No! For example, what if you have but ? It's true that the dimensions always add up, but the direct sum decomposition is a much stronger statement.
Hello. I need confirmation on this: Let be a linear operator on the (finite dimensional)vector sapce V. I know that when T is a proyector,ie, , then the space is the direc sum of the Kernel of T and it's image, . Now it seems to me that this is also true for any operator( ). am I correct?