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Studying the introductory chapters of a Operator Theory book, I have found that the author seem to find a lot of demonstrations "easy" and not worthy of demonstrations. Yet, I don't have such ease as he has... For instance, one that has bugged me (on several books) is the proof that:

range(I-E)=nullspace(E)

nullspace(I-E)=range(E)

where E is a projection and I is the identity matrix.

It always follows stating that its is simple, then, to tell the set of all surjective projections of a linear transformation L.

How are they related? I can't seem to find any way to prove/answer the above and its been a couple of days now.

Can anyone shed some light in this? I'd really appreciate it!

Thanks!

Felipe

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# Demonstrations on Projections

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