Nilpotent Matrices: Show Jordan Form w/Linear Independence

[the_kid]

Homework Statement

Suppose that N is a nilpotent mxm matrix, N$^{m}$=0, but N$^{m'}$$\neq$0 for m'<m. Show that there exists a basis in which it takes the form of a single Jordan block with vanishing diagonal elements. Prove that your basis set is linearly independent.

Homework Equations

The Attempt at a Solution

So I've recognized the fact that N$^{m'}$$\neq$0 for m'<m means that N$^{m'}$ does not annihilate every vector in V. I'm just not really sure where go from here...

 

[Dick]

Pick a basis for the vectors that are annihilated by N. Then add a basis for the vectors that are annihilated by N^2 but not by N. Continue. Eventually you'll get a basis for the whole space, right? What does N look like in that basis?