x 3 Me stoiqei deic prˆxeic mporoôme na fèroume ton epauxhmèno pðnaka [A B C] sth morf /5(3c

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1 Grammik 'Algebra II, IoÔnioc 2009 Jèmata prohgoômenwn exetastik n Didˆskousa: Qarˆ Qaralˆmpouc Grammikˆ Sust mata. 'Estw , X = 2 3 x x 2, B = 5 0, C = c c 2. 5 x 3 Me stoiqei deic prˆxeic mporoôme na fèroume ton epauxhmèno pðnaka [A B C] sth morf 0 3 /5(3c 8c 2 + 0c 3 ) /5( 9x + 29y 5z) c 2c 2 + c 3 a) Na lôsete to sôsthma AX = 0 kai na breðte mða bˆsh gia ton mhdenoq ro tou A, Null(A). b) Na lôsete to sôsthma AX = B. g) Na breðte sunj kh ètsi ste to sôsthma AX = C na mhn èqei lôsh kai na breðte ènan pðnaka C ètsi ste to sôsthma AX = C na mhn eðnai sumbatì. 2. An o pðnakac A eðnai 2 3 poièc eðnai oi dunatèc diastˆseic tou null(a)? D ste apì èna parˆdeigma sth kˆje perðptwsh. Na breðte tic lôseic tou sust matoc x + 2x 2 = 0 kx + λx 2 = 0 giˆ ìlec tic dunatèc timèc twn k kai λ. Idiotimèc, IdiodianÔsmata, Qarakthristikì polu numo. 'Estw 2i 0 0 i i a) Na breðte to qarakthristikì polu numo tou A kai na deðxete ìti oi idiotimèc tou A eðnai Ðsec me 2i, 3, 3. b) Na breðte touc idioq rouc g) Na breðte ènan antistrèyimo pðnakac P kai ènan diag nio pðnaka D ètsi ste D = P AP. 2. 'Estw h grammik sunˆrthsh φ : V V kai èstw ìti to v eðnai idiodiˆnusma me idiotim 3 kai ìti to v 2 eðnai idiodiˆnusma me idiotim 4. Na apodeðxete analutikˆ ìti ta dianôsmata v kai v 2 eðnai grammikˆ anexˆrthta. An to 3 eðnai idiotim gia ton pðnaka A na deðxete ìti to 27 eðnai idiotim gia ton pðnaka A 3.. c 3

2 2 'Estw A M n (K) kai λ eðnai mða idiotim Na apodeðxete ìti λ 3 2λ + eðnai idiotim tou pðnaka A 3 2A + I 4. Na d sete sunj kec gia ta a, b, c C ètsi ste o pðnakac a b 0 a c 0 0 na eðnai diagwniopoi simoc. 3. 'Estw φ : R 3 R 3 h grammik sunˆrthsh pou peristrèfei ta dianôsmata tou R 3 gôrw apì thn eujeða pou orðzetai apì to v me gwnða peristrof c θ = π/9 kai katìpin ta pollaplasiˆzei me to 2. QwrÐc na upologðsete ton pðnaka thc φ na brejoôn ta idiodianôsmata thc φ kai oi antðstoiqec idiotimèc. EÐnai h φ diagwniopoi simh? EÐnai h φ orjog nia? 4. 'Estw f : R 3 R 3 h grammik sunˆrthsh pou antikatoptrðzei ta stoiqeða tou R 3 wc proc to epðpedo pou orðzetai apì ta dianôsmata v 2, v 3. Na breðte tic idiotimèc kai touc idioq rouc thc f. EÐnai h f diagwniopoi simh? EÐnai h f orjomonadiaða? 5. Na breðte endomorfismì me idiotimèc kai 2 kai anðstoiqa idiodianôsmata (, ) kai (2, ). 'Estw ìti to qarakthristikì polu numo tou A M 3 (C) eðnai to polu numo c(x) = x 3 + 5x 2 + x. Na apodeðxete ìti o pðnakac A den eðnai antistrèyimoc. 'Estw D o diag nioc pðnakac D = 'Estw S antistrèyimoc pðnakac me st lec v, v2, v3, kai SDS. EÐnai o A diagwniopoi simoc? Na brejoôn oi idiotimèc Dikaiolog ste pl rwc thn apˆnthsh sac. 'Estw ìti o pðnakac A M 3 (R) èqei idiotim to, me algebrik pollaplìthta 2. Gia kˆje mða apì tic parakˆtw peript seic na d sete, (an eðnai dunatìn), èna parˆdeigma enìc tètoiou pðnaka A, ètsi ste : () to na èqei gewmetrik pollaplìthta 2 (2) to na èqei gewmetrik pollaplìthta (3) to na èqei gewmetrik pollaplìthta 'Estw ìti p A (x) = x 4 2x 3 + x 2 3x + 2 eðnai to qarakthristikì polu numo Na brejeð det(a). Na brejeð Tr(A). Na grˆyete ton pðnaka A wc grammikì sunduasmì dunˆmewn Na grˆyete ton pðnaka A 4 wc grammikì sunduasmì dunˆmewn Na grˆyete ton pðnaka A 5 wc grammikì sunduasmì dunˆmewn 7. Na diagwniopoi sete monadiaða ton autoprosarthmèno pðnaka [ 0 ] i + i 0

3 8. 'Estw () Na brejoôn oi idiotimèc tou A kai to qarakthristikì polu numo (2) Na brejoôn oi idioq roi (3) Na brejeð an eðnai dunatìn orjog nioc pðnakac P kai diag nioc pðnakac D ètsi ste D = P AP. (4) Na grafeð to diˆnusma (,, ) san grammikìc sunduasmìc twn idiodianusmˆtwn 9. 'Estw ìti to qarakthristikì polu numo tou A eðnai to polu numo x 4 + 5x 3 + 2x + 3i. () Na grˆyete (qwrðc na upologðsete) ton antðstrofo tou A san grammikì sunduasmì dunˆmewn tou A qrhsimopoi ntac to je rhma twn Cayley- Hamilton. (2) ApodeÐxte ìti to ginìmeno twn idiotim n isoôtai me 3i. Na brejeð to ˆjroisma twn idiotim n. (3) EÐnai o A autoprosarthmènoc? An φ eðnai o endomorfismìc pou antistoiqeð ston A, eðnai o φ isometrða? Eswterikì ginìmeno, Fasmatikì Je rhma. 'Estw, to eswterikì ginìmeno sto R 3 pou orðzetai apì tic timèc e, e = 2, e, e 2 = 3, e, e 3 =, e 2, e 2 = 4, e 2, e 3 = 0 kai e 3, e 3 = ìpou B = {e, e 2, e 3 } h sun jhc bˆsh. 'Estw u = (, 2, 3), v = (, 0, 5). Na upologðsete u, v. Na upologðsete genik tera to (a, b, c ), (a 2, b 2, c 2 ). 2. JewroÔme ton Ermhtianì q ro C 3 efodiasmèno me to sônhjec eswterikì ginìmeno. Gia ta stoiqeða x = ( + 2i, 2, i) kai y = ( + i, 2, 2i) na upologðsete tic timèc x, y kai x. JewroÔme ton Ermhtianì q ro C 2 wc proc to sônhjec eswterikì ginìmeno. Na apodeðxete ìti ta stoiqeða (w, ) kai (w, ) eðnai orjog nia, ìpou w C kai w 203 =. 'Estw V ènac Ermhtianìc q roc me eswterikì ginìmeno, kai v, u dôo stoiqeða tou V ètsi ste u = 2, v = 3 kai u, v = i. Na upologðsete to 4u v. 3. DÐnetai to epðpedo U = S({v, v 2 }) tou R 3, wc proc to sônhjec eswterikì ginìmeno, ìpou v = (, 0, 2) kai v 2 = (,, 3). Na brejeð mða orjog nia bˆsh gia to U. Na epektajeð se orjog nia bˆsh gia to R 3. Na brejeð mða orjokanonik bˆsh gia to U. Na brejoôn oi suntetagmènec tou (, 2, 5) wc proc th bˆsh tou deôterou erwt matoc. Na grˆyete to stoiqeðo (, 2, 5) san ˆjroisma enìc stoiqeðou apì to U kai enìc stoiqeðou apì to U. Na brejeð h elˆqisth apìstash tou (, 2, 5) apì to U. 4. Na breðte thn probol tou v = (0,, i) pˆnw sto diˆnusma v. 3

4 4 'Estw U = Γ(A) ìpou Na brejeð o U. Na apodeðxete ìti [ + i 2 ] i i 3 (V + V 2 + V 3 ) = V V 2 V (V + W ) = V W. 5. 'Estw v, v 2, v 3 mða orjokanonik bˆsh gia to C 3 kai f : C 3 C 3 h grammik sunˆrthsh, pou orðzetai apì tic sqèseic f(v ) = iv 2, f(v 2 ) = v 3, f(v 3 ) = v. Na apodeðxete ìti h f eðnai monadiaða. 6. 'Estw A M n (C) orjomonadiaðoc pðnakac. Na deðxete ìti det A = kai ìti an λ eðnai idiotim tou A tìte λλ =. 7. Na elègxete an h grammik sunˆrthsh f : C 3 C 3, f(x, y, z) = / 2(x+ z, 2 y, x z) eðnai monadiaða. 'Estw u = (,, 0), u 2 = (,, ), u 3 = (,, 2). 'Estw f : R 3 R 3 orjog nia sunˆrthsh, kai f(u ) = (0, 2, 0), f(u 2 ) = ( 3, 0, 0). Na upologðsete ìlec tic dunatìthtec gia f(u 3 ). 8. DÐnetai h grammik sunˆrthsh φ : C 2 C 2, φ(x, y) = ((2+3i)x+5y, 4ix+2y). Na upologisjeð h φ. 'Estw v = (, i). Na brejeð diˆnusma w C 2 ètsi ste < φ(u), v >=< u, w >, u C 2. Na elègxete thn apˆnthsh sac gia dôo u thc epilog c sac. 9. 'Estw φ : C 3 C 3 o endomorfismìc tou ErmhtianoÔ q rou C 3 me to sônhjec eswterikì ginìmeno pou orðzetai apì th sqèsh φ(a, b, c) = (a + ic, b, 2a + b). a) Na brejeð φ (a, b, c). b) Na brejeð diˆnusma w ètsi ste giˆ kˆje v C 3, na isqôei < (, 0, i), φ(v) >=< w, v >. 0. 'Estw φ : C n C n autoprosarthmènh grammik sunˆrthsh, dhlad φ = φ. Gia poiˆ k eðnai h sunˆrthsh kφ autoprosarthmènh? 'Estw ìti ψ : C n C n eðnai epðshc autoprosarthmènh. Na apofasðsete an ψ + φ eðnai autoprosarthmènh. An A M n (C) na deðxete ìti AA eðnai autoprosarthmènoc. An h φ : C n C n eðnai autoprosarthmènoc na deðxete analutikˆ ìti λ = + 3i den mporeð na eðnai idiotim tou φ.. 'Estw A M n (C) me idiotim λ = 2i. Na deðxete ìti λ = + 2i eðnai idiotim tou A. Na apofasðsete an ja mporoôse AA = I n. Na apofasðsete an ja mporoôse A Na kataskeuˆsete ènan tètoio pðnaka A M 2 (C) (me idiotim 2i) pou na diagwniopoieðtai monadiaða. 2. () JewroÔme ton R-dianusmatikì q ro R 3 me to sônhjec eswterikì ginìmeno. 'Estw v, v 2, v 3 R 3 tètoia ste < v i, v j >= 0 an i j en v i = 2 gia i =, 2, 3. 'Estw A o pðnakac me st lec ta dianôsmata v i. Na apodeðxete ìti A = /4A T. 3.

5 (2) Na perigrˆyete to fasmatikì je rhma. (3) An A eðnai orjog nia diagwnopoi simoc apodeðxte ìti A 3 eðnai orjog nia diagwnopoi simoc. (4) 'Estw A M 3 (C) me idiotimèc, ω kai ω 2 ìpou ω 3 = kai ω 2. Na apodeðxete ìti A 3 = I n. 3. a) Na breðte mða orjog nia bˆsh gia ton upoq ro U tou R 3 pou parˆgetai apì ta dianôsmata (, 2, ) kai (, 2, 3). Na breðte to U. b) Na deðxete ìti an h grammik sunˆrthsh φ : V V twn R dianusmatik n q rwn eðnai isometrða, tìte oi idiotimèc thc eðnai ±. Na apodeðxete ìti to antðstrofo den isqôei pˆnta. g) Na apodeðxete ìti an o φ eðnai autoprosarthmènoc, tìte oi idiotimèc tou φ eðnai pragmatikoð arijmoð. d) 'Estw A M 3 (R) me idiotimèc, ω, ω 2 ìpou ω 3 =, ω. Na apodeðxete ìti A 3 = I 3. Na breðte to qarakthristikì polu numo 4. 'Estw i a) Na breðte to qarakthristikì polu numo tou A kai tic idiotimèc b) Na breðte touc idioq rouc g) Na breðte ènan orjomonadiaðo pðnakac P kai ènan diag nio pðnaka D ètsi ste D = P AP. d) Poiˆ idiìthta tou A egguˆtai ìti o A diagwniopoieðtai orjomonadiaða? 'Estw λ C kai B = λa. EÐnai o B orjomonadiaða diagwniopoi simoc? e) Na grafeð to diˆnusma (a, b, c) san grammikìc sunduasmìc twn idiodianusmˆtwn 5

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