1. ## Find Kernel...

3) Let T: P2 --> R be the linear transformation defined by int [p(x)] from 0 to 1
a) (2) Find the ker(T)
b) (2) Find a basis for basis for ker(T).

for part a, what i did is:

p(x) = a+bx+cx^2

then, int [ a+bx+cx^2] from 0 to 1 and i got: a+b/2+c/3. then set this equation =0

now let a=s, b=t, therefore, c= -3s - 3/2 t

Thus, I conclude that Ker(t) = {(-3s - 3/2 t): where s and t are any real numbers}

Is this process correct?

to continue the problem, for part b

I just simply substitute a set of number, say s=0, t=1, i get -3/2, then is this the basis for the kernel? or did i miss anything there?

2. Originally Posted by shuaige
3) Let T: P2 --> R be the linear transformation defined by int [p(x)] from 0 to 1
a) (2) Find the ker(T)
b) (2) Find a basis for basis for ker(T).

for part a, what i did is:

p(x) = a+bx+cx^2

then, int [ a+bx+cx^2] from 0 to 1 and i got: a+b/2+c/3. then set this equation =0

now let a=s, b=t, therefore, c= -3s - 3/2 t

Thus, I conclude that Ker(t) = {(-3s - 3/2 t): where s and t are any real numbers}

Is this process correct?

to continue the problem, for part b

I just simply substitute a set of number, say s=0, t=1, i get -3/2, then is this the basis for the kernel? or did i miss anything there?

The kernel are all the polynomials $s+tx+(-3s-\tfrac{3}{2}t)x^2 = s(1 - 3x^2) + t(x - \tfrac{3}{2}x^2)$.
We see that $1-3x^2, x - \tfrac{3}{2}x^2$ spam the kernel.