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Math Help - image & kernel

  1. #1
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    image & kernel

    When does the kernel of a function equal the image?
    Thanks in advance
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  2. #2
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    Re: image & kernel

    I think you mean a linear transformation, not a function. They're not the same thing.

    I haven't thought about this question, but I have a notion. By the rank-nullity theorem, the rank would have to be zero in that case so the nullity would be n (where n is the number of columns). Any pivot column would mean a rank greater than 0. So there must be no pivot columns. The only way I can picture that is a matrix with all zeroes.
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  3. #3
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    Re: image & kernel

    Quote Originally Posted by zhandele View Post
    I think you mean a linear transformation, not a function. They're not the same thing.

    There are several meanings for kernel of a function,

    The OP should have defined terms.

    Post Script: Here is a second reference.
    Last edited by Plato; March 21st 2013 at 02:45 PM.
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    Re: image & kernel

    If this had said "null space", rather than "kernel", then I would say that a vector space, and so "linear transformation", but "kernel" is more general.

    The "image" of a function, f, is \{y | y= f(x) for some x}, the "kernel" is \{ x| f(x)= 0\}. If the kernel is equal to the image, then, for all x, f(x) is i the image and so f(f(x))= 0 and, conversely, if f(y)= 0, there exist x such that f(x)= y.
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  5. #5
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    Re: image & kernel

    Quote Originally Posted by jojo7777777 View Post
    When does the kernel of a function equal the image?
    Thanks in advance
    From my understanding of the terms; this could only happen if the function was the 0 map (the map which maps all elements in the domain to 0) from (say) X into X.
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  6. #6
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    Re: image & kernel

    In my question I refer to linear transformation , and I meant to ask whether there are some features in the image that can tell me about the fact that the null-space and image are equal?
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