Let T R3 rightarrow P2 be defined by T a b c a b c a bx

Let T: R^3 rightarrow P_2 be defined by: T (a, b, c) = (a + b + c) + (a + b)x + ax^2 Determine the range R (T) of T. Determine ker(T) and the nullity of T. Is T one-to-one?

Solution

T(a,b,c)=a+b+c+(a+b)x+ax^2=a(1+x+x^2)+b(1+x)+c

Hence, range, R(T)=span{1+x+x^2,1+x,1}=P2

Let,

T(a,b,c)=0=a+b+c+(a+b)x+ax^2

Hence, a+b+c=0,a+b=0,a=0

HEnce, a=0,a+b=0 gives b=0

a+b+c=0 gives c=0

Hence, ker(T)={0}

nullity =dim ker(T)=0