The linear transformation $T: \mathbb{R}^5 \to \mathbb{R}^5$ below is nilpotent of nilpotency index 2.
$T(x_1, x_2, x_3, x_4, x_5) = (x_4 - x_3 - x_5, 0, -x_2, 0, x_2)$
The dimensions of the kernels of $T^m$ are
$\bullet \dim(\text{Ker}(T)) = 3$
$\bullet \dim(\text{Ker}(T^2)) = 5$
Thus it's V-board looks like
Compute the following. (Click to open and close sections below).
(A) Basis vectors $v_1, v_2$
Find $v_1$ and $v_2$ so that {$v_1, v_2$} is a basis of Ker($T^2$) over Ker(T) and then compute their images under $T^m$
$v_1 = $
$T(v_1) = $
$T^2(v_1) = $
$v_2 = $
$T(v_2) = $
$T^2(v_2) = $
(B) Basis vector $v_3$
Find $v_3$ so that {$T(v_1), T(v_2), v_3$} is a basis of Ker(T) and then compute its image under $T^m$
$v_3 = $
$T(v_3) = $
(C) Matrix J$
F = {$v_1, T(v_1), v_2, T(v_2), v_3$} is a Jordan Basis for T. Write the matrix $M_F(T)$.
$M_F(T) = $
