Let L be a linear transformation on P3 given by Lpx p0x p1

Let L be a linear transformation on P_3 given by L(p(x)_= p(0)x + p(1). Find the kernel and range of L.

Solution

p(x)=ax^3+bx^2+cx+d

p(0)=d

p(1)=a+b+c+d

L(p(x))=p(0)x+p(1)=dx+a+b+c+d

ker(L) means set of p(x) so that:

L(p(x))=0

dx+a+b+c+d=0

d=0,

a+b+c+d=0

a+b+c=0

a=-b-c

Hence,

ker(L)={-(b+c)x^3+bx^2+cx: b,c are real numbers}

For any ,p(x)=ax^3+bx^2+cx+d

Set:b=c=0

L(p(x))=p(0)x+p(1)=dx+a+d

So we can vary a,d so as to get all polynomials in P1 is all polynomials of degree less than equal to 1.

Hence,

range(L)=P1