Image and kernel of iterated linear transformation intersect trivially

[pugnet]

Homework Statement

Given a linear transformation f:V -> V on a finite-dimensional vector space V, show that there is a positive integer m such that im(f^m) and ker(f^m) intersect trivially.

Homework Equations

The Attempt at a Solution

Observe that the image and kernel of a linear transformation f:V -> V are each subspaces of V.
The kernels of the successive iterations of the transformation form an increasing chain of subspaces.
The images of the successive iterations of the transformation form a decreasing chain of subspaces.
Since V is finite-dimensional, the chains of the kernels and images must eventually stabilize, say at m1 and m2 respectively. Let m be max{m1,m2}. So the kernel and image of the transformation are stable after m iterations.
Now let v be some element contained in both the image and the kernel of f^m.
This means that some element w in V with f^m(w) = v, and that f^m(v) = 0.

I need to obtain a contradiction, but I have been unsuccessful so far.

Thanks for your help.

 

[Dick]

If the image is stabilized then f^m is a 1-1 map of Im(f^m)->Im(f^m). f^m(f^m(w))=f^m(v)=0. Seems like there is a contradiction there to me if you assume v is nonzero. What is it?

 

[pugnet]

Thank you, Dick. I am kind of dumb, so it took me a while to see what you meant.