Question

8. Let L: P2 → P be the linear transformation defined by Lar? +bt + c) = (a + b)t +(b - c). (a) Find a basis for ker L. (b) Find a basis for range L.

Answer 1

L:P₂ Pi such that Leat²tbttc) = Carbst+ (6-c) @ kerk = { fc) € Pz / LC4)=0} > Llat?+b+c) = 0 > Cath) t+ 6-2) = azob, bac 20 atbro & bac (b=c=-al Cokerla b {athata} a span {aC+t-1)} .. basis of Kerl is į Et-1}. ② Range La {LCafestoc) & attenere Ele} @+b)t +6–c)} = { attb@+1) +c)} - {at + b (E+) + CEJ aspanht, tt), -1} but the set h to tt1,-1} is Linearly dependant since ttl= tt is linear combination of * the set & tt1,-1} still span Rangel and which is Linearly Indepandant. to Basis of Ranget is {-1, tti} Cor) we have to ttl} and h tinig a also bases of Range L ++,-1 Con also are