (x) "* f (y) % q : "+ }" ! x + y TW #%$)(+*-,2( $ £ ¤47¨ , /Z.z>@FE3ULC J .9J.5O.=H. 5H5H/4d<>@FH=HJ. >HD; 2 /<;Ibd4/4=H.D3 r f (x, y) = (x + y,FHy) f5E:/ZJR.02/→ R c s 0 Y V9F`V.9/4J.= ;.9/4.9= =E> V X 8 . j / O 5 . Z d 4 / 9 . H F H = P U I z . Z / T = H > 9 . 3 7 5 9 . E = . ¸ u = (x, y) v = (x0 , y 0 ) R2 0 S D=H=eJ/4dZI¼ c f (λu) = f (λx, λy) = (λx + λy, λy) = λ(x + y, y) = λf (x, y) = λf (u) ; f (u + v) = f (x + x0 , y + y 0 ) = (x + x0 + y + y 0 , y + y 0 ) = (x + y, y) + (x0 + y 0 , y 0 ) = f (x, y) + f (x0 , y 0 ) = f (u) + f (v) . µm.9=ODFE;Q]/<;Ia>@/4. 5H5H/4d4>HFH=HJ d4/Z=H.D3 /4. 5H5H/4d4>@FE=HJ f (x, y) = (2x,2y) f (x, y)5H5H=/4d4>@(x + s 07c s /<;IkD57.93'=E/4LC Id4/4=H.D3 >HD J/4dZI /4. FE=HJ S 0§c s 2, y + 2) f (0) = f (0,0) = (2,2) = 6 0 /4;I§DFELCz=H/<LC I2d4/4=H.D3 >HD ln±H3 S fJ/4(x, y) = (xy, y) f (u+v) 6= f (u)+f (v) u, v ∈ (R\{0})2 dZI¼ /Z.bV9FE32.3U;I.= Q G7Q=H.=IU.J.9C 3/4J.rq=HJd4.9/<LC)FH=HJm/4;I 0 c ² Y ´ (x+x )(y +y 0 ) 6= xy +x0 y 0 0 EN #%$)(+*-,2( $ £ ¤421 , /43B57.uIU3UDLC I.9= >H/Z.3..9d4dZ. ¸ 5O.=H. Dd<;p.9P IQ33DF ±H57.93 D=H= /<;I`>@/4. V F = R>@2.= R 0i5Hc 5H/4d4>HFH=HJ 8i/Z. 1 Y 3.C FE=HJ F ./Z=E.9=kln.;I.= /Z=EP.9d [ FHd4dZGEFH=HP IX./Z=H.qdZ/4=H.D3. f Y α Y c 1 s S D=zDFE;}A:/4d4>'w D5Hd4.;.9=¦P!D=H= /43}8:Qd4dZ.=z>@/4. 5E5H/Zd<>@FH=EJ =)FH=]6587¹ 99;:j5O.;LCE3./Z57.9= f Y 043 0X1 s 0 \Q= /ZI FH=O> DV9Fp=H.9C .=p8i/43b>@/Z.z- ID=E>ED3>@5ED;/<; 2 c Y Y 0 {e , e } R e = (1,0) e = (0,1) 2 /<;Ia>HD=E=.9/4=H.k1hj/4=H2.D3PQ 5H/Z=OD!I/4Q= 1 >@.93 < .>@.3.PIUQ3 2 Y (x, y) ∈ R (x, y) = x e + ye 1 1 2 AbD;/4;\.P IQ3U.9= .9J.9=`>@.93gh}/Z=E.D3U/ZIUf!Ig>@.3 5H5E/Zd<>@FH=HJ FE;U;i>HDCH.93 01 s f Y f (x, y) = f (x1 e1 + ye2 ) = xf (e1 ) + yf (e2 ) J .d_IU.9= /Z.%.9P IQ3.= FH=E> d<D;U;.=T;/<LCTd4.9/<LCI¦DFE;mA:/Zd<>w D5Hd4.;.9= d<;Q 0c 0= 0Ks f (e1 ) f (e2 ) J/4dZI f (x, y) = x(cos α, sin α) + y(− sin α, cos α) = (x cos α − y sin α, x sin α + y cos α) . < .>@.¦d4/Z=E.D3U. s 5H5E/Zd<>@FH=HJ f :V →W >@.8>E=E/Z.3IiV98:.9/}dZ/4=H.D3.¦gfF ¹¹_»º ½ Null f = {x ∈ V : f (x) = 0} ⊆ V ¾ 0 @#A A BDCFEHGJILKMGNKMO4PHQSRTVU"EWPWXYTUWGNKMO4ZKMP+[\O4G Y^]VR_]^] f (x, y) f ((x, y)) ` Y .t¿+;/Z.CH.aAX/4d<>ew ¼ 0? À . (-sin α, cos α) (0,1) α UR !" (cos α, sin α) α . (1,0) 6 F*: "+U «!M9" e1 = (1,0) "* e2 = (0,1) W" R2 W V 0 0 u ¹ !»º A f . t FbUX y@0 Bild f f(u) TW Null f " W " «U α . Null f "*)U 9 Bild f Bild f = {f (x) : x ∈ V } ⊆ W 0 {}|@~jZ , p( $)*(+7* E¢!$)£}¢p£+(+2$ | ¢!$`¬t22( £+®2¯2°7$ -)./Z.= FH=E> V8X./).9=E>@d4/<LC@¥ >H/ .9=O;/4Q=EDdZ.:d4/Z=H.D3U.:qfF .:±H5O.3.9/4=H. 3G7.93 FH=O> Y Y Y ²q´ V W . /Z=E.md4/4=H.D3. F ;.9/ 5E5H/Zd<>@FH=EJ M;I ;QkJ/ZdZI s 0 f :V →W dim V = n S dim Bild f = n − dim Null f . EN #%$ &'$)(+*, -)./ FH=E> ;./ .9/4=H.AbD;/4;k\Q= Q3UQd4d4D3kw w [ DLC ² 0x k = dim Null f v ,...,v Null f 0 P =H=H.= 8i/43T>@/4.;.AbD;/<;p5E/4;`VF 1 ./Z=E.93`kAbD;/4; >H.;eJ.;D IU.9= ´ Y v , . . . , v , v , . . . , v 1 k k+1 gDF .; .38:.9/ZI.3= [email protected]/ /43t5O.CEDFEG@nI.= >HD;; ./_¥ Y 0 01 S V B = {f (v k+1 ), . . . , f (vnFH)} B =H.eAbD;/4;'\Q= /<;I 8XQ3UDFE; =O>N;Q /_I6DFOLC S Y Bild f |B| = dim Bild f n = k + |B| = lnQd4JI 0 dim7Null f + dim Bild f /4;IdZ/4=H.D3§FH=ED5ECEf=HJ/ZJ =HJ.=HQ .9= .;2J/45@I.9/4=H.bhj/4=H.D3PQ 5E/Z=EDI/4Q= FV9.9/4J.=j¼ 0s YkY S Y B n X λi f (v i ) = 0 i=k+1 /_I Y Y P n /4=E>@.;I.=E;%.9/4=H. Y /Z= i=k+1 λi v i f (x) = f Null f S n X i=k+1 λ 6= 0 021 >Hi.9=H=e.;bJ/ZdZI .9J.=?>@.36hj/4=H.D3/ZIUfI\Q= n X λi f (v i ) = 0. λi v i = i=k+1 f d4/Z.JI6>@.93%.9P IQ3 x = UR _"9 a .9P IQ3.= ./Z=H.`AbD;/<;\Q= 5 /Zd<>@.9I P!D=E= H D= >@.= D>H/Z.`.3U;I.= k v ,...,v c Y Pk Null f >H9. 3k.93;S IU.9= .9P IUQ3 DFOLCNDd<;%.9/4=H.Bh}/Z=E.1D3UPQ 5Hk/4=EDI/4Q= 9. P IQ3.= x x = i=1 λi v i k Y >HD3U;I.dZd4.9= -@Q /_Ig.3CODdZI.9=B8i/43 0 Y k X λi v i = x = i=1 n X λi v i i=k+1 D?D5O.3`>@/Z. Q@>@.93`fFE/Z\!Dd4.9= I λ1 v 1./Z+ · · · + λ v − λ d4>@.= v k+1±O− · · · − λ v = 0 0ic .9P IUQ3U.9= =H.kAbD;/4;\kQ= k 5H/4k+1 ;;.9=Dd4dZ. n Q).nkV/Z.= I.9=FH=E>>HD /ZI v , . . . , v n;.9/4= V S¤Y ² Y DFOLC Jd4.9/<1LC [ FHd4d} /<>@.3U;GH3UFELC ¸ /4= λi 0 1 0 7F V9.9/4J.=j¼ -).9/ ln±H3e./Z= /43`;IU.9d4dZ.= Bild f ⊆ span(B) 0 f (x) ∈ Bild f x ∈ V 0i1 Pn >@FH3LCe>@/Z.zAbD;/<;\.9P IQ3.=]>HD3iFH=E>p.93UCEDdZI.= x= i=1 λi v i -)Q j4¨} Y /ZIg/<;I TW f (x) = f (λ1 v 1 + · · · + λk v k + λk+1 v k+1 + · · · + λn v n ) = λ1 f (v 1 ) + · · · + λk f (v k ) + λk+1 f (v k+1 ) + · · · + λn f (v n ) = 0 + · · · + 0 + λk+1 f (v k+1 ) + · · · + λn f (v n ). f (x) Dd4;b./Z=H.zh}/Z=E.D3UPQ ¡pjE¢!®2(+ |@~ $) Y 5H/4=EDI/4Q=]>@.3i.9P IQ3.=p/4= B >HD3UJ.;IU.9d4d_I 0 .93m=EfLCO;IU.%-@DIV'.3P)d<f3Iq8bD3UF ./ZJ.9= Id4/<LCT.9/4=H.AbD;/<;qDFELCuAbD;/<;%J.9=ED=H= I8i/43U>¤¼ < .>@.3 c Y .PIUQ3i/<;I9 : pDd<;b.9/4=H.zhj/4=H.D3PQ 5H/Z=OD!I/4Q=p>@.93gAbD;/<;\.9P IQ3.=]>HD3;I.9d4d45ED3 A Y 0 78:.9/¤d4/Z=E.D3U.qfF . FH=E> ;/4=E><9!6!»7 l+DdZd<;b.;:.9/4=H.9 9: !'dZ/4=H.D3. 5H5E/Zd<>@FH=HJ V W Y S s J/Z5HI f :V →W 0 {}|@~TjZ , -)./ .9/4=B.=E>@d4/4LCH¥ >@/ .9=E;/4Q=EDdZ.3:d4/4=H.D3.3bqDF ±E5O.3g.9/4=H. Y Y ./Z=H.zAbD;/<;b\Q= ;Q6d4f;;Ig;/<LC%.>@.3g.9P ½ 0 M;I B⊆V VS ./4;.aDd<;b.9/4=H.zhj/4=H.D3PQ 5H/Z=OD!I/4Q=p>@.93i.PIUQ3U.9=eDFE; 1 Y ;Q%/<;I M;I VFB>H. .P IQ3U3DF /<;Q dim V = n S V y 0 H Y Y Fn Y EN V 3G7.93 F0 Y ²q´ o IQ3 DF@l 97º A x∈V >ED3;IU.9d4dZ.= B 0 Q3UGHC 0 u ./Z=.9=E>HdZ/<LCH.93 3G7.93 /_I M=E;57.;Q=E>@.3.'J/4d_I¼OM;I ¸ dZ. .9= I.= ;Qt57.;I.C IX.>@.93m.9P IQ3¥ ²q´ Y Y S F.9O qF H > 9 . 3 / = ; 4 / Q = D E F b ; J . E = D .P IQ3U.9= 3DF c Y 0 Y V ⊆ Fm n (n ≤ m) |V | = q n #%$ &'$)(+*, 7F¿ ½À ¼¤-)./ /_I ./Z=H.%AbD;/4;i\Q= FH=O>T;./ .9= B = {v . ., = v n } Y n = dim V V xP ∈ V 0Oc 1 ,8:. .9.J .9P IUQ3 P!D=E= D=B Dd4;:./Z=H.ah}/Z=H.D3UPQ 5H/4=ED!IU/ZQ= n x Y V = span(B) Y x = =E>@.9aFHiIv/4J i >@.3¦AbD;/<;\.9P IQ3.=>HD3U;I.dZd4.9= r V9FV9./ZJ.9= >HD;;m>@/4.;. D3;I.9d4d4FH=HJeDFELC.9/4i=1 0 Y S c Pn /<;I ;./ ./Z=E.m57.9d4/4.95H/4J. D3;IU.9d4dZFE=HJ\Q= D=H=BJ/4d_I x = i=1 bi v i x 0Ec S c 0=x−x= [ FE=]/<;I B n X i=1 ai v i − n X bi v i = i=1 Dd<;:A:D;/<;d4/4=H.D3FH=ED5HCEf=HJ/4J n X (ai − bi )v i . S i=1 Dd<;Q Y FE;U; ai = bi ln±H3bDd4d4. i = 1, . . . , n J.d_IU.9= 0
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