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Math Help - ker f

  1. #1
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    ker f

    Let f:X\to Y be a lin transformation. f is monic iff \text{ker}f=\{0\}

    I have shown (\Rightarrow)

    I am struggling with (\Leftarrow)

    Let x\in\text{ker}f. Then f(x)=\{0\}.

    Now what?
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  2. #2
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    Re: ker f

    If I am not mistaken, the fact that f is monic means that f\circ g_1=f\circ g_2 implies g_1=g_2 for all g_1,g_2. Well, let f(g_1(x))=f(g_2(x)) for all x. Use the fact that f is linear and that its kernel is {0}.
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  3. #3
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    Re: ker f

    Quote Originally Posted by dwsmith View Post
    Let f:X\to Y be a lin transformation. f is monic iff \text{ker}f=\{0\}

    I have shown (\Rightarrow)

    I am struggling with (\Leftarrow)

    Let x\in\text{ker}f. Then f(x)=\{0\}.

    Now what?
    You have to show that f is monic.

    So, you have to show that f(x) = f(y) implies that x = y.

    Now f(x) = f(y) \Rightarrow f(x-y) =0 \Rightarrow x-y \in Ker(f) \Rightarrow  x-y = 0 \Rightarrow  x=y.
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