Question

1) (a) The conjugation function on C" is NOT a linear transformation when the scalar field is C, for any positive integer n. However, it IS true when the scalar field is R. Show that the conjugation function T:C" C", where T () = 7 is a linear transformation for the vector space Cover R. (b) Show that CR2m as vector spaces over R.

Answer 1

Tlautpre) = (au Proof 1(a) Giren mapping wwere Tichych T2 = 0 Let and xiß ER ł and te Ech u Now (A+B) = Āti AB = à LA) =ăà ELLER 1:2,BER 2=a Ř = B (dub) +(18) =all the = a + B Ed Tiū) +27(77) Tlæū+Bw')=d.Tlū) +B.712) Tes liwcas ie Huis Pliows Heat Asansformation.

two proof e(6) we have to show that chuir an as as a rector space over IR we know that veetos spaves are isomoophic only if there crist a bijection between the two we can defindet linear map Tic" R af and the an as bi '9 +6, i 02 T datbyć b2 antbri lan Lon clearly üt E¢ tlen ر71 =رet T (ن) if Let th Tlüte ) = Tlů)+T(20) & TICO) = Tlu). 71%) & T es homo mosphes Hürtan นาน (iii) & ŞER?n & ue&" 8-t Tla)=ý so Poom (i) fin a pii) we that T is isomoothision .ie T is one one & T is onto can say chy Ran Jeenee provel