Suppose $V$ is a vector space over any field with $dim(V) = 2$ and $L$ is a linear transformation such that $L^2 = 0$ but $L \neq 0$. Let $v \in V$ be a vector such that $L(v) \neq 0$.
Show that $v$ and $L(v)$ are independent. (Hint: Consider a linear combination which is zero and apply $L$).
It follows from the above that $v$, $L(v)$ is a basis of $V$. Find the matrix of $L$ with respect to this basis.
Let $V = \mathbb{R}^3$ and consider the linear transformation $L : V \to V$ given by $L(x, y, z) = (0, 0, x)$.
Show that $L \neq 0$ and $L^2 = 0$.
Find the matrix of $L$ with respect to the standard basis of $\mathbb{R}^3$.
Let $V = \mathbb{R}^5$ and consider the linear transformation $L : V \to V$ given by $L(x_1, x_2, x_3, x_4, x_5) = (0, x_1, x_2, 0, x_4)$.
Show that $L^2 \neq 0$ and $L^3 = 0$.
Find the matrix of $L$ with respect to the standard basis of $\mathbb{R}^5$.