Question

Let T be the linear transformation of R* to R that takes and to and takes and - a) Explain why there is one and only one linear transfomation from R to R that does this.

Answer 1

. Given ther TAR" >R4 be a team formation which maps the following. - ... (3) 4) 12 ГР Т она на олова за правени т): к 7 | :(): ( 1 we add 0,0,0,0, ou the equanims. | | 116 1 10 by subs freaching ♡ from ©, we get t - The , т ооо mly we gerT 1/2 з , , \ts / hence, I TIR" >R“ so that, | Те = ke + ke, + 4 - 54 Те, е + 4 + 3 -те, Тея +2, + 4 е, + + + - 4 | Те, - 3 е-+ 5 + 3 - 4e4 & hence, warrix anociated to T is , M 4/3 | У % \-3 -3 43 % 1/3 Y3 | Tment be % ) ivan 3 ) Коом сумаа.

Yaw, if there exists some other linean Krasn termations san's fying the same properties, then that travas formation must map the stand and bonis vectors to the same veras and hence, the marrix anociated to the lineen transformation would be exactly same, which implies that the two lineas trom formation would be identical. Thune fore, I one and only one lines transformation from IRS to RS sans fying the stated properties, o [car part done] say, (a,b,c,d) E kent الإبن حات) پانا يانا wino win cola with در حالی لري ماندا دابيا مدير oooo الما zſ (a+b) -Ź (c+d) = 0 7(a+b) = 2(6+d) 4 (a+b) + § (c+d) = 0 1-4 (a+b) = (Ctd). Ź (a+b) + $ (Ctd) = 0 MM for -3 (a+b) + & (Ctd) = 0 = atb=Ctd = 0 = baca, da-c hence, Kent = {(2,-2,@,-c) | ace R} => , Kent = span {(2,-1,0,0),(0,0,1,-2)}| sanction of T, | Range T = span { (5$ 3,-3),(3,3,3,3)} . by com fruction of 5