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Math Help - Linear map

  1. #1
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    Linear map

    Let s:l^2\to l^2 by the map
    s(x_0,x_1,\cdots )=(0,x_0,\cdots )

    Is it linear?
    Let x_i,y_i\in l \ \text{and} \ \lambda,\mu\in F
    s(\lambda x_i +\mu y_i)=s(\lambda x_0 + \mu y_0, \lambda x_1 +\mu y_1,\cdots ) =(0,\lambda x_0 + \mu y_0, \lambda x_1 +\mu y_1,\cdots )
    =(0,\lambda x_0,\cdots )+(0,\mu y_0,\cdots)=\lambda(0,x_0,\cdots)+\mu(0,y_0,\cdots  ) =\lambda s(x_i)+\mu s(y_i)
    Yes, the map is linear.

    Is it monic?
    If s(x_i)=s(y_i), then (0,x_0,\cdots)=(0,y_0,\cdots)
    (0,x_0,\cdots)-(0,y_0,\cdots)=(0,x_0-y_0,\cdots)=0
    \Rightarrow x_i-y_i=0\Rightarrow x_i=y_i
    Yes, the map is monic.

    How do I show it is an epimorphism?
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  2. #2
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    Re: Linear map

    it doesn't look like it is (although i am not sure what l^2 is, you don't say).

    what pre-image could (1,0,0,...) possibly have?
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    Re: Linear map

    Quote Originally Posted by Deveno View Post
    it doesn't look like it is (although i am not sure what l^2 is, you don't say).

    what pre-image could (1,0,0,...) possibly have?
    I don't know what l^2 is either. What you see is 100% verbatim. If you don't see it, then I don't have the information either.

    Wouldn't it have (0,0,0,....)
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  4. #4
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    Re: Linear map

    well it looks like it might be some kind of hilbert space, but that's just an educated guess.

    pre-image, not image...
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  5. #5
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    Re: Linear map

    Quote Originally Posted by Deveno View Post
    well it looks like it might be some kind of hilbert space, but that's just an educated guess.

    pre-image, not image...
    How do I show it isn't an epimorphism then?
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  6. #6
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    Re: Linear map

    \ell^2 is presumably the space of all square summable complex sequences. It's surely not an epimorphism as [b]Deveno[/tex] pointed out, it's completely analogous to \displaystyle \int:\mathbb{C}[x]\to\mathbb{C}[x].
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