Question

12. Let L : R² + RP be a linear transformation such that L | (3) - -(5)-(1) Find I (*) Find the standard matrix representing L. Find the dimensions of the kernel and the range of L and their bases.

Let L : R2 → R3 be a linear transformation such that L 1 1 = 1 2 3 and L 1 2 = 2 1 3 . Find L 2 1 Find the standard matrix representing L. Find the dimensions of the kernel and the range of L and their bases.

Answer 1

Given that, 1.12² m3 is given by To find L(²), net ,, l th such that a(1) +62 ()-(?) i citl= 22 = 2 =21 +2 = 1 e, 23 - 10%)-1(311) -1.(!)). =31()-1.111) Now, for fir tandand matrix We have to find ::1/3) - (5) 1o) and 2 ). Clearly, 4(!) = -(2(!)-(:)) - 2.1(1)-1()) el!) - 2.(:)-(1)-(:)-(3). 6). 117)= 0 (5) + 3. (8.)+31(8) 2 (0)=(11)-(:)) - 113)-1(!) --(0)= (s)-(1)-(3) (0)-1(0)-) +0.()

[2] 31 ur The Standard marrin for L will be 10 ( 30 ) there are two linearing independent column are there. here, dim (range L) = 2. Live, rank (2)) = 2) and by for a marrin Lmon, we know from rank - nulling theorem that. yomk (l) + Nun (L) an so here we get. 2 + Nudd (L) =2 Hence, Null (L) =0 ne, dim (ker L) =0.