Question

13. A linear transformation T takes Nº into f". T[ +y - y 2.1 + 3y y (a) Is T one to one? Justify your answer. If not, then give two vectors with the same image. (b) Is T onto? Justify your answer. If not then give a vector in R? that is not an image.

Answer 1

>W be a linear transformation И. And we Кто (R Rauk nullity Theorem ar va Let Tiv then dim v= dim (Kert) + dim (Imt) where korter T() = 0 mut nult vee tor inw Sy моле. Колт- Хvv : T(2) =о ты in w} TwТ. 5 {w WEW : I LEV for for which} т(v) = Tis one to one iff & Kott - { of orie. dim (kert) = 0 In our problem, T: R² т[5] **. ay Let Cary) ER? Д. - +34 21-1 * * * 2) а = -1 Solution 20 Тъоли Кола Кот: X®»), d'm (kert) =) T is one to one 1tvo Such that I ? у - 244=°, 2-3 2. - D Ол с 2 - 1) осо is s only abocation — О.

b ) By =) 2 0 onto Rank mullity theorom, in over problem dim (IR²) dim kert) - dim (QmT) + dim (Amt) =) dim (Imt) 2 ť dim (R²) = 3 So, I is not Counter example which Let a rector (6) image T x+y=1 ,a-y=0, 2x+3y = - 4 from 1st too eque &ty=1, aay=0 get 2=y=42 which does not Satisfy Satisfy equation za 2x+3y= ER 3 T is not an becase se 3rd Ĉ1