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Math Help - Hint on how to approach this Kernal and Range problem

  1. #1
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    Hint on how to approach this Kernal and Range problem

    Hi I know how to do problems most problems on Kernel and Range but this one for some reason I can't do

    Please guide me towards how to do this one

    Hint on how to approach this Kernal and Range problem-2nd-post.jpg
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  2. #2
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    Re: Hint on how to approach this Kernal and Range problem

    suppose p(x) is in ker(T). what does this mean? it means T(p(x)) = 0 (the polynomial, not the number).

    since T(p(x)) = x2p(x), we have 2 possibilities:

    a)x2 is the 0-polynomial. is this ever true?

    b) p(x) is the 0-polynomial. what does this say about dim(ker(T))?

    note that dim(P2) = 3 (the "2" is misleading). what does the rank-nullity theorem tell you about dim(im(T))?
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  3. #3
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    Re: Hint on how to approach this Kernal and Range problem

    a) x^2 would equal zero only if x=0
    b) if p(x) is the zero polynomial then wouldn't that mean that dim(ker(T)) = 0

    i don't understand what you mean by im(T)

    The R_N says dim(range) +  dim(kernel) = dim(domain)

    I hope I am saying anything right
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    Re: Hint on how to approach this Kernal and Range problem

    I am sorry if i am saying something really stupid and it might make me look like an idiot, but I am just trying to read this lesson now and do these questions on the study guild.
    so please excuse anything out of this world i might say
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    Re: Hint on how to approach this Kernal and Range problem

    im(T) is the image, or range, of T. it's just an abbreviation (im(T) looks cooler than rang(T)).

    your conclusion for (a) is incorrect. do not confuse the expression x2, where x is a real number, with the POLYNOMIAL x2 (which is a FUNCTION, not a number).
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    Re: Hint on how to approach this Kernal and Range problem

    oh yeah I knew that's where i probably went wrong, so then for (a) x^2 can't ever be 0 polynomial

    so then are we only dealing with part (b) here when it comes to the kernel so far it seams the only thing in the kernel is the {0}, so its dim is 0.
    so that would leave the range to be dim(domain) -dim(kernel). so its 3
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    Re: Hint on how to approach this Kernal and Range problem

    can you think of a 3-dimensional subspace of P4 whose elements all contain an x2 term?

    (hint: perhaps {x2,x3,x4} might form a basis?)
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    Re: Hint on how to approach this Kernal and Range problem

    I am really sorry I don't know if its me or the time since its 1 AM but I am lost
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    Re: Hint on how to approach this Kernal and Range problem

    do you know any bases at all for polynomial spaces?
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    Re: Hint on how to approach this Kernal and Range problem

    well when I was given a polynomial like say (3x^2+2x+5) I would look at it in terms of R^3 (3,2,5) and from there i would work with questions regarding polynomials so i never really had to know more then that till now i guess
    can you please elaborate on what i should know, I know I am taking alot of your time, and i really appreciate your efforts so far
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    Re: Hint on how to approach this Kernal and Range problem

    ok, that's good. now if you're identifying 3x2+2x+5 with (3,2,5) what is your basis?

    that is you have (fill in the blanks): 3___ + 2____ + 5____ ? the stuff in the blanks are your basis vectors.
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  12. #12
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    Re: Hint on how to approach this Kernal and Range problem

    oh so my x^2, x,1 are my bases vectors.
    i see
    so to go back to what we were doing you state " perhaps {x2,x3,x4} might form a basis?" if its for P4 or in R^5 (0,0,1,1,1)
    would that be enough to form a bases, since what i know is that a bases would have to span and isn't it to small to span all of R^5 or P4

    and i think i lost though of how we are relating this to the original question about Kernal and Range, since i still don't know the answer
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  13. #13
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    Re: Hint on how to approach this Kernal and Range problem

    things you should know (if you don't, re-read your texts)

    1. a subset of a linearly independent set is linearly independent

    corollary: a subset of a basis is linearly independent

    2. the span of a linearly independent set is a subspace for which the LI set forms a basis

    corollary: a subset of a basis determines a subspace

    P4 and R5 have the same dimension so they "act alike" (as vector spaces). but they are not "the same" there's just a CORRESPONDENCE between the two:

    ax4+bx3+cx2+dx+e <--> (a,b,c,d,e)

    inducing the following correspondence between the two bases:

    x4 <--> (1,0,0,0,0)
    x3 <--> (0,1,0,0,0)
    x2 <--> (0,0,1,0,0)
    x <--> (0,0,0,1,0)
    1 <--> (0,0,0,0,1)

    *********

    in terms of your linear mapping for your problem, the range of T is {f(x) in P4: f(x) = ax4+bx3+cx2}

    that is f(x) = x2(p(x)) , where p(x) = ax2+bx+c.

    if you want to think of this as the subspace of R5 spanned by (1,0,0,0,0) ("x4"), (0,1,0,0,0) ("x3") and (0,0,1,0,0) ("x2") that's "sort of" ok, just realize you're using an EQUIVALENCE, not an EQUALITY.
    Thanks from ohYeah
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