Determine Ker(L ◦ L) and Im(L o L)

Question

Give the canonical base $C$ and the base $B = \{ (2,1, 1) , (0, 0, 1) , (1, 1, 3)\}$ Consider the linear application $L : \mathbb{R}^3 \to\mathbb{R}^3$ $$ M_B^C(L) = \left[ \begin{matrix} 3 & 1 & 0 \\ 1 & 2 &-1 \\ 1 & -3 &-2\\ \end{matrix} \right] $$ Determine $Ker(L \circ L)$ and $Im(L \circ L)$.

I don't quite know how to find it i tried to search on the web but nothing similar came up.

Answer 1

Let $F_L:\Bbb R^3 \rightarrow \Bbb R^3 \quad F_L(x)=Lx$. $F_L$ represents the linear application of $L$. Now we can see that $$(F_L \circ F_L)(x)=F_L(F_L(x))=F_L(Lx)=L^2x$$ so $F_L \circ F_L$ is just $F_{L^2}$. Are you able to go on from that?