# Math Help - Linear Operator question

1. ## Linear Operator question

Hello. I need confirmation on this: Let $T:V\rightarrow V$ be a linear operator on the (finite dimensional)vector sapce V. I know that when T is a proyector,ie, $T^2=T$, then the space is the direc sum of the Kernel of T and it's image, $V=ker(T)\oplus Im(T)$. Now it seems to me that this is also true for any operator( $T^2\neq T$). am I correct?

2. No! For example, what if you have $T^2=0$ but $T \neq 0$? It's true that the dimensions always add up, but the direct sum decomposition is a much stronger statement.

3. Thank you. With your help I think I found the mistake in my proof.