Question

Consider the following linear transformation T: RR where TX, 22, X, X, Xs) - (******4,2x1******+2x4,2x:+36-32+x) (al Determine the standard matrix representation A of Tix) (b) Find a basis for the kernel of T(x). (c) Find a basis for the range of Tix) (d) Is Tix) one-to-one? Is Tix) onto? Explain. le) is Tix) invertible? Explain

Answer 1

0 TIRS [03 124 3ة -2%+221 (3) 22, 3, 7,76) = (21-X3174 - 22, +373-374175) T(۱, 0, 0, 0) = (۱) 93-9) 2 رو را ره) = (0 زهر هراره )T (3را-را-) = (هه ر ( ره )1 = (ه راره رهه )T اره ره=راره ره رهه )1- (3- ر2 را) 2 0 0 2 so standard metore og O A: a 2 0 - ( 2 3 -3 . 0 20

b Let (21,22,23,24,25 ) e ker I 3 T(2., 22, 23, 24, 25)=0 (301-2314, 221 +22–237274–22, +323–38412)2 0 th solve vel this system 80 o o - 2 q o -2 o 3 =3 1,0 0 Loo R22 R2 ER, R3 R3TER, 6 o 1 - 1

No ом 14, Х is free let Х4. >k Frk. 2 22 – 4t X, то г ) 23 24--X- 37 (ki-ka) до +23 =0 с 2 2 -23 с (2 Moz k2-КІ 22 — 24 і 2 Ку-kz -k 1 2 -ко

so ken Tz { -K2 Karki Ki-K 2 Ki K2 K1, K2 GIR o Keitz ki tka 1 Kiska FIR 0 Kennal T= so basis af (0,-1,110), (1,1,1,0,19 (4,190g It (9,6,4)&( any such ferat (a,b,c) E Range T So I [2, 22, 23, 24, 25) such that T(2,89, X3, X4, 25) = (0,6sy

Kernal t = 2 dimension & dimension of Rang Tz 5-2 23 so basis af Range Tz {(1,0,0), (0.1A), (0.01)} No, T is not one - the as KeTE {o} Yes T is onto be caure Range T= 1R3 invertiable as e No T is not T is not one-one.