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Here's the question I'm stuck in on my latest problem set:
Suppose P is an operator on V and [itex]P^2=P[/itex]. Prove that [itex]V = null P \oplus range P[/itex].
I'm pretty sure that P is simply the projection operator (consisting of the identity matrix replacing some 1's with 0's), in which case the conclusion follows easilly. But I've been looking at this one for a while and I can't see how I can prove that P must be the projection operator.
Thanks for your help.
If I recall correctly, P² = P is the definition of a projection operator!
Not all projections have the form you gave: for example, consider in R² the orthogonal projection onto the line y = x given by [x, y] -> (1/2)[x+y, x+y]
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