Question

For each transformation below, find the closed form of the transformation. 1) Let T be a linear transformation from R$ to M22 (R) [i Let B=1 0:00 [. :] [11] [12] [0 ] Let C= 12 41 -17 -5 65 -27 92 Let M = be the matrix transformation of T from basis B to C 17 58 -15 -51 81 The closed form of the transformation is Tb 3-1 2) Let T be a linear transformation from P3(R) to M22 (R). Let B= (1+2x+(-3) x2 + 3x3),(-1+(-1)2 + 2x2 + (-3) 2*), (2 + 3.c +(-4) x2 + 5.r),(-1+(-1)2 + 3x2 + (-3) 23). Let C= [ ][01] [1 ] [ ] -6 -6 8 8 Let M 8 -9 28 58 13 - 15 46 96 be the matrix transformation of T from basis B to C. -23 -50 -23 -50 The closed form of the transformation is T(a + bx + cx+ d.23) =

Answer 1

Page 1 T(arbat cx²fd83) = (-4at btc) 8 -4 28 57 + (34-c-d) -6 8 -23 -40 + (3a-d) ) 13 46 १५ + (-2a+b+c+d) -6 8 -23 -1 । उप-184+4+कि 35 न -390+gb 180 360-4b-१८ 1120+28b1990 -2284+526+57c + -18a +6 -194 -690+236+4bd 24 - 8C-16d. -1440+48८+960 + 94-13d 1380-46d -45a+15d 3894-94d +120-6b-6c-60 -1ba+8[bchi ५६६ -93 (b+c+d) ५२० -1(bid) +Ma+26+8cd ---१c+d 30+56+280-23d, -490136648५८नव T (atbat ca² tdas) 2 4+2b8cIN 34+ 5 b12-23d -a-b-१c+nd 4+ 36b+ 846 -ad

Page) (2 Subtract from 9 gd-8 +38-8-oto-28+8=bra € 8= 3a-d) at 8z bra x + 3ard = bra La-za+d+b-a 2-4atb+d L Put value of a Brin a=-4athtd-3a+c+ gd + 2(Band) - S a = a +b totd-s. S. 2 - atbicada 182 - 2a+b+c+d of Put values ß, 8, 8 in A atbaten²+da3 = (-4a+b+d) ( 1+ 2x - 3x2 + 3x3) + (3a-e-ad) (-1-30 +2312-373) + ( 39-4 ( 2734-402+523)+-2ath +c+d) (1-x+3x2+373) T(atbx+ca? +223) = 6446+C) T(1+24–3x2+333 +(30-C-20) TH-x+222 - 383) H3and) T( 2+36-40124 583)+(-8a+b+cted) T(-1-x+3x2-3x3)

Now a+batca²+da3 « (H+ 2x - 3x2+373) +B(+42 +223 - 323) (273%-482+593) + 8 (-1-2 + 372–373) A a+bx+cx² + c3 = (-B + 980s) (1) + (32-B737-'s ) x + (-38 +23-48+38)382 + (32-33 +58-38)x3 11 a b с d x-ß tar-8 82-3+37-8 -34 +23-40738 = 36-33+ 57-38 Multiply 13' f thin subtract from @ , weget 34-38+58-7/5-z«+38-68+28 = d-34 -8=d-3a by 8a3a-d 3 & add Again. Multiply ① by '3' by with ), weget 34-35+ 60-76 =34+98-48433 = 3a76 Star a zata Batc 8-3and' -B +2(3a-d) Be Di 39-dd- 30-6 = Ga-39-0-21 TB = 3a-c-ad

T (14x+(-3)x2 + 335) -- 8 st 10 101 2 [1] 42 [Jont [3] T너+UA tg (-3)13) Le} T50) -6)) 10 4 (0) 이 (23) 0 - ㅏ 8 93 -48 T ( 4 3x +(-)x²4 573) = (13) 1 (15) | 01 f(96) 96+ 13 6 15. 4 T-) + (4)x4 32 1 3₂3) (-6) 10 8t 01 + (23) 0 7423 + (Sol 이 0 ] -6 8 -93 -9

+(-39 +5b-2014 -17 T a 50-6b+2) 7-5 12 - 3 ul -10 +(a-2b+c) 38 -27 65 -16 T نه دی a b с 35a-4ab+140 Goa - 19 b +940 -85a +3ob-loc - 150+ 18b-6c f]-72a + 120b-486 123a + 205b-820 +51a-85b +346 + 300-50b+200 + 380-766 + 386 650 - 130b+650 -270+54b-270 -169 +32b-160 T V a fab+46 gat3 b4 70 -a-b-30 — 48с. 0 B2 I (1 +92 +6+2)x2 + 3x3), (+1+(1)2 + 2x2 +43)13 ) (9+32+(-4)382 +503), (+14)2 + 3x2 +6-33313) 3)"} Hlo Ho M be matrix transformation of T' from basis 'B' to'c Ca 1o

7 3 p +66 - 28-86-36-3a-2c + 2b B - 30 +56 oC 1 B = -3a45b-ac Put value ol 4 p' from 6 & a=X+21-30+5b-ac) +2la-obtc) of in (0) a=X-60+ lob-40 taa- 46 ta c x= atoa-lobtuc-2a+ub-ac X = 5a-6btac put value d', 'B', gl in 9 50-66+ac +(-3a45b-g) a g 1 3 B с + (9-2b+c) 9 4 ㅋ + (-3a +5b-20) T T = (5a-6b+20) T و ماه 9 b 3 2 + a-967) I 4 7

CD g c = a = 2+ GB + 28 b X+33 +48 at up + 78 ☺ ☺ we get ar-of-38-40 كما / fx 42p tax = a-b -B-28 = a b B + 28 = b-a 4 2 3 we 2 get X + 3 8 +48 -of-up-785 b-c $-37= ß +38-c-b -38=b-c 5 (5 we get B + 28=b-a (- B+38:26-b -gob-a-cthi gb-a-c N = a-gbtc 6 by '3' 3 & 6 Multiply Subtro et by al 4 thin

BE Solution - 9 3 4 2 4 7 oo C2 10 CALE M be matrix transformation oft I from B to c. + (-5) T 12 +17/0o Hesloo To Il 01 7 - 5 L12-3. 41 -17 45800 51 Oo Tla 3. L4 84 -17 ul 10. 65110 -27 i 192 Logo 8110 Dorot 4 17 38.-27 1ss 16 a # B folg A Now 2 3 b 4 с ^ a b. at aptar 0+334 4mg at UB +78