1. ## 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?

2. ## 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?

3. ## Re: Linear map

Originally Posted by Deveno
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,....)

4. ## 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...

5. ## Re: Linear map

Originally Posted by Deveno
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?

6. ## 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 $\displaystyle \int:\mathbb{C}[x]\to\mathbb{C}[x]$.