Find the Kernel & Image of A: $\mathbb{R}^\infty \rightarrow \mathbb{R}^\infty$

[arestes]

Homework Statement

Find the kernel and image of the linear function $$A: \mathbb{R}^\infty \rightarrow \mathbb{R}^\infty$$ defined on the vector space (with usual operations) of sequences of real numbers $$x \in \mathbb{R}^\infty$$, $$x = (x_1, x_2,...)$$. given by $$A(x) = (y_1, y_2, ...)$$ with $$y_k = x_{2k+1} - 2x_k$$.

Homework Equations

$$Ker(A) = \{ x \in \mathbb{R}^\infty : Ax = (0, 0, 0,...) \}$$
and the standard notation for the image of A

The Attempt at a Solution

For the kernel I tried to check formulas to solve finite difference equations, since I need to solve $$x_{2k+1} - 2x_k =0$$ but I can only find the usual method when we have k+1, k+2 or k+r terms involved, not when one of the terms is of the form $$x_{ak+b}$$ when a is not 1.

I did it by hand, plugging in values and I see a pattern, and I know that after a while i could come up with a general form for a basis of the (as I can see) infinite dimensional Kernel. I would like to solve it more decently.

For the second part (the image of A) I am still lost, since I was hoping to get a general solution and from there see which sequences are not allowed, somehow... Does anyone know how to solve this recurrence equation?
Or at least how to solve this problem, if the solving of the equation is not absolutely necessary? (but I would love to learn how to solve these equations).
Thanks!

 

[tiny-tim]

hi arestes!

(try using the X₂ icon just above the Reply box )

the key is to find the pattern for the indices:

k, 2k+1, 2(2k+1) + 1 etc

 

[arestes]

Hi Tiny Tim, I was doing what you said for the kernel. Any ideas for the image?
thanks! :D

 

[tiny-tim]

hi arestes!

Hi Tiny Tim, I was doing what you said for the kernel. Any ideas for the image?
thanks! :D

ooh, i didn't get that far …

i sort of assumed that if you managed to find the kernel, the image would be easy …

would you like to show us how far you've got? ​