Thread: 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?

pick a basis B = {b₁,...,b_n} for V.

thus we can write any v in V as v = c₁b₁+...+c_nb_n.

define T(c₁b₁+...+c_nb_n) = c_nb₁.

T is non-zero because T(b_n) = b₁ ≠ 0, since b₁ is a basis vector.

but T²(v) = T(T(v)) = T(T(c₁b₁+...+c_nb_n)) = T(c_nb₁) = 0

(because c_nb₁ = c_nb₁ + 0b₂+....+0b_n).

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