Direct Sum Computation

Question

Let $T: V\rightarrow V$ be a linear transformation. Let $B_n=\ker T^n$ and $C_n=im T^n$.Let $B=\bigcup B_i$ and $C=\bigcup C_i$. If $V$ is finite dimensional, is it true that $V=B\oplus C$.

Answer 1

Note that $B_n$ is an increasing sequence of subspaces of $V$ but that $C_n$ is a decreasing sequence of subspaces of $V$ so your $C$ is in fact just $C_1$. As written, the statement is not true (consider for example the operator whose matrix is given by

$$ \left( \begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} \right) $$

for which $\dim C = \dim B = 2$). However, if you replace the union $\bigcup_{i=1}^{\infty} C_i$ with the intersection $\bigcap_{i=1}^{\infty} C_i$ then the statement will hold. To prove it, prove first the the sequence ${B_i}_{i=1}^{\infty}$ stabilizes so you have $B_N = B_{N + k}$ for all $k \in \mathbb{N}$. Choose a minimal such $N$ and show that $V = B_N \oplus C_N$. By construction, $B = B_N$ so to finish, show that you also have $C_{N + k} = C_N$ for all $k \in \mathbb{N}$ showing that $C = C_N$.