Is P = P² the definition of a projection operator?

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SUMMARY

The discussion centers on the mathematical definition of a projection operator, specifically addressing the equation P = P². Participants confirm that this equation characterizes projection operators, which can be represented by matrices that replace certain elements with zeros. The conclusion drawn is that for any operator P satisfying P² = P, the vector space V can be expressed as the direct sum of the null space of P and the range of P. An example of a non-standard projection operator in R² is also provided, illustrating that not all projections conform to the simple identity matrix form.

PREREQUISITES
  • Understanding of linear algebra concepts, particularly vector spaces and operators.
  • Familiarity with the definition and properties of projection operators.
  • Knowledge of null spaces and ranges of linear transformations.
  • Basic proficiency in matrix representation and manipulation.
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  • Study the properties of projection operators in linear algebra.
  • Learn about null spaces and ranges of linear transformations in depth.
  • Explore examples of non-standard projection operators in various dimensions.
  • Investigate the implications of the equation P² = P in different mathematical contexts.
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Mathematics students, educators, and professionals in fields requiring linear algebra knowledge, particularly those focused on operator theory and vector space analysis.

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Ignore this post (it was irrecoverable due to bad LaTeX markup). See the one below.
 
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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.
 
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.
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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