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Math Help - Find Kernel...

  1. #1
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    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?


    Thanks in advance
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  2. #2
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    Quote Originally Posted by shuaige View Post
    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?


    Thanks in advance
    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.
    Now show that they are linearly indepedent which will mean you got a basis.
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