fylladio5-2005 - FULLADIO 05 Grammik 'Algebra SHMMU 2005-06...

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Unformatted text preview: FULLADIO 05 Grammik 'Algebra SHMMU 2005-06 1. H isìthta x ∗ y = x + y + x2 y 2 orÐzei mÐa prˆxh ∗ sto R. Na breÐte to oudètero stoiqeÐo thc 1 prˆxhc ∗ kai na deÐxete ìti kˆje stoiqeÐo x ∈ R∗ me x < √ èqei dÔo summetrikˆ stoiqeÐa, en¸ 3 4 1 kˆje x ∈ R me x > √ den èqei summetrikì stoiqeÐo, wc proc thn prˆxh aut . Ta stoiqeÐa 3 4 1 0, √ èqoun summetrikˆ kai poiˆ? 3 4 2. JewroÔme to sÔnolo a −b M= : a, b ∈ R , b a me prˆxeic thn {+} prìsjesh pinˆkwn kai {·} pol/smì pinˆkwn. Na deÐxete ìti: i) H algebrik dom (M, +) eÐnai antimetajetik omˆda. ii) H algebrik dom (M ∗ , · ) eÐnai omˆda. iii) H algebrik dom (M, +, · ) eÐnai s¸ma. Katìpin na lÔsete sto M thn exÐswsh 2 + I = O. X 3. 'Estw h omˆda (G, · ). i) Na deÐxete ìti gia kˆje a, b ∈ G isqÔei: (a · b)−1 = b−1 · a−1 , ìpou me x−1 sumbolÐzoume to summetrikì (antÐstrofo) tou x ∈ G. ii) An isqÔei h sqèsh (a · b)2 = a2 · b2 gia kˆje a, b ∈ G, na deÐxete ìti h omˆda G eÐnai antimetajetik . 4. O daktÔlioc (∆, +, · ) lègetai daktÔlioc tou Boole an isqÔei: x2 = x gia kˆje x ∈ ∆. Na deÐxete ìti kˆje daktÔlioc tou Boole eÐnai antimetajetikìc kai isqÔei x + x = 0 gia kˆje x ∈ ∆. 5. 'Estw (G, · ) mÐa omˆda kai H ⊆ G, H = ∅. To H lègetai upoomˆda thc G an to H eÐnai omˆda wc proc thn prˆxh · thc G. Na deÐxete ìti èna uposÔnolo H = ∅ thc omˆdac G eÐnai upoomˆda thc G an kai mìno an isqÔei x · y −1 ∈ H gia kˆje x, y ∈ H , ìpou y −1 eÐnai to summetrikì tou y ∈ G. S. Karanˆsioc m r fˆy† h hg x cge •— c† d ygd †y„ ¤gƒ s‚wy…–!…i€˜˜ef “y r i‰sAsy c #‡ m ey fˆy† h hg “s„ ƒ es‚yy…–x „ ƒ egwj“€‚— ˜™eˆ r ”d ƒ ffdce h f†• ey d f c‘y d aˆ q ff a h f†•— › ‘ †y„ Asƒkeƒn Au ekdjy“€˜¤r t ”¢sAsy c ‰‡ m h d r fˆy† h hg x e…f f† • ey †y„ 6“y ”ƒ es‚yy…–™…¤“€˜— ˜£…sy c gd l c— h xs˜yge u Y m h fy l† aˆ l† f e…f f†•— › ‘ †y„ dl¢˜ekeo˜–dxc r …ey“€˜¤r t ”¢s…y c ‰‡ m d x g e a f f † l e Ÿ lm  ‘ x  –…iec eˆ ƒ kd#yh gˆ u Wxs–x€dxa Œ † {djwz r r A˜s˜e¦kdjg”c “† f h ¡eˆ†yg†f a hfd a kdf t r t € m › h†y es…„ ge g˜y rž“€ˆ u ’iEd„ ƒ s‚yy…–Š“y ‚€wY v m f† • g c › fˆy† h hg x e y x ey fd a l‘y a† fey x eˆ†y—fge ey dˆ “y”c “† Apde“€d•g“€Te˜s‚…gŠ˜‰‡ v m †s…„ u e } y h”ec “† fda †y Ÿ l‘y f†• r …„ Tm Ap‰žr“€ˆŒ u g cg!œde“€’•”˜€Qe˜—e ˜—ˆ s‚…ge m ˜€wY v m n ƒ e|y“‚wkhgq”d ƒ gec “Šxsgd „ eˆ u {d”e u š‘ ˜–™‘ ir ˆ u d › a† fey x †y—fg y x fy† zdy† fycd f fd a† l ‡c e a e eyx †y„ s…sy c‡V• VUS ‰oŽW¨TR n ƒ ‚€˜w”ec “† ’–¦ƒ‰‡ v fˆ†y  x† fd a – ‘ dˆ f ca df ”d ƒ xsc y ƒ 6gd ƒ geˆyc f q feˆyc „ he e…f fd a† ‘xdy ˆ gs˜— ƒ ”e iƒ … ƒ gs˜† ƒ yi”…¨gec “Š€g˜€x r gd u l‘y d ‘ dˆ Ap¦n ƒ f „ eˆ t ’e‰‡ v m n hp dhn• …‰yA”qˆ l…eh fˆ …iyyz egd l… ff f h lgey dfd a †y …ie ƒ egd t j“…g˜o„ u A”˜’y“† q s…e u d i“#€x Aiƒ € m ’l“eid „ gey ‘ rq a†c ‘e dˆ ‡y fd a† l a hdcgfe f l‘y l†y‘y f f f Aˆ u xp6”c “#ikdjie…ei–dxAp#˜s…pA{„kfy‰v m h d hdc…fe ffdy „ nygd ‘x—•x € V  V U S 6yie…g–dxs”˜‚† ir ˆ t pg¤€˜€–Qx$W¨TR m gef g˜— u f u † gey a† zg† c‘y † h ad i˜&kd˜y“˜‰ApŽywegwh ieAe ir x”c “‹€g˜€x r gd u ldc Œ e fd a† ‘xdy ˆ l‘y d ApŠn ƒ f „ eˆ t € m n kdgy‰v v feh p d hn• ˆ ey ‰yA”qT˜€x dˆ ‡y fdy† a ‘ xdy ˆ l‘y d exp–”˜‚c ”d „ –i˜–x r id u As†n ƒ f „ eˆ t € m n h d hdc…fe f fdy „ nygd ‘ x— • x € V ~ V U S 6wge…g–dxy”˜‚† gr ˆ t pgƒ–‚€6‚yW¨TR m m‘ e €q f } ˜y ‘y— f ff † h c † x— f ff c y f g dyf d x„ l† x„yf d a† d z l a hdcgfe f f† rq f e l c— h p|ekˆgkdwbx‚–˜kˆ”{dqx‚#d u ge q „ u ‰˜egd ƒ 6iw˜–gs”c “#yjikdjie…g–dxxdl“–x Aiƒ ”xxs˜yge u Y α α |z | =2 ⇔ α|z |2 + (α2 + z∈ : |z |2 + 1 α +1 1 1)|z | + α = 0 ⇔ |z | = α |z | = α (0, 0) 1 α α x, y ∈ R x + y, x2 y 2 ∈ R ⇒ x + y + x2 y 2 ∈ R ∗ R e ∗ ∀x∈R x = x ∗ e = x + e + x2 e2 ⇔ e(1 + e) = 0 (1) e=0 ∗ e=0 x x∈R x ∗ x = 0 ⇔ x + x + x2 x 2 = 0 (2) x=0 x ∆ = 1 − 4x3 √ −1 ± 1 − 4x3 3>0 • 1 − 4x x1,2 = 2 1 x3 < 1/4 ⇔ x < √ x x 1 , x2 3 4 √ 1 1 √ • x= √ x =−32 3 3 4√ 4 −32 1 • 1 − 4x3 < 0 ⇔ x > √ x 3 4 C∗ |w| |w|2 + 1 |z |2 |z | = +1 (x, y ) ∈ S ⇔ x3 = x2 (y + 2) ⇔ x2 (y − x + 2) = 0 ⇔ {x = 0 y − x + 2 = 0} yy y−x+2=0 [0], [1], ..., [ν − 1] (x, y ) ∈ S ⇔ x2 − y 2 ≤ 0 ⇔ (x − y )(x + y ) ≤ x − y ≥ 0, x + y ≤ 0} ⇒ x − y = (k1 + k2 ) ν + v1 − v2 ⇐⇒ v1 − v2 = 0 ⇔ v1 = v2 . (1) dege i”ƒ kdf t m f ur †y„y ‘ rq f a c‘yx cg nc s…ss€x Aiƒ kdekf¦Ap€o…ge ƒ sgd ‰‡ c eyx l a hdcgfe f ‘x—•x fd a “€#ikdjie…g–dx™€˜€–6”ec “† ‘ dˆ ’‰‡ v Z x − y = k ν (1) ≡ (mod ν ) k∈Z lh xw‰v v m 0 ⇔ {x − y ≤ 0, x + y ≥ 0 ν x = k 1 ν + v1 y = k2 ν + v2 x, y x − x = 0 = 0 ν ⇒ x ≡ x(mod ν ) x ≡ y (mod ν ) ⇒ x − y = k ν, k ∈ Z ⇒ y − x = (−k ) ν, −k ∈ Z ⇒ y ≡ x(mod ν ) x ≡ y (mod ν ) y ≡ z (mod ν ) x−y = kν (+) k, λ ∈ Z =⇒ x − z = (k + λ) ν, k + ν ∈ Z ⇒ x ≡ z (mod ν ) y − z = λν †y„ s…sy f gd ƒ y x v ‚€bY wr u rt rp fd a Y VX V U S sqihgec b`6¦W¨TR PB QEDCI'B HGFEDCA97 164¢¢10(&%¦$#!¨¦¤¢ B B @8 ) 53 2  ) '   ¥ £ " £     ©§ ¥ £ ¡ m l‘y d e g c— † d he fd a ey dˆ d he fd a eyf Apsef r hgge u #lx˜yhie u 6f”d ƒ f r i¤”c “† ˜’Œ ‰‡ v m ef r g6”ec “† “–q”d ƒ … ge „ d ey f†… • f dy‘ f f fy f dy† e ˜€x¤˜…€6’x™˜…py…k„kf¦n ƒ €dxkˆfg˜‚–xgeˆ u € m gƒ t ‘ ir ˆ u Au Ap#ieˆ u x’€dx˜† gƒ gd ƒ gec “† ˜y c‘y le l  „y f f fd a e nf i q ‘ } † yz r kdf A˜— q a u Y m ce rd f d f †yz ”ƒ kdf t ˆ r gd ƒ †…„ ·m r yŸ … ge „ d e‚£˜€eˆ u Eg› dc— f† • r yx ˜€wY v h eˆy a cd ey f ld he f ld he l‘y fˆe ey d ex† fd a† … zg† V ¶ V U S 6e „ s–’xeysgT“ƒ{d¨ª m ief r gw”d ƒ gf r ige u g!ApŽ„ykhi¦cA“ƒ„ u T–˜yh r ”c “’y“˜Te Ÿ ŽW¨TR m † hg •— r › y…ge€˜bAEµ› } rs³pŠ„ u ‡ m f ”d a † e ˆ — h e y u u ¯ ® ƒ¬ « ­ mc “$lg˜y’˜yˆ†‡ …ie„e kˆ”“d‡a ”u± jr¬ yy}°›u« ”g´„ ju –~‡ hg c r t› m h†hge ew…g€•˜— a u † r f Œ e!„ u Wrs³p†cApbg˜c q e ¨m r ›t †f d › ‘y ldy ² ¦ h†hge ey…i€•˜— a u † r m ‚€dxeˆg„ u d!s³p&cApy’lg˜sc q e ¦ ”jwy””QjgX ˆ†y f d r › ‘ dy ² ±¬ « ° ¯®­ ¬ « hi”dc “† fa †r f yz ”ƒ kd%ª † r fª V¨ VUS yz ”ƒ {d¨1w©W¨TR m x∈H x −1 = 1 · x−1 ∈ H x∈H⇒x H H⊆G G 1 = x · x−1 ∈ H x, y ∈ H ⇒ x · y = x · (y −1 )−1 ∈ H << · >> H G H H=∅ x∈∆ (x + x)2 = x + x ⇒ (x + x) · (x + x) = x + x ⇒ +++ =x+x⇒x+x+x+x=x+x⇒x+x=0 x, y ∈ ∆ (x + y )2 = x + y ⇒ x2 + x · y + y · x + y 2 = x + y ⇒ x + x · y + y · x + y = x + y ⇒ x · y + y · x = 0 ⇔ x · y = −y · x (1) x x2 · y = −x · y · x ⇒ x · y = x x · y · x = −y · x2 ⇒ −x · y · x = y · x (3) −x · y · x x·y =y·x y · x + y · x = 0 ⇒ y · x = −y · x (5) x·y =y·x x2 x2 x2 h†hge• ey…i€˜— x2 (a · b)−1 = b−1 · a−1 ⇔ (a · b) · (b−1 · a−1 ) = (b−1 · a−1 ) · (a · b) = 1 = a · [(b · b−1 ) · a−1 ] = a · (1 · a−1 ) = a · a−1 = 1 (b−1 · a−1 ) · (a · b) = 1 (ii) (a · b)2 = a2 · b2 ⇔ (a · b) · (a · b) = a2 · b2 ⇔ (a−1 · a) · [(b · a) · (b · b−1 )] = (a−1 · a2 ) · (b2 · b−1 ) ⇔ b·a=a·b x2 − y 2 = −1 2xy = 0 01 X= −1 0 kdgy‰v v m feh ⇔ = m lh xw‰v v m x −y y x n eydc…f ˜i…eid ih”c bY fda (i) x −y y x m (a · b) · (b−1 · a−1 ) V§VUS ¢W¨TR x = 0, y 2 = 1 y = 0, x2 = −1, −1 0 0 −1 0 −1 ⇔ x = 0, y = ±1 ⇔ X = 1 0 X 2 + I = O ⇔ X 2 = −I ⇔ ⇔ m yx€6”ec “† dh‡xfda d eˆ‰‡ v m fy„ †y a†ˆ‘ydˆ ¦ d ce A¢s€dx“Apieid `m ef r Ayh †h y$lgdef q …sy ƒ ef$i„ ƒ ‚†y“|†ykhid£i„ ƒ f‚efAwh¦‘w–f”c “† d l fy zdy fyc l c† ‘ h d a fy„ f d he fy† zdy† fycd fd a …–”d ƒ ef r g!n ƒ ‚y˜‚ykhi–”c “† } ihgec b$y¦W¨TR fdaY V¥ VUS f „ fd † a† dˆ ‡ y…¤”˜y‚c€d•“ef™expyŽd q e ƒ wY m … ey fˆy† h hg x f† •— › c‘y “s„ ƒ e|yy…–¤“€˜qr t ™AsŽ„ u d eŸ 10 01 xy −y x 00 ∈M 00 (M, +, ·) 1 x −y ∈ M∗ ⇒ A−1 = 2 y x x + y2 ∈M O= x =0 ∈ M∗ (M, +) m (M, +, ·) A= I= x=0 ...
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This note was uploaded on 10/02/2009 for the course G 001 taught by Professor Shmmygr during the Spring '07 term at National Technical University of Athens, Athens.

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