100305_ProblemsofIsom_Solutions

# 100305_ProblemsofIsom_Solutions - Rsswms 03¢ Isonthwvs...

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Unformatted text preview: Rsswms 03¢ Isonthwvs- Sawtzm | 1. Consider the subset Saxﬁ R) C M'gxﬁ R) consisting of the symmetric matrices= that is= those which equal their transpose. Show that ngg( R) is actually a subspace of Mrgxgt: R) and then determine the dimension and a basis for this subspace. Construct an isomorphism based on your basis from Shy: R) to R“. POD 4 0 O D ‘ _ . O ___ 0 O O '3 31Msz 6 S O 0 D If A63, BéS (As—376.; Ange A+B gs (CAjC=CAtz—OA 65 Hence. 8 is a Slabs/00ch 0]; M_ J: F07 W7 A68, a be A: 10 aka caf- 2. Show that the polynomials P1 = 2 — x: P; =1 + x + x3: and P3 = 3 x — 2 x3 from P; are linearly independent. Hence they form a basis of P; Construct an isomorphism based on this basis from P1 to R“. Hoof, La; 0424) +190 +x—r—X7’) +c,(3><—2,><")>o r9 20L+Jp :—o ——OL-t'io +5clo [0 «gaze 2 I D —\ r 3 S /4L—é——2=~/2_—'t=0 o l "‘2 Hana. 171% Me lit/L. l—I/Lokl 1.»er fix): 0w bx+c><"-= Clé'xttGW’Fx‘) + Cs (Sx—ax‘) 2— ( Diox —| [3:19 o I‘Zg 0 Yr; -\I 3 b I—I‘S -b 1' 0 “ﬂ 0 3 6 Grab o (2' o o lea C. C! ': “b +5 01+ = ’10 1- ;L (we +3 9+ :5 (web —3 o) qu-Pﬁb +%C> h, ﬂ (7%): 23-stst won-’2 7i aw} (a \$0) 3, Show that the space gonoratcd by the row vectors of a matrix is isomorphic to the space gcncrat cd by its columnvcctors. ooj“. B? ‘le' 3.6, W oil-menéiﬂl 030 £6544 facets :Vm cf A _ -=> M We {Sanka/[bksz 4. Show that T: P1 —> 11%? dcfmcd byTﬁa+ bx] = [a — b, b) is an isomorphism. Catt you find thcbasis that this isomorphismis based on? Fr- ism. ow (RJJOtIMCFiZFZ. \ . We 0an W tb 8W JUN" W” [34122, TOuﬁbx) —=O 5; iii—7'0 __.__; Sb NC‘U Sfasj - ' T is 1-4. Suﬁ;st mow) =5; (was , ,8: f m 7:; ﬁ(‘7i):<l, b) —_9 d—jozlgft:oi A \S 5. Checkifthefollowing are isomorphism: ajTﬁa1,a2,a3)= a1 + [:11 + agjx + (a1 + a2 + (13))? 13') Ttalrﬂﬂrag) = '11 +311 + ‘12)”? C} Ttalﬁmaaj = (‘11 + [12412 + Haﬁz + (11,011 + [12 + ‘13) d)T(a1ra2’a3)= (‘11 + ‘12 + (13:91 + a2rﬂ'1 + ‘12) - 1R3 —9 Ema) alga air/'0 T( a" I all as) :0 g a“"'&.>,=o % arr—’0 at+a¢+0g=o “5:0 T [S 1"[ Tea. 5) "25 —> Rail) dim ((23) 4: ohm (303) N? C) R339 gf ND 0!} H? 9,91_ — T(a.,az.,az,):3 % )5 §,':g::gs-o aI+QL=o I l l 1 . O :01 ob \$[oo—{,o 1 I D 0 a o o 'f o ...
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100305_ProblemsofIsom_Solutions - Rsswms 03¢ Isonthwvs...

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