Question

10. Let T : P P , be the linear transformation defined by T(P) = (a) What is the kernel of T? (b) According to the concept of the rank theorem, what is the dimension of the range of T? (C) (needs an idea from earlier in the semester) If we represent P, by coordinate vectors rela- tive to it's standard basis (1.1.1-.1') and P, by coordinate vectors relative to it's standard basis (1,1,1"), find the standard matrix A of the transformation T.

Answer 1

1. - where Pg={at Por {9+9+ + + + g / 2,939,9 6P 3 P= { 9.49,qil 9,9,196F3. defined linear Transformation TCP()) = d pats lex Pse? 6 f. ou ADS Formatio >> Plt=P = 0.49,87927 ge? »), 7 (Pus) = h (pet) = (4.79€t qe ngop) = 0,4 25+ 394 @ To find Kernal of T ie To find K (T) K(7) - { P«t) PIT (PPD) - 03 = { pitze Pg / dpt=} : : SAH GA I Plt) = a; }(dp) con P(t) =k {a 19,6F}. Kis const) K(T) a set of all constant Polynomial in p

D Nextst At, to find dimension of the range of T by useing rank Theorm. w rank theorm . 'Tiv> W V w be linear Transformation then Jimmor va sim (ternador 1) - Jim? (range oft) A T: Ph T(P) d dim" off is 3 and timor fk()} = tim" of kernal OFT is = 1. Next to find time of rance puting all values in Rank Theorm wo 3=4+ dim" (range of T) », dim" of range of T =2) © Nexi, to find unear Transformation Matrix of - T with respective to the basis of pois {let, t it?? and basis of P is { 1st, 23 respectivally. To find TID

We set oiol next write o is linear combination flitit's 0:0.1 +0.4+0.d? 7 0 .is first column of zhe linear Transformation Matrix of I say a lly, we find. T(H)=1 - 1 = 1.1+ otto.f? [1,0,0]? is second column of the hit Matrix of I say A. Nexty T (E)= 2t > 2t = 0.1+ 2t tot => [0, 2,o] is 3rd column of the LT Matrix S OF T say A Next, TR)= 3+3 » 3t = oltott 3ť » [0,0, 33T is Fourth column of the Li Matrix of I say A. - by above information linear Truns Formation Matrix of I say A is becomes Az oo 2 0 LOOo3 a