Linear transformation question

I've got a question like this:

Let V be a finite dimensional vector space with dimV >= 2 . Show

that there exists a linear transformation T:V to V such that T not equal to zero but

T^2 = 0 .

I have figured out the "T not equal to zero but T^2 = 0 ." part already, rankT>=1, dimNulT>=rankT>=1, satisfies with dimV>=2 by rank theorem, but how to present it nicely?

Re: Linear transformation question

pick a basis B = {b_{1},...,b_{n}} for V.

thus we can write any v in V as v = c_{1}b_{1}+...+c_{n}b_{n}.

define T(c_{1}b_{1}+...+c_{n}b_{n}) = c_{n}b_{1}.

T is non-zero because T(b_{n}) = b_{1} ≠ 0, since b_{1} is a basis vector.

but T^{2}(v) = T(T(v)) = T(T(c_{1}b_{1}+...+c_{n}b_{n})) = T(c_{n}b_{1}) = 0

(because c_{n}b_{1} = c_{n}b_{1} + 0b_{2}+....+0b_{n}).

basically we are using the linear transformation that has the matrix (relative to the basis B):

$\displaystyle \begin{bmatrix}0&0&\dots&0&1\\0&0&\dots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\0&0&\dots&0&0 \end{bmatrix}$